intermediate algebra, third edition

C H A P T E R PPrerequisites: Fundamentals of Algebra

Section P.1 The Real Number System . . . . . . . . . . . . . . . . . .72

Section P.2 Operations with Real Numbers . . . . . . . . . . . . . . .74

Section P.3 Properties of Real Numbers . . . . . . . . . . . . . . . . .78

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80

Section P.4 Algebraic Expressions . . . . . . . . . . . . . . . . . . . .81

Section P.5 Constructing Algebraic Expressions . . . . . . . . . . . . .84

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89

C H A P T E R PPrerequisites: Fundamentals of Algebra

Section P.1 The Real Number SystemSolutions to Odd-Numbered Exercises

72

1.

(a) Natural numbers:

(b) Integers:

(c) Rational numbers:

(d) Irrational numbers:H2!5, !3, 2pJH210, 22

3, 214, 0, 58, 1, 4, 6J

H210, 0, 1, 4, 6J

H1, 4, 6J

H210, 2!5, 223, 21

4, 0, 58, 1, !3, 4, 2p, 6J 3.

(a) Natural numbers:

(b) Integers:

(c) Rational numbers:

(d) Irrational numbers:H!5, 3pJH23.5, 2!4, 21

2, 20.3, 0, 3, 25.2JH2!4, 0, 3J

H3J

H23.5, 2!4, 212, 20.3, 0, 3, !5, 3p, 25.2J

5. H25, 24, 23, 22, 21, 0, 1, 2, 3J 7. H1, 3, 5, 7, 9J

9. (a) The point representing the real number 3 lies between 2 and 4.

(b) The point representing the real number lies between 2 and 3.

(c) The point representing the real number lies between

(d) The point representing the real number lies between

03 2 1

5.2

456

26 and 25.25.2

012

27

34

24 and 23.272

4

25

3210

52

4

3

3210

11.

21 < 3

a 5 21, b 5 3 13.

292 < 22

a 5 292, b 5 22

15. because 2 is to the left of 5 on the number line.2 < 5 17. because 10 is the right of 4 on the number line.10 > 4

19. because is to the left of on the number line.222727 < 22

21. because is to the left of on the number line.222525 < 22

23. because is to the right of on the number line.14

13

13 > 1

4 25. because is to the left of on the number line.122

582

58 < 1

2

Section P.1 The Real Number System73

31. Distance 5 10 2 4 5 6 33. Distance 5 7 2 s212d 5 7 1 12 5 19

35. Distance 5 18 2 s232d 5 18 1 32 5 50 37. Distance 5 0 2 s28d 5 0 1 8 5 8

39. Distance 5 35 2 0 5 35 41. Distance 5 s26d 2 s29d 5 s26d 1 9 5 3

51. 2|234| 5 2

34 53. 2|3.5| 5 23.5 55. |2p| 5 p

43. |10| 5 10 45. |2225| 5 225 47. 2|285| 5 285 49. 2|16| 5 216

57. |26| > |2| since 6 > 2. 59. |47| > |227| since 47 > 27.

61. 2|216.8| 5 2|16.8| since 216.8 5 216.8. 63. |234| > 2|4

5| since 34 > 245.

99. False. |3 1 s22d| 5 1 Þ 5 5 |3| 1 |22|

101. The set of integers includes the natural numbers, zero, and the negative integers.

103. Yes, the nonnegative real numbers include 0.

105. Place them on the real number line. The number on the right is greater.

65. Opposite:

Absolute value: 34

234 67. Opposite: 160

Absolute value: 160

69. Opposite:

Absolute value: 311

311 71. Opposite:

Absolute value:54

254

73. Opposite:

Absolute value: 4.7

24.7 75.

7

84048

7

|7| 5 7 77.

0 2 4 6−2

−5 5

−4−6

|25| 5 5

79.

53

1

53

1 0

|235| 5

35 81. Opposite of

−2 −1 0 1 2

5 53 3

−

53 is 25

3. 83. Opposite of

0−2−4−6 2 4 6

−4.25 4.25

24.25 is 4.25.

85. Opposite of 0.7 is

0 1 2−1

−0.7 0.7

−2

20.7. 87. x < 0 89. x ≥ 0

91. 2 < z ≤ 10 93. p < $225 95. True 97. False. is not an integer.23

27. because is to the right of on the number line.21032

232

23 > 2

103

29. because 2.75 is to the left of on the number line.p2.75 < p

74 Chapter P Prerequisites: Fundamentals of Algebra

11.

5 220

5 2s12 1 8d

5 212 1 s28d

5 2s16 2 4d 1 s28d

4 2 16 1 s28d 5 4 1 s216d 1 s28d 13.

5 0.7

5 1s1.1 2 0.4d

5 20.4 1 1.1

5 2s6.2 2 5.8d 1 1.1

5.8 2 6.2 1 1.1 5 5.8 1 s26.2d 1 1.1

15.

554

5108

38

178

53 1 7

817.

524

512

34

214

53 2 1

419.

51

10

56 2 5

10

56

102

510

35

1 12122 5

3s2d5s2d 2

1s5d2s5d

21.

51

24

515 1 6 2 20

24

51524

16

242

2024

58

114

256

55s3d8s3d 1

1s6d4s6d 2

5s4d6s4d 23.

51058

or 1318

546 1 59

8

523s2d4s2d 1

59s1d8s1d

534

1 738

5234

1598

25. 85 2 |225| 5 85 2 25 5 60 27.

5 45.95

2s211.325d 1 |34.625| 5 11.325 1 34.625

29.

5 228

5 228.000

5 215.667 1 s212.333d

2|215.667| 2 12.333 5 215.667 2 12.333 31. 4s5d 5 5 1 5 1 5 1 5

33. 3s24d 5 s24d 1 s24d 1 s24d 35. 14 1

14 1

14 1

14 1

14 1

14 5 6s1

4d

37. s215d 1 s215d 1 s215d 1 s215d 5 4s215d 39. 5s26d 5 230

41. s28ds26d 5 48 43. 26s12d 5 272 45. s258ds24

5d 512

Section P.2 Operations with Real Numbers

1. 13 1 32 5 45 3. 213 1 32 5 1s32 2 13d 5 19 5. 13 1 s232d 5 2s32 2 13d 5 219

7.

5 2s7 1 15d 5 222

27 2 15 5 27 1 s215d 9. 213 1 s28d 5 2s13 1 8d 5 221

Section P.2 Operations with Real Numbers75

53.21823

5 6 55.24816

5 23 57. 63 4 s27d 56327

5 29

59. 245

48

255 2

45

?258

5 252 61. 121

3 2 4 1256 2 5 121

34

256 2 5 121

3?

265 2 5

25

63.

5 4617

5 234 ? 8

17

534 4 21

8 5234 4

178 65.

5 1112

5 338 ? 2

9

418 4 33

2 5338 4

92

67. 43 5 s4ds4ds4d 69. 1234 2

4

5 1234 2123

4 21234 2123

4 2

71. s20.8d6 5 s20.8ds20.8ds20.8ds20.8ds20.8ds20.8d 73. s27d 3 s27d 3 s27d 5 s27d3

75. s25ds25ds25ds25d 5 s25d4 77. 2s7 3 7 3 7d 5 273 79.

5 16

s22d4 5 s22ds22ds22ds22d

81.

5 28

s22d3 5 s22ds22ds22d 83.

5 264

243 5 2s4ds4ds4d 85.

5 64125

s45d3

5 s45ds4

5ds45d

87.

5 132

5 2s2 132d

2s212d5

5 2s212ds21

2ds212ds21

2ds212d 89.

5 0.027

s0.3d3 5 s0.3ds0.3ds0.3d

91.

5 20.32

5 5s20.064d

5s20.4d3 5 5s20.4ds20.4ds20.4d 93.

5 0

5 10 2 10

16 2 6 2 10 5 s16 2 6d 2 10

47. 2321

852 5

22410

5212

549.

121

162 5

112

51.

5 213

298 116

2721122 5

29 ? 2 ? 8 ? 18 ? 9 ? 3 ? 2

95.

5 4

5 24 2 20

5 24 2 s5 ? 4d

24 2 5 ? 22 5 24 2 5 ? 4 97.

5 22

5 7 1 15

28 4 4 1 3 ? 5 5 s28 4 4d 1 s3 ? 5d

99.

5 6

5 14 2 8

14 2 2s8 2 4d 5 14 2 2s4d 101.

5 57

5 45 1 12

45 1 3s16 4 4d 5 45 1 3s4d


103.

5 3

5 2 1 1

2 1 f8 2 s14 4 2dg 5 2 1 f8 2 7g 105.

5 27

5 25 1 2

5 25 2 2f21g

52 2 2f9 2 s18 2 8dg 5 25 2 2f9 2 10g

107.

5 135

5 125 1 10

53 1 |214 1 4| 5 125 1 |210| 109.8 1 7

12 2 155

1523

5 25

111.

5 26

5 1 2 7

51111

2 7

42 2 5

112 7 5

16 2 511

2 7 113.

5 1

51212

524 2 12

12

6 ? 22 2 12

32 1 35

6 ? 4 2 129 1 3

115.

5 161

5 5.6f28.75g

5.6f13 2 2.5s26.3dg 5 5.6f13 1 15.75g 117. 56 2 3s400d 5 15,625 2 1200 5 14,425

119.500

s1.055d20 5500

2.91775755 171.36448 < 171.36

121.

Thus:

5 1 2 1 45180

140

1801

18180

160

1802 5 1 2 145 1 40 1 18 1 60180 2 5 1 2

163180

5180180

2163180

517

180.

x 5 1 2 114

129

11

101

132

14

129

11

101 x 1

13

5 1

123. $2618.68 1 $1236.45 2 $25.62 2 $455.00 2 $125.00 2 $715.95 5 $2533.56

125. (a)

(b) the sum of the daily gains and losses. The sum of the daily gains and lossesis equal to the difference of the value of the stock on Friday and the value of the stock on Monday. This sum could bedetermined from the graph by $524 svalue on Fridayd 2 $500 svalue on Mondayd 5 $24.

s15d 1 s18d 1 s25d 1 s116d 5 124 5

Day Daily Gain or Loss

Tuesday

Wednesday

Thursday

Friday 116

25

18

15

Section P.2 Operations with Real Numbers77

127. (a)

(b)

The fund would have $27,018.72.

(c) 27,018.72 2 10,800 5 $16,218.72

50311 10.0912 2216

2 1411 112

0.092 < 27,018.71558 < 27,018.72

$50s12ds18d 5 $10,800 129.

square meters A 5 5 ? 3 5 15

A 5 lw

l 5 5m, w 5 3m

131.

square inches A 512s8ds5d 5 20

A 512bh

b 5 8 cm, h 5 5 cm 133. V 5 l ? w ? h 5 14" ? 18" ? 42" 5 10,584 in3 4 1728 in3 5 6.125 ft3

135. (a)

Student incorrectly added the 3 and the 18 insteadof multiplying the 20 and the 3 first. Order ofoperations must be followed.

(b) UPC of 07673720012 9

1.

2.

3.

4. Next highest multiple of 10which is the check digit

Yes, it checks.

(c) UPC of 04180048700 3

1.

2.

3.

4. Next highest multiple of 10which is not the check digit

No, it does not check.60 2 56 5 4

5 60

36 1 20 5 56

4 1 8 1 0 1 8 1 0 5 20

s0 1 1 1 0 1 4 1 7 1 0d 3 3 5 36

70 2 61 5 95 70

39 1 22 5 61

7 1 7 1 7 1 0 1 1 5 22

s0 1 6 1 3 1 2 1 0 1 2d 3 3 5 39

20 3 3 1 18 5 60 1 18 5 78

20 3 3 1 18 5 20 3 21 5 420 137. True. A rational number is an integer divided by aninteger. The reciprocal of such a number is still aninteger divided by an integer, and thus, a rationalnumber.

141. Yes. For example,

27 < 23 and 27 < 24

s23d 1 s24d 5 27

143. If the numbers have like signs, the product or quotientis positive.

If the numbers have unlike signs, the product or quotientis negative.

145. Evaluate additions and subtractions from left to right.

not

3 5 21

6 2 3 5 1 2 2

6 2 s5 2 2d 6 2 5 2 2 5 s6 2 5d 2 2

147. Only common factors (not terms) of the numerator anddenominator can be canceled.

149. The squaring of the four must be performed beforemultiplying by the three by order of operations.

5 48

3 ? 42 5 3 ? 16

139. False. If a negative number is raised to an odd power,the result will be negative.

Section P.3 Properties of Real Numbers


1.Commutative Property of Addition3 1 s25d 5 25 1 3 3.

Additive Inverse Property25 2 25 5 0 5.

Commutative Property ofMultiplication

6s210d 5 210s6d

7.Multiplicative Identity Property7 ? 1 5 7 9.

Commutative Property of Addition25 1 35 5 35 1 25 11.

Associative Property of Addition3 1 s12 2 9d 5 s3 1 12d 2 9

13.Distributive Propertys8 2 5ds10d 5 8 ? 10 2 5 ? 10 15.

Associative Property of Additions10 1 8d 1 3 5 10 1 s8 1 3d 17.

Associative Property ofMultiplication

5s2ad 5 s5 ? 2da

19.Multiplicative Identity Property1 ? s5td 5 5t 21.

Additive Identity Property3x 1 0 5 3x 23.

Associative Property of Addition4 1 s3 2 xd 5 s4 1 3d 2 x

25.Distributive Property3s6 1 bd 5 3 ? 6 1 3 ? b 27.

Distributive Property6sx 1 3d 5 6 ? x 1 6 ? 3 29. 3s6yd 5 s3 ? 6dy

31. 15s23d 5 s23d15 33. 5s6 1 zd 5 5 ? 6 1 5 ? z 35. 25 1 s2xd 5 s2xd 1 25

37. sx 1 8d ? 1 5 sx 1 8d 39. (a) Additive Inverse:

(b) Multiplicative Inverse: 110

210 41. (a) Additive Inverse: 16

(b) Multiplicative Inverse:21

16

43. (a) Additive Inverse:

(b) Multiplicative Inverse:16z

26z 45. (a) Additive Inverse:

(b) Multiplicative Inverse:1

x 1 1

2sx 1 1d or 2x 2 1

47. sx 1 5d 2 3 5 x 1 s5 2 3d 49. 32 1 s4 1 yd 5 s32 1 4d 1 y 51. 3s4 ? 5d 5 s3 ? 4d5

53. 6s2yd 5 s6 ? 2d ? y 5 12y 55. 20s2 1 5d 5 20 ? 2 1 20 ? 5

57. 5s3x 2 4d 5 5 ? 3x 2 5 ? 4 or 5 ? 3x 1 5 ? 24 59. sx 1 6ds22d 5 x ? s22d 1 6 ? s22d

61. 26s2y 2 5d 5 26s2yd 1 s26ds25d 63. 3sx 1 5d 5 3x 1 15

65. 22sx 1 8d 5 22x 2 16 67. Original equation

Multiplication Property of Equality

Commutative Property of Multiplication

Associative Property of Multiplication

Multiplicative Inverse Property

Multiplicative Identity Property a 5 b

1 ? a 5 1 ? b

11c

? c2a 5 11c

? c2b

1cscad 5

1cscbd

1csacd 5

1csbcd

ac 5 bc, c Þ 0

Section P.3 Properties of Real Numbers79

69. Original Equation

Addition Property of Equality

Associative Property of Addition

Additive Inverse Property

Additive Identity Property x 5 22

x 1 0 5 22

x 1 s5 1 s25dd 5 3 2 5

sx 1 5d 1 s25d 5 3 1 s25d

x 1 5 5 3

71. Original equation




Additive Identity Property

Multiplication Property of Equality



Multiplicative Identity Property x 5112

1 ? x 5112

s12 ? 2dx 5

112

12s2xd 512s11d

2x 5 11

2x 1 0 5 11

2x 1 s25 1 5d 5 11

s2x 2 5d 1 5 5 6 1 5

2x 2 5 5 6

73.

5 28

5 32 2 4

16s1.75d 5 16s2 214d 5 16s2d 2 16s1

4d 75.

5 434

5 420 1 14

5 7s60d 1 7s2d

7s62d 5 7s60 1 2d

77.

5 62.82

5 63 2 0.18

5 9s7d 2 9s0.02d

9s6.98d 5 9s7 2 0.02d 79. Distributive Propertyasb 1 cd 5 ab 1 ac,

81. Use the graph to approximate the dividend paid in 1995.According to the graph, the dividend paid in 1995 wasapproximately $0.60.

83. Dividend per share

2000 dividend per share5 0.08s10d 1 0.21 5 $1.01

5 0.08t 1 0.21

85. Given two real numbers a and b, the sum a plus b is the same as the sum b plus a.

87. The multiplicative inverse of a real number is the number The product of a number and its multiplicative inverse is

the multiplicative identity 1. For example, 8 ?18

5 1.

1a

.asa Þ 0d

89. is the Multiplicative Property of zero.0 ? a 5 0


91. let

So, the Commutative Property is not true.

4 Þ 5

2 ? 1 1 2 Þ 2 ? 2 1 1

1 ( 2 Þ 2 ( 1

a ( b Þ b ( a

a 5 1 & b 5 2 let

So, the Associative Property is not true.

9 Þ 11

9 Þ 8 1 3

9 Þ 2 ? 4 1 3

2 1 7 Þ 4 ( 3

2 ? 1 1 s2 ? 2 1 3d Þ s2 ? 1 1 2d ( 3

1 ( s2 ( 3d Þ s1 ( 2d ( 3

a ( sb ( cd Þ sa ( bd ( c

a 5 1, b 5 2, c 5 3

Mid-Chapter Quiz for Chapter P

1.

−7 −6 −5 −4

−4.5−6

24.5 > 26 2.

−1 0 1 2

3 34 2

34 < 3

2

3. |23.2| 5 3.2 4. 2|5.75| 5 25.75 5. |215 2 7| 5 |222| 5 22

6.

5 |23.75| 5 3.75

|s210.5d 2 s26.75d| 5 |210.5 1 6.75| 7. 32 1 s218d 5 14

8. 210 2 12 5 s210d 1 s212d 5 2s10 1 12d 5 222 9.34

174

53 1 7

45

104

552

10.

54 2 1

65

36

512

223

216

546

216

11. s212ds24d 5 4812.

5s24ds15ds5ds32d 5 2

38

124521

15322 5

s24ds15ds5ds32d21

1

3

8

13.

5s7ds6ds12ds5d 5

s7ds6ds12ds5d 5

710

712

456

57

12?

65

14.

5s23ds23ds23d

s2ds2ds2d 5 2278

12322

3

5 1232212

32212

322 15.

5 4

5 21 1 5

5 3 2 4 1 5

3 2 22 1 25 4 5 5 3 2 4 1 25 4 5

1

2

16.

5 2

542

518 2 14

36 2 s34d

18 2 2s3 1 4d

62 2 s12 ? 2 1 10d 518 2 2s7d

36 2 s24 1 10d 17. (a) Distributive Property

(b) Additive Inverse Property10x 2 10x 5 0

8su 2 5d 5 8 ? u 2 8 ? 5

Section P.4 Algebraic Expressions 81

18. (a) Associative Property of Addition

(b) Multiplicative Identity Property2x ? 1 5 2x

s7 1 yd 2 z 5 7 1 sy 2 zd

19. 1522.76 2 328.37 2 65.99 2 50.00 1 413.88 5 $1492.28

20. s$30ds2ds12ds5d 5 $3600 21.

The sum of the parts of a circle is equal to 1.

724 5 x

2424 2

824 2

624 2

324 5 x

1 213 2

14 2

18 5 x

1 513 1

14 1

18 1 x

Section P.4 Algebraic Expressions

1. Terms: 10x, 5 3. Terms: 23y2, 2y, 28 5. Terms: 4x2, 23y2, 25x, 2y

7. Terms: x2, 22.5x, 21x

9. The coefficient of is 5.5y3 11. The coefficient of 234

t2 is 234

.

13. illustrates theCommutative Property of Addition4 2 3x 5 23x 1 4 15. illustrates the

Associative Property ofMultiplication

25s2xd 5 s25 ? 2dx 17. 5sx 1 6d 5 5x 1 30

19. 6x 1 6 5 6sx 1 1d 21. x3 ? x4 5 x ? x ? x ? x ? x ? x ? x

23. z2 ? z5 5 z ? z ? z ? z ? z ? z ? z 25. s25xds25xds25xds25xd 5 s25xd4

27. sx ? x ? xdsy ? y ? yd 5 x3y3 29.

5 227

223 ? 24 5 22314

31. x5 ? x7 ? x 5 x51711 5 x13 33. 33y4 ? y2 5 33y412 5 27y6

35. s24xd2 5 s24d2 ? x2 5 16x2 37.

5 216x2

24s2xd2 5 24s4x2d

39.

5 2125z6

5 s25 ? 25 ? 25dsz21212d

s25z2d3 5 s25z2ds25z2ds25z2d 41.

5 6x3y4

5 6 ? sx112d ? sy113d

s2xyds3x2y3d 5 s2 ? 3d ? sx ? x2d ? sy ? y3d

43.

5 210y9

s5y2ds2y4ds2y3d 5 s5 ? 21 ? 2dsy21413d 45.

5 23125z8

5 s25 ? 625dsz414d

25z4s25zd4 5 25z4s625z4d


63.

5 22z4 1 5z 1 8

5 s23 1 1dz4 1 s6 2 1dz 1 8

23z4 1 6z 2 z 1 8 1 z4 5 23z4 1 z4 1 6z 2 z 1 8 65.

5 uv 1 4u2v2

5 s2 2 1duv 1 s5 2 1du2v2

2uv 1 5u2v2 2 uv 2 suvd2 5 s2uv 2 uvd 1 s5u2v2 2 u2v2d

67. 4s2x2 1 x 2 3d 5 8x2 1 4x 2 12 69. 23s6y2 2 y 2 2d 5 218y2 1 3y 1 6

71.

5 12x 2 35

5 s10 1 2dx 1 s230 2 5d

5 s10x 1 2xd 1 s230 2 5d

10sx 2 3d 1 2x 2 5 5 10x 2 30 1 2x 2 5 73.

5 27y 2 7

5 s29 1 2dy 2 7

5 29y 1 2y 1 3 2 10

23s3y 2 1d 1 2sy 2 5d 5 29y 1 3 1 2y 2 10

75.

5 6 1 y3

5 s23 1 3dy2 1 6 1 y3

23sy2 2 2d 1 y2sy 1 3d 5 23y2 1 6 1 y3 1 3y2 77.

5 2y3 1 y2 1 y

5 sy3 1 y3d 1 sy2d 1 syd

y2sy 1 1d 1 ysy2 1 1d 5 y3 1 y2 1 y3 1 y

79.

5 26x 1 96

5 3f22x 1 32g

3f2x 2 4sx 2 8dg 5 3f2x 2 4x 1 32g 81.

5 2x 1 12x2

5 8x 2 6x 1 12x2

5 8x 1 3xf22 1 4xg

8x 1 3xf10 2 4s3 2 xdg 5 8x 1 3xf10 2 12 1 4xg

83.

5 22b2 1 4b 2 36

5 s22b2d 1 s6b 2 2bd 1 s230 2 6d

5 6b 2 30 2 2b2 2 2b 2 6

2f3sb 2 5d 2 sb2 1 b 1 3dg 5 2f3b 2 15 2 b2 2 b 2 3g

47.

5 64a7

5 s28 ? 28dsa611d

s22a2d3s28ad 5 s28a6ds28ad 49.

5 254u5v3

5 s9 ? 26d ? su213d ? sv211d

5 s32 ? 26d ? su2 ? u3d ? sv2 ? vd

s3uvd2s26u3vd 5 s32u2v2ds26u3vd

51. sxnd4 5 xn?4 5 x 4n 53. xn11 ? x3 5 xn1113 5 xn14

55. 3x 1 4x 5 s3 1 4dx 5 7x 57. 9y 2 5y 1 4y 5 s9 2 5 1 4dy 5 8y

59.

5 8x 1 18y

5 s3 1 5dx 1 s22 1 20dy

3x 2 2y 1 5x 1 20y 5 s3x 1 5xd 1 s22y 1 20yd 61.

5 6x2 2 2x

5 s7 2 1dx2 2 2x

7x2 2 2x 2 x2 5 7x2 2 x2 2 2x

Section P.4 Algebraic Expressions 83

85.

5 24x4 2 50x3

5 2s10 2 60dx3 2 4x4

2xs5x2d 2 4x3sx 1 15d 5 10x3 2 4x4 2 60x3 87. (a) Substitution:

Value of expression: 3

(b) Substitution:


5 2 3s5d

5 2 3s23d

89. (a) Substitution:

Value of expression:

(b) Substitution:

Value of expression: 10 2 4s14d 5 10 2 1 5 9

10 2 4s12d2

10 2 4 5 6

10 2 4s21d2 91. (a) Substitution:


(b) Substitution:


332 1 1

002 1 1



(b) Substitution:


3s26d 1 2s29d

3s1d 1 2s5d



(b) Substitution:


5 7

s23d2 2 s23ds22d 1 s22d2 5 9 2 6 1 4

5 7

22 2 s2ds21d 1 s21d2 5 4 2 s22d 1 1



(b) Substitution:


|22 2 s22d| 5 |22 1 2| 5 |0| 5 0

|5 2 2| 5 |3| 5 3 99. (a) Substitution:


(b) Substitution:


35s4d

40s514d

101.

5 90

5 12s15ds12d

A 512s15ds15 2 3d

A 512bsb 2 3d

b − 3

b

103.

A 5 2x2 1 3x

A 5 s2x 1 3dx

A 5 lw

105. Graphically, the sales in 1995 is approximately $2800million.

Let

Sales

5 $2800.34

5 969.45 1 1830.89

5 193.89s5d 1 1830.89

t 5 5.

107. Graphically, the median sale price in 1995 isapproximately $134 thousand.

Let

Sale price

thousand 5 $135.5

5 29.5 1 106.0

5 5.9s5d 1 106.0

t 5 5.


109. 2 2

Area

Area

square feet 5 1440

5 1200 1 240

5 12s100d 1 240

5 12s60 1 40d 1 20 ? 12

5 312

? hsb1 1 b2d4 1 2312

? b ? h4

Area oftriangle

?1Area oftrapezoid

?5Total area

113. To combine like terms add (or subtract) their respective coefficients and attach the common variable factor.Example: 5x4 2 3x4 5 s5 2 3dx4 5 2x4

115. The Distributive Property is used to simplify as follows: .5x 1 3x 5 s5 1 3dx 5 8x5x 1 3x

117. It is not possible to evaluate is undefined.x 1 2y 2 3

when x 5 5 and y 5 3 because 70

111. (d)

(e) , the next-highest multiple of 10 will be 90.

No. The work shows the only possible answer. 2a 5 25 90 2 81 2 a 5 4

a 5 5 9 2 a 5 4 90 2 s81 1 ad 5 4

81 1 a

s0 1 3 1 1 1 2 1 6 1 7d3 1 s4 1 8 1 9 1 3 1 ad

Section P.5 Constructing Algebraic Expressions

1. The sum of 8 and a number n is translated into thealgebraic expression 8 1 n.

3. The sum of 12 and twice a number n is translated intothe algebraic expression 12 1 2n.

5. Six less than a number n is translated into the algebraicexpression n 2 6.

7. Four times a number n minus 3 is translated into thealgebraic expression 4n 2 3.

9. One-third of a number n is translated into the algebraic

expression 13

n.

11. The quotient of a number x and 6 is translated into the

algebraic expression x6

.

13. Eight times the ratio of N and 5 is translated into the

algebraic expression 8 ?N5

.

15. The number c is quadrupled and the product is increasedby 10 is translated into the algebraic expression 4c 1 10.

17. Thirty percent of the list price L is translated into thealgebraic expression 0.30L.

19. The sum of a number and 5 divided by 10 is translated

into the algebraic expression n 1 5

10.

21. The absolute value of the difference between a numberand 5 is translated into the algebraic expression |n 2 5|.

23. The product of three and the square of a number decreasedby 4 is translated into the algebraic expression 3x2 2 4.

Section P.5 Constructing Algebraic Expressions85

25. A verbal description of is a number decreased by 2.t 2 2 27. A verbal description of is the sum of a numberand 50.

y 1 50

29. A verbal description of is the sum of three times anumber and two.

3x 1 2 31. A verbal description of is the ratio of a number

to two.

z2

37. A verbal description of is the sum of a number

and ten, divided by three.

x 1 103

41. Verbal Description: The amount of money (in dollars) represented by n quarters

Label: number of quarters

Algebraic Description: amount of money (in dollars)0.25n 5

n 5

43. Verbal Description: The amount of money (in dollars) represented by m dimes

Label: number of dimes

Algebraic Description:m10

m 5

33. A verbal description of is four-fifths of a number.45x 35. A verbal description of is eight times the

difference of a number and five.8sx 2 5d

39. A verbal description of is some number times thesum of the same number and seven.

xsx 1 7d

45. Verbal Description: The amount of money (in cents) represented by m nickels and n dimes

Label: number of dimes

number of dimes

Algebraic Description: 5m 1 10n

n 5

m 5

47. Verbal Description: The distance traveled in t hours at an average speed of 55 miles per hour

Label: number of hours

Algebraic Description: 55t

t 5

49. Verbal Description: The time to travel 100 miles at an average speed of r miles per hour

Label: average speed

Algebraic Description:100

r

r 5

51. Verbal Description: The amount of antifreeze in a cooling system containing y gallons of coolant that is 45% antifreeze

Label: number of gallons

Algebraic Description: 0.45y

y 5

53. Verbal Description: The amount of wage tax due for a taxable income of I dollars that is taxed at the rate of 1.25%

Label: number of dollars

Algebraic Description: 0.0125I

I 5


63. Verbal Description: The product of two consecutive even integers, divided by 4

Labels: first even integer

second even integer

Algebraic Description:

5 nsn 1 1d

2ns2n 1 2d

45

4nsn 1 1d4

2n 1 2 5

2n 5

65.

5 s2

5 s ? s

Area 5 side ? side 67.

5 0.375b2

5 12sbds0.75bd

Area 512sbasedsheightd 69. Perimeter

Area 5 2w ? w 5 2w2

5 2s2wd 1 2swd 5 4w 1 2w 5 6w

71. Perimeter

Area 5 sx ? 3d 1 s3 ? 2xd 5 3x 1 6x 5 9x

5 3 1 2x 1 6 1 x 1 3 1 x 5 4x 1 12 73. area of billiard table

The unit of measure for the area is feet or square feet.2

l ? sl 2 6d 5

75.n 0 1 2 3 4 5

2 7 12 17 22235n 2 3

Differences 5 5 5 5 5

77. The third row difference for the algebraic expressionwould be a.an 1 b

79. The phrase reduced by implies subtraction.

55. Verbal Description: The sale price of a coat that has a list price of L dollars if the sale is a “20% off” sale

Label: number of dollars

Algebraic Description: 0.80L

L 5

57. Verbal Description: The total hourly wage for an employee when the base pay is $8.25 per hour plus 60 cents for each ofq units producted per hour

Label: number of units produced

Algebraic Description: 8.25 1 0.60q

q 5

59. Verbal Description: The sum of a number n and three times the number

Label: the number

three times the number

Algebraic Description: n 1 3n 5 4n

3n 5

n 5

61. Verbal Description: The sum of two consecutive odd integers, the first of which is

Labels: first odd integer

second odd integer

Algebraic Description: s2n 1 1d 1 s2n 1 3d 5 4n 1 4

2n 1 3 5

2n 1 1 5

2n 1 1.

81. is the equivalent to (a) x multiplied by 4 and (c) theproduct of x and 4.4x

83. Using a specific case may make it easier to see the form of the expression for the general case.

Review Exercises for Chapter P 87

Review Exercises for Chapter P

1.

0 2 4−2−4−6

25 < 3 3.

0 1−1−3 −2

25

85

−−

285 < 2

25

5.

5 11

5 |11| d 5 |9 2 s22d| 7.

5 7.3

5 |27.3| 5 |213.5 1 6.2|

d 5 |213.5 2 s26.2d| 9. |25| 5 5

11. 2|27.2| 5 27.2 13. 15 1 s24d 5 11 15. 340 2 115 1 5 5 230

17. 263.5 1 21.7 5 241.8 19. 421 1

721 5

1121 21. 2

56 1 1 5 2

56 1

66 5

16

23. 834 2 65

8 5354 2

538 5

708 2

538 5

178 25. 27 ? 4 5 228 27. 120s25ds7d 5 24200

29.38

?2215

526120

52120

31.25624

5 14 33. 27

154 2

730

5 27

15?

3027

5 2

35. s26d3 5 s26ds26ds26d 5 2216 37. 242 5 21 ? 4 ? 4 5 216 39. 2s212d 5 21 ? s21

2ds212ds21

2d 518

41.

5 20

5 120 2 100

120 2 s52 ? 4d 5 120 2 s25 ? 4d 43.

5 98

5 8 1 90

5 8 1 3f30g

5 8 1 3f36 2 6g

8 1 3f62 2 2s7 2 4dg 5 8 1 3f36 2 2s3dg

45. Additive Inverse Property justifies 13 2 13 5 0. 47. Distributive Property justifies 7s9 1 3d 5 7 ? 9 1 7 ? 3.

49. Associative Property of Addition justifies 5 1 s4 2 yd 5 s5 1 4d 2 y.

51. illustrates the CommutativeProperty of Multiplication.su 2 vds2d 5 2su 2 vd

53. illustrates the DistributiveProperty.8sx 2 yd 5 8 ? x 2 8 ? y 55. 2s2u 1 3vd 5 u 2 3v

57. 2ys3y 2 10d 5 23y2 1 10y 59. x2 ? x3 ? x 5 x21311 5 x6

61.

5 23x3y4

sxyds23x2y3d 5 23 ? x112 ? y113 63. s5abds25a3d 5 125a4b


65.

5 5x

7x 2 2x 5 s7 2 2dx 67.

5 5v

3u 2 2v 1 7v 2 3u 5 s3u 2 3ud 1 s22v 1 7vd

69.

5 5x 2 10

5sx 2 4d 1 10 5 5x 2 20 1 10 71.

5 5x 2 y

3x 2 sy 2 2xd 5 3x 2 y 1 2x

73.

5 18b 2 15a

5 3b 1 15b 2 15a

3fb 1 5sb 2 adg 5 3fb 1 5b 2 5ag 75. (a)

Substitute:


(b)

Substitute:


s0d2 2 2s0d 2 3

x 5 0

32 2 2s3d 2 3

x 5 3

77. 200 2 3n 79. n2 1 49 81. The sum of twice a number andseven

83. The difference of a number and five, all divided by four 85. tax on I dollars at 18%0.18I 5

87. area of rectangle with length l and width sl 2 5dl ? sl 2 5d 5

89. Combined expenditures5 12.1 1 10.8 1 38.6 1 9.2 1 40.3 5 $111.0

91. Difference between the airports with the greatest and smallest passenger volumes

million5 15.6

5 30.8 2 15.2

93. Airports from greatest to smallest volume:

1. Atlanta/Hartsfield 30.8

2. Chicago/O’Hare 30.5

3. Dallas/Ft. Worth 26.6

4. Los Angeles 22.7

5. Denver 15.2

6. San Francisco 15.2

Pas

seng

ers

(in m

illio

ns)

Atla

nta

Chi

cago

Dal

las

Los

Ang

eles

Den

ver

San

Fra

ncis

co

40

30

20

10

Chapter Test for Chapter P 89

Chapter Test for Chapter P

1. (a)

(b) 223 > 2

32

252 > 2|23| 2. d 5 |26.2 2 5.7| 5 11.9

3.

5 220

5 25 2 15

214 1 9 2 15 5 s214 1 9d 2 15 4. 23 1 s27

6d 546 1 s27

6d 5 236 5 2

12

5. 22s225 2 150d 5 22s75d 5 2150 6.

5 60

s23ds4ds25d 5 s212ds25d

7. s2 716ds2 8

21d 516 8 5

18 4158 5

518 ? 8

15 5427

9. 12352

3

5227125

10.

5 15

5 2 1 13

5105

1 13

42 2 6

51 13 5

16 2 65

1 13

11. (a) demonstrates theAssociative Property of Multiplication.

(b) demonstrates the Multiplicative

Inverse Property.

3y ?13y

5 1

s23 ? 5d ? 6 5 23s5 ? 6d 12. 5s2x 2 3d 5 5s2xd 2 5s3d

13. s3x2yds2xyd2 5 s3x2ydsx2y2d 5 3x4y3 14. 3x2 2 2x 2 5x2 1 7x 2 1 5 22x2 1 5x 2 1

15.

5 a2

as5a 2 4d 2 2s2a2 2 2ad 5 5a2 2 4a 2 4a2 1 4a 16.

5 11t 1 7

5 4t 1 7t 1 7

5 4t 2 f27t 2 7g

4t 2 f3t 2 s10t 1 7dg 5 4t 2 f3t 2 10t 2 7g

17. Evaluating an expression means to substitute numerical values for each of the variables in the expression and then to simplifyaccording to the rules for order of operations.

(a) (b)

212

4 2 164

4 2 s4d24 2 s0d2

x 5 3 ⇒ 4 2 s3 1 1d2x 5 21 ⇒ 4 2 s21 1 1d2


18. “The product of a number n and 5 is decreased by 8” istranslated into the algebraic expression .5n 2 8

19. Perimeter

Area 5 ls0.6ld 5 0.6l2

5 2sld 1 2s0.6ld 5 2l 1 1.2l 5 3.2l

20. Verbal Description: The sum of two consecutive even integers, the first of which is 2n.

Labels: first even integer

second even integer

Algebraic Description: 2n 1 s2n 1 2d 5 4n 1 2

2n 1 2 5

2n 5

21. Verbal model: 9

Equation:

feet n 5 16

9 ? n 5 144

Total length5Length of each piece?

22. Verbal model:

Equation:

cubic feet

Verbal model: 5

Equation:

cubic feet 5 640

V 5 5 ? 128

Volume of 1 cord?5Volume of 5 cords

V 5 128

V 5 4 ? 4 ? 8

Height?Width?Length5Volume of 1 cord

C H A P T E R 1Linear Equations and Inequalities

Section 1.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . .92

Section 1.2 Linear Equations and Problem Solving . . . . . . . . . . .99

Section 1.3 Business and Scientific Problems . . . . . . . . . . . . .105

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

Section 1.4 Linear Inequalities . . . . . . . . . . . . . . . . . . . . .116

Section 1.5 Absolute Value Equations and Inequalities . . . . . . . .122

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

92

C H A P T E R 1Linear Equations and Inequalities

Section 1.1 Linear EquationsSolutions to Odd-Numbered Exercises

1. (a)

No

27 Þ 2

3s0d 2 7 5?

2

x 5 0 (b)

Yes

2 5 2

9 2 7 5 2

3s3d 2 7 5?

2

x 5 3

7. (a)

No

21 Þ 3

14s24d 5?

3

x 5 24 (b)

Yes

3 5 3

14s12d 5?

3

x 5 12

3. (a)

Yes

12 5 12

4 1 8 5?

3s4d

x 5 4 (b)

No

4 Þ 212

24 1 8 5?

3s24d

x 5 24

5. (a)

Yes

230 5 230

233 1 3 5 2s215d 3s211d 1 3 5

?2s211 2 4d

x 5 211 (b)

No

18 Þ 2

15 1 3 5 2s1d

3s5d 1 3 5?

2s5 2 4d

x 5 5


Distributive Property

Subtract from both sides.

Simplify.

No solution since 23 Þ 0.

23 5 0

3x 3x 2 3 2 3x 5 3x 2 3x

3x 2 3 5 3x

3sx 2 1d 5 3x



Combine like terms.

Identity since both sides equal.

5x 1 15 5 5x 1 15

5x 1 15 5 2x 1 3x 1 15

5sx 1 3d 5 2x 1 3sx 1 5d 13. is linear since variable has exponent 1.3x 1 4 5 10

15. is not linear since variable has exponent

not 1.

214x

2 3 5 5 17. Original equation

Subtract 15 from both sides.

Combine like terms.

Divide both sides by 3.

Simplify. x 5 25

3x3

5215

3

3x 5 215

3x 1 15 2 15 5 0 2 15

3x 1 15 5 0

Section 1.1 Linear Equations 93


Subtract 5 from both sides.

Combine like terms.

Divide both sides by

Simplify. x 5 272

22. 22x22

57

22

22x 5 7

22x 1 5 2 5 5 12 2 5

22x 1 5 5 1221. Check:

x 5 3

0 5 0 x 2 3 1 3 5 0 1 3

3 2 3 5?

0 x 2 3 5 0

23.

x 5 4

3x3

5123

3x 5 12 Check:

12 5 12

3s4d 5?

12 25.

y 5 20.7

26y26

54.226

26y 5 4.2 Check:

4.2 5 4.2

26s20.7d 5?

4.2

27.

x 5 223

x 5 246

6x6

5246

6x 5 24

6x 1 4 2 4 5 0 2 4

6x 1 4 5 0 Check:

0 5 0

24 1 4 5?

0

612232 1 4 5

?0

29.

u 5 21

22u22

52

22

22u 5 2

22u 1 5 2 5 5 7 2 5

22u 1 5 5 7 Check:

7 5 7

2 1 5 5?

7

22s21d 1 5 5?

7

31.

x 5 21

4x4

5244

4x 5 24

4x 2 7 1 7 5 211 1 7

4x 2 7 5 211 Check:

211 5 211

24 2 7 5?

211

4s21d 2 7 5?

211

33.

x 5 2

23x23

54623

23x 5 46

23x 2 4 1 4 5 42 1 4

23x 2 4 5 42 Check:

42 5 42

46 2 4 5?

42

23s2d 2 4 5?

42

35.

t 5 2103

3t3

5210

3

3t 5 210

3t 1 8 2 8 5 22 2 8

3t 1 8 5 22 Check:

22 5 22

210 1 8 5?

22

312103 2 1 8 5

?22

94 Chapter 1 Linear Equations and Inequalities

37.

22 5 t

212

65

6t6

212 5 6t

8 2 20 5 20 1 6t 2 20

8 5 20 1 6t

8 2 5t 1 5t 5 20 1 t 1 5t

8 2 5t 5 20 1 t Check:

18 5 18

8 1 10 5?

18

8 2 5s22d 5?

20 1 s22d

39.

x 5 2

2x2

542

2x 5 4

2x 2 5 1 5 5 21 1 5

2x 2 5 5 21

4x 2 2x 2 5 5 2x 2 2x 2 1

4x 2 5 5 2x 2 1 Check:

3 5 3

8 2 5 5?

4 2 1

4s2d 2 5 5?

2s2d 2 1

41.

13

5 x

721

521x21

7 5 21x

7 2 8x 1 8x 5 13x 1 8x

7 2 8x 5 13x Check:

133

5133

213

283

5? 13

3

7 283

5? 13

3

7 2 81132 5

?1311

32

43.

No solution

0 Þ 3

0 5 3

4y 2 4y 5 4y 1 3 2 4y

4y 5 4y 1 3

4y 2 3 1 3 5 4y 1 3

4y 2 3 5 4y

47.

No solution

24 Þ 0

24 5 0

29y 1 9y 2 4 5 29y 1 9y

29y 2 4 5 29y

49.

x 5 11

8x8

5888

8x 5 88

8x 2 64 1 64 5 24 1 64

8x 2 64 5 24

8sx 2 8d 5 24 Check:

24 5 24

8s3d 5?

24

8s11 2 8d 5?

24

51.

t 5 22

24t24

58

24

24t 5 8

24t 2 8 1 8 5 0 1 8

24t 2 8 5 0

24st 1 2d 5 0 Check:

0 5 0

24f0g 5?

0

24fs22d 1 2g 5?

0

45.

t 5 0

8t8

508

8t 5 0

28t 1 16t 5 216t 1 16t

28t 5 216t Check:

0 5 0

28s0d 5?

216s0d


53.

x 5 292

24x24

51824

24x 5 18

24x 2 12 1 12 5 6 1 12

24x 2 12 5 6

3x 2 7x 2 12 5 7x 2 7x 1 6

3x 2 12 5 7x 1 6

3sx 2 4d 5 7x 1 6 Check:

2512

5 2512

312172 2 5

?2

512

31292

2822 5

?2

632

1122

31292

2 42 5?

712922 1 6

55.

x 565

5x5

565

5x 5 6

5x 1 6 2 6 5 12 2 6

5x 1 6 5 12

8x 2 3x 1 6 5 12

8x 2 3sx 2 2d 5 12 Check:

12 5 12

605

5?

12

485

1125

5?

12

485

2 312452 5

?12

485

2 3165

2105 2 5

?12

81652 2 316

52 22 5

?12

57.

y 5 23

22y22

56

22

22y 5 6

22y 1 9 2 9 5 15 2 9

22y 1 9 5 15

5 2 2y 1 4 5 15

5 2 s2y 2 4d 5 15 Check:

15 5 15

5 1 10 5?

15

5 2 f210g 5?

15

5 2 f26 2 4g 5?

15

5 2 f2s23d 2 4g 5?

15

59.

x 5 23

5x5

5215

5

5x 5 215

5x 1 36 2 36 5 21 2 36

5x 1 36 5 21

12x 1 36 2 7x 5 7x 1 21 2 7x

12x 1 36 5 7x 1 21

12sx 1 3d 5 7sx 1 3d Check:

0 5 0

12f0g 5?

7f0g

12fs23d 1 3g 5?

7fs23d 1 3g

61.

253

5 x

253

53x3

25 5 3x

5 1 20 5 3x 2 20 1 20

5 5 3x 2 20

2x 1 5 2 2x 5 5x 2 20 2 2x

2x 1 5 5 5x 2 20

2x 1 14 2 9 5 5x 2 20

2sx 1 7d 2 9 5 5sx 2 4d Check:

653

5653

923

2273

5? 65

3

21463 2 2 9 5

?5113

3 2

21253

1213 2 2 9 5

?5125

32

123 2

21253

1 72 2 9 5?

51253

2 42


63.

u 5 50

51u52 5 s10d5

u5

5 10 Check:

10 5 10

505

5?

10

65.

t 51910

10t10

51910

10t 5 19

10t 2 4 1 4 5 15 1 4

10t 2 4 5 15

101t 2252 5 13

2210

t 225

532

Check:

1510

51510

1910

24

105? 15

10

1910

225

5? 3

2

67.

t 51023

23t23

51023

23t 5 10

2t 2 5t 5 10

101 t5

2t22 5 s1d10

t5

2t2

5 1 Check:

1 5 1

33

5?

1

223

153

5?

1

10

2151

106

5?

1

2

103

52

2103

25?

1

69.

x 5 2209

27x27

526027

27x 5 260

32x 2 5x 5 260

4s8xd 2 5x 5 260

2018x5

2x42 5 s23d20

8x5

2x4

5 23 Check:

23 5 23

2279

5?

23

2329

159

5?

23

8512

209 2 2

1412

209 2 5

?23

71.

x 5 220

3x3

5260

3

3x 5 260

3x 1 12 2 12 5 248 2 12

3x 1 12 5 248

4x 1 12 2 x 5 x 2 48 2 x

4x 1 12 5 x 2 48

12113

x 1 12 5 1 112

x 2 4212

13

x 1 1 51

12x 2 4 Check:

2173

5 2173

217

35? 25

32

123

220

31

33

5? 220

122 4

13

s220d 1 1 5? 1

12s220d 2 4


73.

2831

5 u

2831

531u31

28 5 31u

100 2 108 5 31u 1 108 2 108

100 5 31u 1 108

100 2 16u 1 16u 5 15u 1 108 1 16u

100 2 16u 5 15u 1 108

100 2 16u 5 15u 1 36 1 72

4s25 2 4ud 5 3s5u 1 12d 1 72

12125 2 4u3 2 5 15u 1 12

41 6212

25 2 4u

35

5u 1 124

1 6Check:

26931

526931

80793

5? 269

31

77593

13293

5? 210

311

9331

118631

253

13293

5? 210

311 3 1 6

25 1

3231

35? 2

4031 1 12

41 6

25 2 4s2 8

31d3

5? 5s2 8

31d 1 12

41 6

75.

x 5 23

3x3

5693

3x 5 69

3x 1 15 2 15 5 84 2 15

3x 1 15 5 84

10s0.3x 1 1.5d 5 s8.4d10

0.3x 1 1.5 5 8.4 Check:

8.4 5 8.4

6.9 1 1.5 5?

8.4

0.3s23d 1 1.5 5?

8.4

77.

x 5 12

12x12

514412

12x 5 144

12x 2 36 1 36 5 108 1 36

12x 2 36 5 108

10s1.2x 2 3.6d 5 s10.8d10

1.2x 2 3.6 5 10.8

1.2sx 2 3d 5 10.8 Check:

10.8 5 10.8

1.2s9d 5?

10.8

1.2s12 2 3d 5?

10.8

79.

x 515

5x5

515

5x 5 1

5x 2 16 1 16 5 215 1 16

5x 2 16 5 215

8x 2 3x 2 16 5 3x 2 3x 2 15

8x 2 16 5 3x 2 15

8x 2 16 5 3x 1 9 2 24

4s2x 2 4d 5 3sx 1 3d 2 24

6323

s2x 2 4d4 5 312

sx 1 3d 2 446

23

s2x 2 4d 512

sx 1 3d 2 4 Check:

2125

5 2125

2125

5? 8

52

205

2312

185 2 5

? 121

165 2 2 4

231

25

2205 2 5

? 121

15

1155 2 2 4

233211

52 2 44 5? 1

2115

1 32 2 4


81.

Labels:

Equation:

n 1 1 5 126

n 5 125

2n2

52502

2n 5 250

2n 1 1 2 1 5 251 2 1

2n 1 1 5 251

n 1 sn 1 1d 5 251

n 1 1 5 second integer

n 5 first integer

VerbalModel: 1 5 251

Firstinteger

Secondinteger

83.

Labels:

Equation:

n 1 2 5 84

n 5 82

2n2

51642

2n 5 164

2n 1 2 2 2 5 166 2 2

2n 1 2 5 166

n 1 sn 1 2d 5 166

n 1 2 5 second even integer

n 5 first even integer


First eveninteger

Second eveninteger

85.

Label:

Equation:

The repairs took 1.5 hours to complete.

n 532

32n32

54832

32n 5 48

162 2 162 1 32n 5 210 2 162

162 1 32n 5 210

n 5 number of hours for the repair

87. The fountain reaches its maximum height when the veloc-ity of the stream of water is zero.

t 532

seconds 5 1.5 seconds

32t32

54832

32t 5 48

0 1 32t 5 48 2 32t 1 32t

0 5 48 2 32t

VerbalModel:

5 210162 1 32 ?Numberof hours

89.

t 5 6 hours

61 t62 5 6s1d

t6

5 1

5t30

5 1

3t30

12t30

5 1

t

101

t15

5 1

91. (a)

—CONTINUED—

1 1.5 2 3 4 5

Width 300 240 200 150 120 100

Length 300 360 400 450 480 500

Area 90,000 86,400 80,000 67,500 57,600 50,000

t

Section 1.2 Linear Equations and Problem Solving99

91. (a) —CONTINUED—

(b) Since the length is times the width and the perimeter is fixed, as gets larger, the length gets larger and the area getssmaller. The maximum area occurs when the length and width are equal.

tt

500 5 l 480 5 l 450 5 l

100 5 w 120 5 w 150 5 w

1200 5 12w 1200 5 10w 1200 5 8w

If t 5 5: 1200 5 2w 1 2s5wd If t 5 4: 1200 5 2w 1 2s4wd If t 5 3: 1200 5 2w 1 2s3wd

400 5 l 360 5 l 300 5 l

200 5 w 240 5 w 300 5 w

1200 5 6w 1200 5 5w 1200 5 4w

If t 5 2: 1200 5 2w 1 2s2wd If t 5 1.5: 1200 5 2w 1 2s1.5wd If t 5 1: 1200 5 2w 1 2w

93.

From the graph, 1993 is the year in which expendituresreached $5500.

2.6 < t

538207

5207t207

538 5 207t

5500 2 4962 5 207t 1 4962 2 4962

5500 5 207t 1 4962 95. A conditional equation is an equation whose solution setis not the entire set of real numbers. An identity is anequation whose solution set is all real numbers.

97. Evaluating an expression means finding its value when its variables are replaced by real numbers. Solving an equation meansfinding all values of the variable for which the equation is true.

99. Equivalent equations have the same solution set. For example, and are equivalent.3x 2 6 5 03x 1 4 5 10

101. False. Multiplying both sides of an equation by zero does not yield an equivalent equation.

Section 1.2 Linear Equations and Problem Solving

1.

Label:

Equation:

x 5 52

x 1 30 2 30 5 82 2 30

x 1 30 5 82

Number 5 x

VerbalModel: 1 5Number 30 82 3.

Labels:

Equation:

$1084.62 5 x

28,200

265

26x26

28,200 5 26x

30,500 2 2300 5 26x 1 2300 2 2300

30,500 5 26x 1 2300

Bonus 5 2300

Amount of each paycheck 5 x

Annual salary 5 30,500

VerbalModel: 15 26 ?

Annualsalary

Amount ofeach paycheck Bonus


5. Percent: 30%

Parts out of 100: 30

Decimal: 0.30

Fraction: 30100 5

310

7. Percent: 7.5%

Parts out of 100: 7.5

Decimal: 0.075

Fraction: 751000 5

340

9. Percent: 12.5%

Parts out of 100: 12.5

Decimal: 0.125

Fraction: 18

11.

Labels:

Equation:

a 5 87.5

a 5 s0.35ds250d

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:

5 Percent ?Basenumber

13.

Labels:

Equation:

a 5 69.36

a 5 s0.085ds816d

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:


15.

Labels:

Equation:

a 5 600

a 5 s0.004ds150,000d

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:


17.

Labels:

Equation:

350 5 b

84

0.245 b

84 5 s0.24dsbd

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:


19.

Labels:

Equation:

400 5 b

42

0.1055 b

42 5 s0.105dsbd

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:


21.

Labels:

Equation:

12,000 5 b

96

0.0085 b

96 5 s0.008dsbd

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:



23.

Labels:

Equation:

3313% 5 p

16505000 5 p

1650 5 s pds5000d

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:


25.

Labels:

Equation:

175% 5 p

21001200 5 p

2100 5 s pds1200d

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel:


27.120 meters180 meters

51218

523

29.36 inches48 inches

53648

534

31.

54

1005

125

40 milliliters

1 liter5

0.04 liter1

33.5 pounds24 ounces

580 ounces24 ounces

5103

35.

x 5 4

x 5 6 ?23

x6

523

37.

t 5 6

t 5 4 ?32

t4

532

39.

t 5152

5 712

5 7.5

t 554

? 6

54

5t6

41.

6 5 y

12 5 2y

4y 5 6y 2 12

4y 5 6sy 2 2d

y6

5y 2 2

443.

4 5 y

16 5 4y

6y 1 6 5 10y 2 10

6sy 1 1d 5 10sy 2 1d

y 1 1

105

y 2 16

45.

Labels:

Equation:

a 5 1140

a 5 s0.38ds3000d

a 5 p ? b

Total enrollment 5 b

Percent 5 p

Freshmen 5 a

FreshmenVerbalModel:

5 Percent ?Total enrollment

47.

Labels:

Equation:

a 5 2

a 5 s0.05ds40d

a 5 s1 2 0.95ds40d

a 5 p ? b

Total students 5 b

Percent 5 p

Students failing test 5 a

Studentsfailing test

VerbalModel:

5 Percent ?Total students


49.

Labels:

Equation:

15.625% 5 p

25

1605 p

25 5 s pds160d

a 5 p ? b

Number of employees 5 b

Percent 5 p

Number laid off 5 a

Numberlaid off

VerbalModel:

5 Percent ?Number ofemployees

51.

Labels:

Equation:

18% < p

0.18 < p

1.558.45

5p ? 8.45

8.45

1.55 5 p ? 8.45

10 2 8.45 5 p ? 8.45

Cost of meal 5 b

Percent 5 p

Tip 5 a

TipVerbalModel:

5 Percent ?Cost ofmeal

53.

Labels:

Equation:

7% 5 p

0.07 5 p

12,250175,000

5p ? 175,000

175,000

12,250 5 p ? 175,000

Price of home 5 b

Percent 5 p

Commission 5 a

CommissionVerbalModel:

5 Percent ?Price ofhome

55.

Labels:

Equation:

200 5 b total parts

3

0.0155 b

3 5 s0.015dsbd

a 5 p ? b

Total parts 5 b

Percent 5 p

Defective parts 5 a

Defectiveparts

VerbalModel:

5 Percent ?Totalparts

57. (a)

Labels:

Equation:

177.78% < p

1.7778 < p

320180

5ps180d

180

320 5 s pds180d

a 5 p ? b

Area of smaller floor 5 b

Percent 5 p

Area of larger floor 5 a

Area oflarger floor

VerbalModel:

5 Percent ?Area ofsmaller floor

(b)

Labels:

Equation:

56.25% < p

0.5625 < p

180320

5ps320d

320

180 5 s pds320d

a 5 p ? b

Area of larger floor 5 b

Percent 5 p

Area of smaller floor 5 a

Area oflarger floor

VerbalModel:

5 Percent ?Area ofsmaller floor


59.

Labels:

Equation:

Monroe: Spring: Washington:

West: Howard: Clark:

21.66% < p 11.30% < p 12.77% < p

0.2166 < p 0.1130 < p 0.1277 < p

321,300

1,483,7005 p

167,7001,483,700

5 p 189,400

1,483,7005 p

321,300 5 ps1,483,700d167,700 5 ps1,483,700d 189,400 5 ps1,483,700d

6.01% < p 17.44% < p 30.84% < p

0.0601 < p 0.1744 < p 0.3084 < p

89,100

1,483,7005 p

258,7001,483,700

5 p 457,500

1,483,7005 p

89,100 5 ps1,483,700d 258,700 5 ps1,483,700d 457,500 5 ps1,483,700d

a 5 p ? b

Total population 5 b

Percent 5 p

County's population 5 a

County’spopulation

VerbalModel:

5 Percent ?Total population

61. Using the bar graph, the decrease in the per capita consumption of beef from 1980 to 1995 is approximately 8 pounds. Theapproximate percent decrease is:

Labels:

Equation:

870 5 p < 11%

8 5 p ? 70

a 5 p ? b

Beef consumption in 1980 5 b

Percent 5 p

Amount of decrease 5 a

Amount ofdecrease

VerbalModel:

5 Percent ?Beef consumptionin 1980

63.

Equation:

x < 11,750 million pounds

x 5 47 ? 250,000,000

Total number of poundsof pork consumed

VerbalModel:

5 ?Number ofpersons

Number of pounds of porkconsumed per capita

65.TaxPay

512.50625

5125

62505

150

67.Expanded volume

Compressed volume5

425 cu cm20 cu cm

5854

69.Area 1Area 2

5ps4d2

ps6d2 516p

36p5

49

71.Total priceTotal units

50.9520

590

20005 $0.0475 per ounce


77. (a)

(b)

The 6-ounce tube is a better buy.

Unit price 52.39

65 $0.3983 per ounce

Unit price 51.69

45 $0.4225 per ounce 79. Proportion:

x 5 5 111

x 5 7 ?4

5.5

x7

54

5.5

81. Proportion:

x 5 3

x 5 6 ?24

x6

524

83. Proportion:

h < 46.9 feet

h 551611

h 56 ? 86

11

h

865

611

85. Proportion:

x < 17.1 gallons

x 51800105

x 55 ? 360

105

5

1055

x360

87.

Proportion:

x 5 $2400 taxes

x 5 160,000 ?1650

110,000

x

160,0005

1650110,000

TaxAssessed value

VerbalModel:

5Tax

Assessed value

89.

Proportion:

x 5 2667 defective units

x 5 200,000 ?175

x

200,0005

175

Defective unitsTotal units

VerbalModel:

5Defective units

Total units

91. Percent means parts out of 100. 93. No,

50% 5 0.5

0.5% 5 0.005

12% Þ 50%.

95. No. It is necessary to know one of the following: the totalnumber of students in the class, the number of boys inthe class, or the number of girls in the class.

97. Mathematical modeling is using mathematics to solveproblems that occur in real-life situations. For examplesreview the real-life problems in the exercise set.


51.6920

5169

20005 $0.0845 per ounce 75. (a)

(b)

The ounce bag is a better buy.1412-

Unit price 50.995.5

5 $0.18 per ounce

Unit price 52.3214.5

5 $0.16 per ounce

Section 1.3 Business and Scientific Problems105

Section 1.3 Business and Scientific Problems

1.

Labels:

Equation:

Labels:

Equation:

40% 5 x

18.3645.97

5 x

18.36 5 x ? 45.97

Cost 5 45.97

Markup rate 5 x

Markup 5 18.36

x 5 $18.36

x 5 64.33 2 45.97

64.33 5 45.97 1 x

Markup 5 x

Cost 5 45.97

Selling price 5 64.33

Markuprate

VerbalModel: ?5 CostMarkup

Sellingprice

VerbalModel: 15 Cost Markup 3.

Labels:

Equation:

Labels:

Equation:

65% 5 x

98.80

152.005 x

98.80 5 x ? 152.00

Cost 5 152.00

Markup rate 5 x

Markup 5 98.80

$152.00 5 x

250.80 2 98.80 5 x

250.80 5 x 1 98.80

Markup 5 98.80

Cost 5 x


Markuprate


Sellingprice

VerbalModel: 15 Cost Markup

5.

Labels:

Equation:

Labels:

Equation:

21% 5 x

4672.50

22,250.005 x

4672.50 5 x ? 22,250.00

Cost 5 22,250.00

Markup rate 5 x

Markup 5 4672.50

$22,250.00 5 x

26,922.50 2 4672.50 5 x

26,922.50 5 x 1 4672.50

Markup 5 4672.50

Cost 5 x

Selling price 5 26,922.50

Markuprate


Sellingprice


Labels:

Equation:

Labels:

Equation:

x 5 $416.70

x 5 225.00 1 191.70

Markup 5 191.70

Cost 5 225.00

Selling price 5 x

x 5 $191.70

x 5 85.2% ? 225.00

Cost 5 225.00

Markup rate 5 85.2%

Markup 5 x

Markuprate


Sellingprice



9.

Labels:

Equation:

Labels:

Equation:

48.5% < x

24.2149.95

5 x

24.21 5 x ? 49.95

List price 5 49.95

Discount rate 5 x

Discount 5 24.21

x 5 $24.21

x 5 49.95 2 25.74

25.74 5 49.95 2 x

Discount 5 x

List price 5 49.95

Sale price 5 25.74

Listprice

VerbalModel:

5 ?Discount Discountrate

Saleprice

VerbalModel:

5 2 DiscountListprice

11.

Labels:

Equation:

Labels:

Equation:

63% 5 x

189.00300.00

5 x

189.00 5 x ? 300.00

List price 5 300.00

Discount rate 5 x

Discount 5 189.00

x 5 $111.00

x 5 300.00 2 189.00

Discount 5 189.00

List price 5 300.00

Sale price 5 x

Listprice

VerbalModel:


Saleprice

VerbalModel:


13.

Labels:

Equation:

Labels:

Equation:

x 5 $33.25

x 5 95.00 2 61.75

Discount 5 61.75

List price 5 95.00

Sale price 5 x

x 5 $61.75

x 5 65% ? 95.00

List price 5 95.00

Discount rate 5 65%

Discount 5 x

Listprice

VerbalModel:


Saleprice

VerbalModel:


15.

Labels:

Equation:

Labels:

Equation:

22% 5 x

251.90

1145.0055 x

251.90 5 x ? 1145.00

List price 5 1145.00

Discount rate 5 x

Discount 5 251.90

$1145.00 5 x

893.10 1 251.90 5 x

893.10 5 x 2 251.90

Discount 5 251.90

List price 5 x

Sale price 5 893.10

Listprice

VerbalModel:


Saleprice

VerbalModel:



17.

Labels:

Equation:

x 5 $22.05

x 5 85 2 62.95

85 5 62.95 1 x

Markup 5 x

Cost 5 62.95

Selling price 5 85

Sellingprice


Labels:

Equation:

Labels:

Equation:

6623 % 5 x

1015 5 x

10 5 x ? 15

Cost 5 15

Markup rate 5 x

Markup 5 10

10 5 x

25 5 15 1 x

Markup 5 x

Cost 5 15

Selling price 5 25

Markuprate


Sellingprice


21.

Labels:

Equation:

x 5 $25

x 5 75 2 50

50 5 75 2 x

Discount 5 x

List price 5 75

Sale price 5 50

Saleprice

VerbalModel:


23.

Labels:

Equation:

Labels:

Equation:

20% 5 x

420 5 x

4 5 x ? 20

List price 5 20

Discount rate 5 x

Discount 5 4

x 5 $4

x 5 20 2 16

16 5 20 2 x

Discount 5 x

List price 5 20

Sale price 5 16

Listprice

VerbalModel:


Saleprice

VerbalModel:


25.

Labels:

Equation:

—CONTINUED—

Length of call 5 9 minutes

8 5 x

4.40 5 0.55x

5.15 5 0.75 1 0.55x

Cost of additional minutes 5 0.55x

Cost of first minute 5 0.75

Total cost 5 5.15

Totalcost

VerbalModel:

15Cost of first minute

Cost of additional minutes


25. —CONTINUED—

Labels:

Equation:

x 5 $3.09

x 5 60% ? 5.15

List price 5 5.15

Discount rate 5 60%

Discount 5 x

Listprice

VerbalModel:


Labels:

Equation:

x 5 $2.06

x 5 5.15 2 3.09

Discount 5 3.09

List price 5 5.15

Selling price 5 x

Sellingprice

VerbalModel:


27.

Labels:

(Each tire costs $19.855, so three tires cost$59.565.)

Equation:

x 5 $54.15

1.10x 5 59.565

x 1 0.10x 5 59.565


Markup 5 0.10x

Cost 5 x

29.

Labels:

Equation:

Labels:

Equation:

Labels:

Equation:

x 5 $3717

x 5 4717 2 1000

Down payment 5 1000

Total bill 5 4717

Amount financed 5 x

x 5 $4717

x 5 4450 1 267

Sales tax 5 267

Cost 5 4450

Total bill 5 x

x 5 $267

x 5 6% ? 4450

Cost 5 4450

Sales tax rate 5 6%

Sales tax 5 x

CostVerbalModel: 1 5Markup

Sellingprice

Salestax

VerbalModel: 5

Sales tax rate ? Cost

Totalbill

VerbalModel: 5

Sales tax1Cost

Amountfinanced

VerbalModel: 5

Totalbill 2 Down payment

31.

Labels:

Equation:

2.5 hours 5 x

8032 5 x

80 5 32x

216.37 5 136.37 1 32x

Number of hours of labor 5 x

Bill for labor 5 32x

Bill for parts 5 136.37

Total bill 5 216.37

Totalbill

VerbalModel: 5

Bill forparts 1

Bill forlabor

33.

Labels:

Equation:

3 hours 5 x

10535 5 x

105 5 35x

380 5 1275 1 35x


Bill for labor 5 35x

Bill for parts 5 275

Total bill 5 380

Totalbill

VerbalModel: 5

Bill forparts 1

Bill forlabor


35.

Labels:

Equation:

100 2 x 5 50 gallons at 60%

x 5 50 gallons at 20%

20.40x 5 220

0.20x 1 60 2 0.60x 5 40

0.20x 1 0.60s100 2 xd 5 0.40s100d

Gallons of final solution 5 100

Percent of final solution 5 40%

Gallons of solution 2 5 100 2 x

Percent of solution 2 5 60%

Gallons of solution 1 5 x


Amount ofsolution 1

VerbalModel:

1 5Amount ofsolution 2

Amount offinal solution

37.

Labels:

Equation:

24 2 x 5 16 quarts at 60%

x 5 8 quarts at 15%

20.45x 5 23.6

0.15x 1 14.4 2 0.60x 5 10.8

0.15x 1 0.60s24 2 xd 5 0.45s24d

Quarts of final solution 5 24


Quarts of solution 2 5 24 2 x


Quarts of solution 1 5 x


Amount ofsolution 1

VerbalModel:



39.

Labels:

Equation:

100 2 x 5 25 pounds at $20 per pound

x 5 75 pounds at $12 per pound

28x 5 2600

12x 1 2000 2 20x 5 1400

12x 1 20s100 2 xd 5 14s100d

Cost per pound of final seed mix 5 14

Number of pounds of final seed mix 5 100

Cost per pound of seed 2 5 20

Number of pounds of seed 2 5 100 2 x

Cost per pound of seed 1 5 12

Number of pounds of seed 1 5 x

Cost ofseed 1

VerbalModel: 1 5

Cost ofseed 2

Cost of finalseed mix 41.

Labels:

Equation:

100 children tickets 5 x

2200 5 22x

2200 5 18x 1 4x

2200 5 6s3xd 1 4x

Price of children tickets 5 4

Number of children tickets 5 x

Price of adult tickets 5 6

Number of adult tickets 5 3x

Total sales 5 2200

Totalsales

VerbalModel: 15

Adultsales

Childrensales

43.

Labels:

Equation:

x 556 gallon

0.60x 5 0.5

2 2 0.40x 1 1.00x 5 2.5

0.40s5d 2 0.40x 1 1.00x 5 0.50s5d

Percent of antifreeze in final solution 5 50%

Number of gallons of final solution 5 5

Percent of pure antifreeze 5 100%

Number of gallons of pure antifreeze 5 x

Number of gallons antifreeze withdrawn 5 x

Percent of antifreeze in original mix 5 40%

Number of gallons of original antifreeze 5 5

Originalantifreezesolution

VerbalModel:

2 5

Someantifreezesolution

1Pureantifreeze

Finalantifreezesolution


45.

Labels:

Equation:

d 5 2275 miles

d 5 650 ? 3.5

Time 5 3.5

Rate 5 650

Distance 5 d

DistanceVerbalModel:

5 ?Rate Time 47.

Labels:

Equation:

10011 hour 5 t

1000110 5 t

1000 5 110 ? t

Time 5 t

Rate 5 110

Distance 5 1000


5 ?Rate Time

49.

Labels:

Equation:

20003

ftysec 5 r

10003y2

5 r

1000 5 r ?32

Time 532

Rate 5 r

Distance 5 1000


5 ?Rate Time 51.

Labels:

Equation:

2.5 hours 5 x

3012

5 x

30 5 12x

Time 5 x

Rate 5 12

Distance 5 30


5 ?Rate Time

53.

Labels:

Equation:

x 5 1440 miles

x 5 480s43d 1 600s4

3dTime 5

43

Rates 5 480 and 600

Distance 5 x


5 ?Rate Time 55.

Labels:

Equation:

17.65 minutes < t

517 hour 5 t

500017,000 5 t

5000 5 17,000 ? t

Time 5 t

Rate 5 17,000

Distance 5 5000


5 ?Rate Time

57.

Labels:

Equation:

234 hours 5 53

4 2 x s2nd part of trip at 52 mphd

3 hours 5 x s1st part of trip at 58 mphd

18 5 6x

317 5 58x 1 299 2 52x

317 5 58x 1 52s534 2 xd

52s534 2 xd 5 52 ? s53

4 2 xd s2nd part of tripd

58x 5 58 ? x s1st part of tripd

Time for second part of trip 5 534 2 x

Rate for second part of trip 5 52

Time for first part of trip 5 x

Rate for first part of trip 5 58

Distance 5 317


5 ?Rate Time 59. (a)

(b)

5 154 units per hour

5 308 units per hour

Shop’s rate 5 30 units in 8 hours

Printer’s rate 5 8 pages per minute


61. (a)

(b)

Labels:

Equation:

157

hours 5127

hours 5 t

1

7y125 t

1 5 1 7122t

1 5 113

1142t

1 5 1132std 1 11

42std

Friend’s time 5 t

Friend’s rate 514

Your time 5 t

Your rate 513

Work done 5 1

Friend’s rate 514 job per hour

Your rate 513 job per hour

Workdone

VerbalModel: 15

Work doneby you

Work doneby friend

63.

EI

5 R

E 5 IR 65.

S

1 2 r5 L

S 5 Ls1 2 rd

S 5 L 2 rL 67.

2h 2 96t

t 2 5 a

2h 2 96t 5 at2

2sh 2 48td 5 at2

h 2 48t 512

at2

h 5 48t 112

at2

69. Common formula:

Equation:

V 5 24 cubic units

V 5 3 ? 4 ? 2

V 5 lwh 71. Common formula:

Equation:

V < 461.8 cubic centimeters

V 5 147p

V 5 ps312d2

12

V 5 pr2h

73.

Labels:

Equation:

0.926 feet < x

3 5 3.24x

3 5 1.24x 1 2x

3 5 2s0.62xd 1 2sxd

Width 5 0.62x

Height 5 x

Perimeter 5 3

75.

Equation:

x 5 43 centimeters

43 5 x

129 5 3x

43 cm

43 cm 43 cm

129 5 x 1 x 1 x

VerbalModel: 1 2Perimeter 5 2 Width Height

VerbalModel: SidePerimeter 5 Side1 Side1


77.

Labels:

Equation:

I 5 $2850

I 5 s5000ds0.095ds6d

Time 5 6

Rate 5 9.5%

Principal 5 5000

Interest 5 I

InterestVerbalModel: 5 ?Principal Rate ? Time 79.

Labels:

Equation:

$3571.43 < P

5000.14

5 P

500 5 Ps0.14d

500 5 sPds0.07ds2d

Time 5 2

Rate 5 7%

Principal 5 P

Interest 5 500

InterestVerbalModel: 5 ?Principal Rate ? Time

81.

Labels:

Equation:

$15,000 5 x

3000.02

5 x

300 5 0.02x

3500 5 0.10x 1 3200 2 0.08x

3500 5 0.10x 1 0.08s40,000 2 xd

Time 5 1

Principal at 8% 5 40,000 2 x

Principal at 10% 5 x

Interest 5 3500

InterestVerbalModel: 5 ?Principal Rate ? Time 83. (a)

From the graph, 1993 was the year when the averagehourly wage was $10.15.

Yes, the result would be the same, 1993.

(b) The average annual hourly raise for bus drivers duringthis 8-year period is $0.307. Determine the averagehourly wage for each year using the model. The differ-ence between each two consecutive years is $0.307.

3 < t

2.9641 < t

0.91 5 0.307t

10.15 5 9.24 1 0.307t

y 5 9.24 1 0.307t, 0 ≤ t ≤ 7

85. The bus drivers’ average salaries were increasing at agreater annual rate at $0.307 compared to $0.209 for thecafeteria workers.

87. Markup is the difference between the cost a retailer paysfor a product and the price at which the retailer sells theproduct. Markup rate is the percent increase of themarkup.

89. If it takes you hours to complete a task, you can com-plete of the task in 1 hour.1yt

t 91. No, it quadruples. The area of a square of side is Ifthe length of the sides is the area is s2sd2 5 4s2.2s,

s2.s

Mid-Chapter Quiz for Chapter 1

1.

x 5 2

4x4

584

4x 5 8

4x 1 3 2 3 5 11 2 3

4x 1 3 5 11 Check:

11 5 11

8 1 3 5?

11

4s2d 1 3 5?

11

2.

z 5 2

z 2 2 1 2 5 0 1 2

z 2 2 5 0

23sz 2 2d

235

023

23sz 2 2d 5 0 Check:

0 5 0

23s0d 5?

0

23s2 2 2d 5?

0

Mid-Chapter Quiz for Chapter 1 113

3.

y 5 2

6y6

5126

6y 5 12

6y 1 6 2 6 5 18 2 6

2y 1 4y 1 6 5 18 2 4y 1 4y

2y 1 6 5 18 2 4y

2sy 1 3d 5 18 2 4y Check:

10 5 10

2s5d 5?

18 2 8

2s2 1 3d 5?

18 2 4s2d

4.

Identity

5t 1 7 5 5t 1 7

5t 1 7 5 7t 1 7 2 2t

5t 1 7 5 7st 1 1d 2 2t

5.

285

5 x

285

55x5

28 5 5x

24 1 4 5 5x 2 4 1 4

24 5 5x 2 4

x 2 x 1 24 5 6x 2 4 2 x

x 1 24 5 6x 2 4

4114

x 1 62 5 4132

x 2 12

14

x 1 6 532

x 2 1 Check:

375

5375

75

1305

5? 42

52

55

141

285 2 1 6 5

? 321

285 2 2 1

6.

u 5127

7u7

5127

7u 5 12

3u 1 4u 5 12

121u4

1u32 5 12s1d

u4

1u3

5 1 Check:

1 5 1

77

5?

1

37

147

5?

1

127

41

127

35?

1

7.

x 5332

22x22

523322

22x 5 233

22x 1 58 2 58 5 25 2 58

22x 1 58 5 25

8 2 2x 1 50 5 25

2s4 2 xd 1 50 5 25

1014 2 x5

1 52 5 101522

4 2 x

51 5 5

52

Check:

52

552

252

1102

5? 5

2

2252

?15

1 5 5? 5

2

82 2

332

51 5 5

? 52

4 2

332

51 5 5

? 52

8.

x 5 6

2x2

5122

2x 5 12

2x 1 3 2 3 5 15 2 3

2x 1 3 5 15

10s0.2x 1 0.3d 5 10s1.5d

0.2x 1 0.3 5 1.5 Check:

1.5 5 1.5

1.2 1 0.3 5?

1.5

0.2s6d 1 0.3 5?

1.5


9.

x 5 229

144< 20.20

x 5 22948

?13

3x3

5 22948

4 3

3x 5 22948

3x 51548

24448

3x 11112

21112

55

162

1112

3x 11112

55

1610.

x < 1.41

24.83x24.83

526.8024.83

24.83x 5 26.80

24.83x 1 6 2 6 5 20.80 2 6

24.83x 1 6 5 20.80

0.42x 1 6 2 5.25x 5 5.25x 2 0.80 2 5.25x

0.42x 1 6 5 5.25x 2 0.80

11. 0.45 is 45 hundredths so which reduces to and since percent means hundredths, 0.45 5 45%.

9200.45 5

45100 12.

Labels:

Equation:

200 5 b

5002.50

5 b

500 5 s2.50dsbd

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber

VerbalModel: 5 Percent ?

Basenumber


52.3512

5235

1200< $0.1958 per ounce 14.

Proportion:

x 5 2000 defective units

x 5 600,000 ?1

300

x

600,0005

1300

Defective unitsTotal units

VerbalModel:

5Defective units

Total units

15. Store computer:

Labels:

Equation:

—CONTINUED—

x 5 $437.50

x 5 s0.25ds1750d

List price 5 1750

Discount rate 5 0.25

Discount 5 x

Listprice

VerbalModel:


Labels:

Equation:

x 5 $1312.50

x 5 1750 2 437.50

Discount 5 437.50

List price 5 1750

Selling price 5 x

Sellingprice

VerbalModel: 25 Discount

Listprice


15. —CONTINUED—

Mail-order catalog computer

Labels:

Equation:

The mail-order catalog computer is the better buy.

x 5 $1274.95

x 5 1250 1 24.95

Shipping 5 24.95

List price 5 1250

Selling price 5 x

Sellingprice

VerbalModel: 15 Shipping

Listprice

16.

Labels:

Equation:

7 hours 5 x

126 5 18x

616 5 490 1 18x

616 5 40s12.25d 1 xs18d

Number of hours 5 x

Overtime wages 5 xs18dRegular wages 5 40s12.25dTotal wages 5 616

Totalwages

VerbalModel: 15

Regularwages

Overtimewages 17.

Labels:

Equation:

50 2 x 5 10 gallons at 50%


20.25x 5 210

25 2 0.25x 5 15

0.25x 1 25 2 0.50x 5 15

0.25x 1 0.50s50 2 xd 5 0.30s50d

Gallons of final solution 5 50


Gallons of solution 2 5 50 2 x


Gallons of solution 1 5 x


Amount ofsolution 1

VerbalModel:



18.

Labels:

Equation:

4.5 hours 5 6 2 x ssecond part of trip at 46 mphd

1.5 hours 5 x sfirst part of trip at 62 mphd

24 5 16x

300 5 62x 1 276 2 46x

300 5 62x 1 46s6 2 xd

Time for second part 5 6 2 x

Rate of second part 5 46

Time for first part 5 x

Rate of first part 5 62

Distance 5 500


5 ?Rate Time


19.

Labels:

Equation:

3.43 hours < 247 5

4814 5 t

1 5 s1448dstd

1 5 s16 1

18dstd

1 5 s16dstd 1 s1

8dstdTime for both 5 t

Rate of second person 518

Rate of first person 516

Work done 5 1

Workdone

VerbalModel: 15

Portion doneby first person

Portion done bysecond person

20.

5 13

Length of side of square III 5 5 1 8

s 5 5

4s 5 20

Perimeter of square I 5 20

5 169 square inches

5 132

Area 5 s2

s 5 8

4s 5 32

Perimeter of square II 5 32

Section 1.4 Linear Inequalities

1. (a)

Yes

11 > 0

21 2 10 > 0

7s3d 2 10 > 0 (b)

No

224 > 0

214 2 10 > 0

7s22d 2 10 > 0 (c)

Yes

152 > 0

352 2

202 > 0

352 2 10 > 0

7s52d 2 10 > 0 (d)

No

2132 > 0

72 2202 > 0

72 2 10 > 0

7s12d 2 10 > 0

3. (a)

No

0 < 236

< 2

0 <156

< 2

0 <10 1 5

6< 2 (b)

Yes

0 < 136

< 2

0 <96

< 2

0 <4 1 5

6< 2 (c)

Yes

0 <56

< 2

0 <0 1 5

6< 2 (d)

No

0 <216

< 2

0 <26 1 5

6< 2

5. Matches graph (d).

x

0 1 2 3 4

4

5

7. Matches graph (a).

−4

−4

−2 64

4

20−6

x

9. Matches graph (f).

x

−2 2

2−1

10−1

11. Matches graph (a).

x

−2 2

1−1

10−1


x

−2 2

1−2

10−1

15.

x

0 321−1−2

x ≤ 2

Section 1.4 Linear Inequalities 117

17.

x

543

3.5

210

x > 3.5 19.

x

420−6

−5

−2−4

3

25 < x ≤ 3 21.

x

543210

4 > x ≥ 1

23.

x

3210−1

32

32 ≥ x > 0 25. or

x

20−6

−5 −1

−8 −2−4

x ≥ 21x < 25 27. or

x

42 860

3 7

x > 7x ≤ 3

29.

215 1 x < 224

23s5 213xd < s8d 2 3

5 213x > 8 31.

x

30 6541 2

x ≥ 4

x 2 4 1 4 ≥ 0 1 4

x 2 4 ≥ 0 33.

x

3210

x ≤ 2

x 1 7 2 7 ≤ 9 2 7

x 1 7 ≤ 9

35.

x

4 620−2

x < 4

2x2

<82

2x < 8 37.

x

0−5 −1−2−3−4

x ≤ 24

29x29

≤3629

29x ≥ 36 39.

x

106 840 2

x > 8

243

? 234

x > 26 ? 243

234

x < 26

41.

x

9875 6

x ≥ 7

21 ? x ≥ 27 ? 21

2x ≤ 27

5 2 x 2 5 ≤ 22 2 5

5 2 x ≤ 22 43.

x

9875 6

x > 7

2x2

>142

2x > 14

2x 2 5 1 5 > 9 1 5

2x 2 5 > 9 45.

x

101

32

2

x > 223

23x23

>2

23

23x < 2

5 2 3x 2 5 < 7 2 5

5 2 3x < 7

47.

x >92

x64

29

20

4x4

>184

4x > 18

4x 2 11 1 11 > 7 1 11

4x 2 11 > 7

3x 2 11 1 x > 2x 1 7 1 x

3x 2 11 > 2x 1 7 49.

x >2011

211x211

>220211

x

3210

1120 211x < 220

211x 1 7 2 7 < 213 2 7

211x 1 7 < 213

23x 2 8x 1 7 < 8x 2 8x 2 13

23x 1 7 < 8x 2 13


51.

x >83

3x3

>83

3x > 8

x 1 2x > 8 2 2 1 2x

x > 8 2 2xx

3210

38 41x

42 > 12 2x224

x4

> 2 2x2

53.

x ≤ 28

5x5

≤ 2405

5x ≤ 240

5x 1 40 2 40 ≤ 0 2 40

5x 1 40 ≤ 0

8x 2 3x 1 40 ≤ 3x 2 3x

8x 1 40 ≤ 3x

8x 2 32 1 72 ≤ 3x

x

0−8 −2−4−10 −6−12

8sx 2 4d 1 72 ≤ 3x

241x 2 43

1 32 ≤ 1x8224

x 2 4

31 3 ≤

x8

55.

x > 215x

−16

−15

−20 0−8 −4−12

2x < 15

2x 2 60 1 60 < 245 1 60

2x 2 60 < 245

9x 2 10x 2 60 < 10x 2 45 2 10x

9x 2 60 < 10x 2 45

1513x5

2 42 < 12x3

2 3215

3x5

2 4 <2x3

2 3 57.

x

86

52

2 4

7

0

52

< x < 7

52

<2x2

<142

5 < 2x < 14

0 1 5 < 2x 2 5 1 5 < 9 1 5

0 < 2x 2 5 < 9

59.

23 ≤ x < 21

21 > x ≥ 23

2

22>

22x22

≥6

22

x

−4 −3 −2 −1 0

2 < 22x ≤ 6

8 2 6 < 6 2 6 2 2x ≤ 12 2 6

8 < 6 2 2x ≤ 12 61.

840−4

−6 6

−8

x

6 > x > 26

s26ds21d > s26d12x62 > s26ds1d

21 < 2x6

< 1

63.

232

< x <92

232

<2x2

<92

23 < 2x < 9

26 1 3 < 2x 2 3 1 3 < 6 1 3

26 < 2x 2 3 < 6x

6

9

4

2

0 2

3

2

2 23 <

2x 2 32

< 3 65.

x

129

10

3 60

1

1 < x < 10

23 1 4 < x 2 4 1 4 < 6 1 4

23 < x 2 4 < 6

1 >x 2 423

> 22


67.

21 < x ≤ 4

x ≤ 4 and x > 21

2x2

≤82 and

2x2

>222

2x ≤ 8 and 2x > 22

2x 2 4 1 4 ≤ 4 1 4 and 2x 1 8 2 8 > 6 2 8

x

0 1 2 3 4 5−2 −1

2x 2 4 ≤ 4 and 2x 1 8 > 6

69.

x ≤ 26

x < 24 and x ≤ 26

3x3

<212

3

3x < 212 and x ≤ 26

7 2 7 1 3x < 25 2 7 and 2x2

≤212

2

7 1 3x < 25 and 2x ≤ 212

7 1 4x 2 x < 25 1 x 2 x and 2x 1 10 2 10 ≤ 22 2 10

x

−4 0 2−2−8−10 −6

7 1 4x < 25 1 x and 2x 1 10 ≤ 22

71.

2` < x < `

x < 10 or x ≥ 8

2212x22 < s25ds22d or

451

54

x2 ≥ s10d

2x2

> 25 or 54

x ≥ 10

6 2 6 2x2

> 1 2 6 or 54

x 2 6 1 6 ≥ 4 1 6

x

7 8 9 10 11

6 2x2

> 1 or 54

x 2 6 ≥ 4

73.

x < 283 or x ≥

52

3x3

< 283 or

4x4

≥104

3x < 28 or 4x ≥ 10

3x 1 11 2 11 < 3 2 11 or 4x 2 1 1 1 ≥ 9 1 1

3x 1 11 < 3 or 4x 2 1 ≥ 9

7x 2 4x 1 11 < 3 1 4x 2 4x or 52

x 132

x 2 1 ≥ 9 232

x 132

x

x25

−4 −3 −2 −1 21 3 40

− 38 7x 1 11 < 3 1 4x or

52

x 2 1 ≥ 9 232

x


75.

y

051015

210 ≥ y

270 ≥ 7y

240 2 30 ≥ 7y 1 40 2 40

230 ≥ 7y 1 40

3y 2 3y 2 30 ≥ 3y 1 4y 1 40

23y 2 30 ≥ 4y 1 40

23sy 1 10d ≥ 4sy 1 10d 77.

20−2−4

−5

−6

x

0 ≥ x > 25

0

23≥

23x23

>1523

0 ≤ 23x < 15

24 1 4 ≤ 24 2 3x 1 4 < 11 1 4

24 ≤ 4 2 3x < 11

24 ≤ 2 2 3x 2 6 < 11

24 ≤ 2 2 3sx 1 2d < 11

79. or

Hx|x < 23J < Hx|x ≥ 2J

x ≥ 2x < 23 81.

Hx|x ≥ 25J > Hx|x < 4J

25 ≤ x < 4 83. or

Hx|x ≤ 22.5J < Hx|x ≥ 20.5J

x ≥ 20.5x ≤ 22.5

85. Hx|x ≥ 27J > Hx|x < 0J 87. Hx|x < 25J < Hx|x > 3J 89. Hx|x > 292J > Hx|x ≤ 2

32J

91. is nonnegative” using inequali-ty notation is x ≥ 0.“x 93. is at least 2” using inequality

notation is z ≥ 2.“z 95. is at least 10, but no more than

16” using inequality notation is10 ≤ n ≤ 16.

“n

97. A verbal description of is is at least 52.

xx ≥ 52 99. A verbal description of

is is at least 3 and less than 5.y3 ≤ y < 5 101. A verbal description of

is is greater than 0and no more than p.

z0 < z ≤ p

103.

Labels:

Inequality:

C ≤ 2600

1900 1 C 2 1900 ≤ 4500 2 1900

1900 1 C ≤ 4500

Total money 5 4500

Other costs 5 C

Transportation costs 5 1900

Transportationcosts

VerbalModel:

1Othercosts

Total moneyfor trip

≤ 105.

The average temperature in Miami, therefore, is greaterthan ( ) the average temperature in New York.>

Temp inMiami

VerbalModel:

> Temp inWashington

Temp inNew York

>

107.

Label:

Inequality:

m < 26,000 miles

0.35m0.35

<91000.35

0.35m < 9100

0.35m 1 2900 2 2900 < 12,000 2 2900

0.35m 1 2900 < 12,000

Operating cost 5 0.35m 1 2900

109.

Labels:

Inequality:

x ≥ 31

x > 30.224525

28.95x28.95

>875

28.95

28.95x > 875

89.95x 2 61x > 61x 1 875 2 61x

89.95x > 61x 1 875

Cost 5 61x 1 875

Revenue 5 89.95x

Operatingcost

VerbalModel:

< $12,000Revenue

VerbalModel:

> Cost


111.

Label:

Inequality:

Since represents the additional minutes after the first minute, the call must be less than6.38 minutes. If a portion of a minute is billed as a full minute, then the call must be lessthan or equal to 6 minutes.

x

x ≤ 5.386667

0.75x0.75

≤4.040.75

0.75x ≤ 4.04

0.96 1 0.75x 2 0.96 ≤ 5.00 2 0.96

$0.96 1 $0.75 ≤ $5.00

Number of additional minutes 5 x

Cost of first minute

VerbalModel:

1Cost of additionalminutes

$5.00≤

113.

Label:

Inequality:

2 ≤ x ≤ 16

42

≤2x2

≤322

4 ≤ 2x ≤ 32

36 2 32 ≤ 2x 1 32 2 32 ≤ 64 2 32

36 ≤ 2x 1 32 ≤ 64

Perimeter 5 2x 1 32

36 ≤VerbalModel:

≤ 64Perimeter 115.

3 ≤ n ≤152

124

≤4n4

≤304

12 ≤ 4n ≤ 30

117.

Labels: First payment plan: $12.50 per hour

Second payment plan: perhour where represents the number ofunits produced.

Inequality:

If more than 6 units are produced per hour, the secondpayment plan yields the greater hourly wage.

n > 6

0.75n > 4.5

8 1 0.75n > 12.5

n$8 1 $0.75n

Second plan> First plan 119.

Label:

Inequality:

t 5 23 → year 1987

t 5 22 → year 1988

t 5 21 → year 1989

t < 20.399

20.276t20.276

<0.11

20.276

20.276t > 0.11

5.890 2 5.890 2 0.276t > 6 2 5.890

5.890 2 0.276t > 6

Air pollutant emission 5 5.890 2 0.276t

Air pollutantemission

VerbalModel: > 6

121. (f)

At most, you can purchase one premium moviechannel.

x ≤ 1.58

x ≤18.811.91

11.91x ≤ 18.8

C 5 31.20 1 11.91x ≤ 50 (g)

At most, you can purchase four pay-per-view movies.

x ≤ 4.00

3.95x ≤ 15.81

C 5 34.19 1 3.95x ≤ 50

VerbalModel:


123. Yes, dividing both sides of an inequality by 5 is the sameas multiplying both sides by

x 5 3 x 5 3

15

? 5x 5 15 ?15

5x5

5155

5x 5 15 5x 5 15

15.

125. The multiplication and division properties differ. Theinequality symbol is reversed if both sides of theinequality are multiplied or divided by a negative realnumber.

127. If then and or 28 < 2t ≤ 5.5 ≥ 2t > 28s21ds25d ≥ s21dstd > s21ds8d25 ≤ t < 8,

Section 1.5 Absolute Value Equations and Inequalities

1.

No

7 Þ 10

|27| 5?

10

|212 1 5| 5?

10

|4s23d 1 5| 5?

10

|4x 1 5| 5 10, x 5 23 3.

Yes

2 5 2

|22| 5?

2

|6 2 8| 5?

2

|6 2 2s4d| 5?

2

|6 2 2w| 5 2, w 5 4 5. or x 2 10 5 217x 2 10 5 17

7. or 4x 1 1 5 2124x 1 1 5

12 9.

or x 5 24x 5 4

|x| 5 4 11.

No solution

|t| 5 245

13.

h 5 0

|h| 5 0 15.

x 5 3 x 5 23

5x 5 15 or 5x 5 215

|5x| 5 15 17.

x 5 21 x 5 11

x 2 16 5 5 or x 2 16 5 25

|x 2 16| 5 5

19.

s 5 11 s 5 214

2s 5 22 2s 5 228

2s 1 3 5 25 or 2s 1 3 5 225

|2s 1 3| 5 25 21.

y 5163 y 5 16

23y 5 216 23y 5 248

32 2 3y 5 16 or 32 2 3y 5 216

|32 2 3y| 5 16

23.

No solution

|3x 1 4| 5 216 25.

x 543

23x 5 24

4 2 3x 5 0

|4 2 3x| 5 0

27.

x 5152 x 5 2

392

2x 5 15 2x 5 239

23x 5 5 23x 5 213

23x 1 4 5 9 or 23x 1 4 5 29

|23x 1 4| 5 9 29.

x 5 18.75 x 5 26.25

x 56

0.32 x 5

220.32

0.32x 5 6 0.32x 5 22

0.32x 2 2 5 4 or 0.32x 2 2 5 24

|0.32x 2 2| 5 4

Section 1.5 Absolute Value Equations and Inequalities123

31.

x 5175 x 5 2

115

5x 5 17 5x 5 211

5x 2 3 5 14 or 5x 2 3 5 214

|5x 2 3| 5 14

|5x 2 3| 1 8 5 22 33.

x 5 253 x 5 2

133

3x 5 25 3x 5 213

3x 1 9 5 4 or 3x 1 9 5 24

|3x 1 9| 5 4

|3x 1 9| 2 12 5 28

35.

x 5 214 x 5

154

24x 5 1 24x 5 215

7 2 4x 5 8 or 7 2 4x 5 28

|7 2 4x| 5 8

22|7 2 4x| 5 216 37.

x 5 3 x 5 2

2x 5 6 2x 5 4

2x 2 5 5 1 or 2x 2 5 5 21

|2x 2 5| 5 1

3|2x 2 5| 5 3

3|2x 2 5| 1 4 5 7

39.

x 5 23

3x 5 29

7 5 x 3x 1 8 5 21

8 5 x 1 1 x 1 8 5 22x 2 1

x 1 8 5 2x 1 1 or x 1 8 5 2s2x 1 1d|x 1 8| 5 |2x 1 1| 41.

x 5 214

32 5 x 4x 5 21

3 5 2x x 1 2 5 23x 1 1

x 1 2 5 3x 2 1 or x 1 2 5 2s3x 2 1d|x 1 2| 5 |3x 2 1|

43.

11 5 x

77 5 7x

13 5 x 45 5 232 1 7x

45 5 32 1 x 45 2 4x 5 232 1 3x

45 2 4x 5 32 2 3x or 45 2 4x 5 2s32 2 3xd|45 2 4x| 5 |32 2 3x| 45.

x 548 5

12

No solution 8x 5 4

210 5 6 8x 2 10 5 26

4x 2 10 5 4x 1 6 4x 2 10 5 24x 2 6

4x 2 10 5 2s2x 1 3d or 4x 2 10 5 22s2x 1 3d|4x 2 10| 5 2|2x 1 3|

47. |x 2 5| 5 3

49. (a)

Yes

2 < 3

|2| < 3

x 5 2 (b)

No

4 < 3

|24| < 3

x 5 24 (c)

No

4 < 3

|4| < 3

x 5 4 (d)

Yes

1 < 3

|21| < 3

x 5 21

51. (a)

No

2 ≥ 3

|2| ≥ 3

|9 2 7| ≥ 3

x 5 9 (b)

Yes

11 ≥ 3

|211| ≥ 3

|24 2 7| ≥ 3

x 5 24 (c)

Yes

4 ≥ 3

|4| ≥ 3

|11 2 7| ≥ 3

x 5 11 (d)

No

1 ≥ 3

|21| ≥ 3

|6 2 7| ≥ 3

x 5 6


53.

23 < y 1 5 < 3

|y 1 5| < 3 55.

or 7 2 2h ≤ 297 2 2h ≥ 9

|7 2 2h| ≥ 9 57. “All greater than and lessthan 5.”

x

6

5

2 42 04

22x

59. “All less than or equal to 4 orgreater than 7.”

x

7

84 620

x 61.

24 < y < 4

|y| < 4 63.

or x ≤ 26 x ≥ 6

|x| ≥ 6

65.

27 < x < 7

214 < 2x < 14

|2x| < 14 67.

29 ≤ y ≤ 9

23 ≤y3

≤ 3

|y3| ≤ 3 69.

22 ≤ y ≤ 6

24 ≤ y 2 2 ≤ 4

|y 2 2| ≤ 4

71.

x > 4 x < 216

x 1 6 > 10 or x 1 6 < 210

|x 1 6| > 10 73.

23 ≤ x ≤ 4

26 ≤ 2x ≤ 8

27 ≤ 2x 2 1 ≤ 7

|2x 2 1| ≤ 7 75.

t ≤ 2152 t ≥ 5

2

t ≤ 2456 t ≥ 15

6

6t ≤ 245 6t ≥ 15

6t 1 15 ≤ 230 or 6t 1 15 ≥ 30

|6t 1 15| ≥ 30

77.

Absolute value is always positive.

2` < x < `

|2 2 5x| > 28 79.

No solution

Absolute value is never negative.

|3x 1 10| < 21 81.

282 ≤ x ≤ 78

280 ≤ x 1 2 ≤ 80

|x 1 2| ≤ 80

|x 1 2|10

≤ 8

83.

2104 < y < 136

2120 < y 2 16 < 120

|y 2 16| < 120

|y 2 16|4

< 30 85.

z < 250 z > 110

z

10< 25

z10

> 11

z

102 3 < 28 or

z10

2 3 > 8

| z10

2 3| > 8 87.

25 < x < 35

210.2

< x <7

0.2

21 < 0.2x < 7

24 < 0.2x 2 3 < 4

|0.2x 2 3| < 4

89.

283

≤ x ≤323

323

≥ x ≥283

232 ≤ 23x ≤ 228

22 ≤ 30 2 3x ≤ 2

20.4 ≤ 6 235

x ≤ 0.4

|6 235

x| ≤ 0.4 91.

Absolute value is always positive.

2` < x < `

|3x 1 6| > 22

22|3x 1 6| < 4 93.

24 ≤ x ≤ 40

40 ≥ x ≥ 24

220 ≤ 2x2

≤ 2

211 ≤ 9 2x2

≤ 11

|9 2x2| ≤ 11

|9 2x2| 2 7 ≤ 4

Review Exercises for Chapter 1 125

95.

Keystrokes: 3 2 4

22 < x < 23

|3x 1 2| < 4

Y5 ABS x X,T, u 1

x < GRAPH

97.

Keystrokes: 2 3 9

or x > 3x < 26

|2x 1 3| > 9

Y5 ABS x X,T, u 1

x

GRAPH>

99.

Keystrokes: 5 3 5

3 ≤ x ≤ 7

|x 2 5| 1 3 ≤ 5

Y5 ABS x X,T, u

x

GRAPH2 1 ≤


x

2 6 84

80

0

0 ≤ x ≤ 8

24 ≤ x 2 4 ≤ 4

|x 2 4| ≤ 4

103. Matches graph (b).

x

−4 12

12−4

840

x > 12 x < 24

x 2 4 > 8 or x 2 4 < 28

|x 2 4| > 8

12|x 2 4| > 4

105.

|x| ≤ 2

f22, 2g

107.

|x 2 19| < 3

23 < x 2 19 < 3

16 < x < 22

s16, 22d 109. |x| < 3 111. |x 2 5| > 6

113.

t

82

90

62

70 8050 60

62 < t < 82

210 < t 2 72 < 10

|t 2 72| < 10 115. (a)

(b)

8516 ≥ x ≥ 79

16

28516 ≤ 2x ≤ 2

7916

2 316 ≤ 41

8 2 x ≤ 316

2 316 ≤ 51

8 2 x ≤ 316

|518 2 x| ≤ 3

16

|s 2 x| ≤ 316 117. The absolute value of a real num-

ber measures the distance of thereal number from zero.

119. The solutions of areand For example,

to solve

x 5 8 x 5 22

x 2 3 5 5 or x 2 3 5 25

|x 2 3| 5 5:x 5 2a.x 5 a

|x| 5 a 121. The graph of can bedescribed as all real numbers lessthan one unit from 4.

|x 2 4| < 1 123. since

0 ≤ x ≤ 6.

23 ≤ x 2 3 ≤ 3

|x 2 3| ≤ 3

Review Exercises for Chapter 1

1. (a)

Not a solution

24 5 3

45 2 21 5 3

45 2 7s3d 5 3 (b)

Solution

3 5 3

45 2 42 5 3

45 2 7s6d 5 3


3. (a)

Solution

1 5 1

1212

5 1

512

17

125 1

3512

71

3512

55 1 (b)

Not a solution

224

12255 1

210

12251 2

141225

5 1

22

2451 2

2175

5 1

2

235

71

2235

55 1

5.

x 5 3

x 1 2 2 2 5 5 2 2

x 1 2 5 5 Check:

5 5 5

3 1 2 5?

5

7.

x 5 212

23x23

53623

23x 5 36 Check:

36 5 36

23s212d 5?

36

9.

x 5 224

s28d1218

x2 5 s3ds28d

218

x 5 3 Check:

3 5 3

218

s224d 5?

3

11.

x 5 3

5x5

5155

5x 5 15

5x 1 4 2 4 5 19 2 4

5x 1 4 5 19 Check:

19 5 19

15 1 4 5?

19

5s3d 1 4 5?

19

13.

x 5 2

27x27

521427

27x 5 214

17 2 7x 2 17 5 3 2 17

17 2 7x 5 3 Check:

3 5 3

17 2 14 5?

3

17 2 7s2d 5?

3

15.

x 5 4

4x4

5164

4x 5 16

4x 2 5 1 5 5 11 1 5

4x 2 5 5 11

7x 2 3x 2 5 5 3x 2 3x 1 11

7x 2 5 5 3x 1 11 Check:

23 5 23

28 2 5 5?

12 1 11

7s4d 2 5 5?

3s4d 1 11

17.

y 5 4

3y3

5123

3y 5 12

3y 2 3 1 3 5 9 1 3

3y 2 3 5 9

6y 2 3y 2 3 5 9 1 3y 2 3y

6y 2 3 5 9 1 3y

3s2y 2 1d 5 9 1 3y Check:

21 5 21

3s7d 5?

9 1 12

3s2s4d 2 1d 5?

9 1 3s4d


19.

y 5 14

22y22

522822

22y 5 228

22y 1 30 2 30 5 2 2 30

22y 1 30 5 2

4y 2 6y 1 30 5 2

4y 2 6sy 2 5d 5 2 Check:

2 5 2

56 2 54 5?

2

56 2 6s9d 5?

2

4s14d 2 6s14 2 5d 5?

2

21.

No solution

220 5 18

12x 2 12x 2 20 5 12x 2 12x 1 18

12x 2 20 5 12x 1 18

4s3x 2 5d 5 6s2x 1 3d

23.

x 5 2

8x8

5168

8x 5 16

8x 2 1 1 1 5 15 1 1

8x 2 1 5 15

10345

x 21

104 5 332410

45

x 21

105

32

Check:

32

532

1510

5? 3

2

1610

21

105? 3

2

85

21

105? 3

2

45

s2d 21

105? 3

2

25.

t 5 24.2

0.5t0.5

522.10.5

0.5t 5 22.1

0.5t 1 2.1 2 2.1 5 0 2 2.1

0.5t 1 2.1 5 0

1.4t 1 2.1 2 0.9t 5 0.9t 2 0.9t

1.4t 1 2.1 5 0.9t Check:

23.78 5 23.78

25.88 1 2.1 5?

23.78

1.4s24.2d 1 2.1 5?

0.9s24.2d

27. 29.

Labels:

Equation:

a 5 65

a 5 1.30 ? 50

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Parts outPercent of 100 Decimal Fraction

87% 87 0.87 87100

Comparednumber


Basenumber


31.

Labels:

Equation:

3000 5 b

645

0.2155 b

645 5 0.215 ? b

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber


Basenumber 33.

Labels:

Equation:

1.25 5 p or 125%

250200

5 p

250 5 p ? 200

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber


Basenumber

35.16 feet4 yards

516 feet12 feet

51612

543

37.45 seconds5 minutes

545 seconds

300 seconds5

45300

53

20

39.

y 572

y 5288

8y 5 28

78

5y4

41.

b 5103

b 5309

9b 5 30

15b 5 30 1 6b

15b 5 6s5 1 bd

b6

55 1 b

15

43.

Labels:

Equation:

Labels:

Equation:

50% < x

49.9899.95

5 x

49.98 5 x ? 99.95

Cost 5 99.95

Markup rate 5 x

Markup 5 49.98

$49.98 5 x

149.93 2 99.95 5 x

149.93 5 99.95 1 x

Markup 5 x

Cost 5 99.95


Markuprate


Sellingprice


Labels:

Equation:

Labels:

Equation:

54% < x

44.1381.72

5 x

44.13 5 x ? 81.72

Cost 5 81.72

Markup rate 5 x

Markup 5 44.13

$81.72 5 x

125.85 2 44.13 5 x

125.85 5 x 1 44.13

Markup 5 44.13

Cost 5 x


Markuprate


Sellingprice



47.

Labels:

Equation:

Labels:

Equation:

25% < x

17.9971.95

5 x

17.99 5 x ? 71.95

List price 5 71.95

Discount rate 5 x

Discount 5 17.99

x 5 $17.99

x 5 71.95 2 53.96

53.96 5 71.95 2 x

Discount 5 x

List price 5 71.95

Sale price 5 53.96

Listprice

VerbalModel:


Saleprice

VerbalModel:


49.

Labels:

Equation:

Labels:

Equation:

30% 5 x

598.65

1995.505 x

598.65 5 x ? 1995.50


Discount rate 5 x

Discount 5 598.65

x 5 $1396.85

x 5 1995.50 2 598.65

Discount 5 598.65


Sale price 5 x

Listprice

VerbalModel:


Saleprice

VerbalModel:


51.

x 512s7y 2 4d

x 572 y 2 2

2x 5 7y 2 4

2x 2 7y 1 4 5 0 53.

V

pr2 5 h

V 5 pr2h 55.

x

543210

x ≤ 4

x 2 5 1 5 ≤ 21 1 5

x 2 5 ≤ 21

57.

x

20−2−4−6−8

x > 26

25x25

>3025

25x < 30 59.

x

3 420 1

x > 3

5x > 15

5x 1 3 > 18 61.

x

0−2 2 4 6 8

x ≤ 6

22x22

≤21222

22x ≥ 212

22x 1 1 2 1 ≥ 211 2 1

22x 1 1 ≥ 211

8x 2 10x 1 1 ≥ 10x 2 10x 2 11

8x 1 1 ≥ 10x 2 11

63.

70

y

22232425

3

y > 2703

23y < 70

2 2 3y < 72

13

212

y < 12 65.

27 ≤ x < 22

214

2≤

2x2

<242

x

0−2−4−6−8

−7 214 ≤ 2x < 24

26 2 8 ≤ 2x 1 8 2 8 < 4 2 8

26 ≤ 2x 1 8 < 4


67.

x

1

048121602

216 < x < 21

215 < x 1 1 < 0

5 >x 1 123

> 069.

x

0 2−2−4

−323 < x < 2

x < 2 and x > 23

5x5

<105

3x3

>293

5x < 10 3x > 29

5x 2 4 1 4 < 6 1 4 3x 1 1 2 1 > 28 2 1

5x 2 4 < 6 and 3x 1 1 > 28

71.

x ≤ 23

x

0 2−2−4 −1 1−3−5 2x2

≤262

2x ≤ 26

212 1 12 1 2x ≤ 218 1 12

212 1 2x ≤ 218

212 1 8x 2 6x ≤ 6x 2 6x 2 18

212 1 8x ≤ 6x 2 18

24s3 2 2xd ≤ 3s2x 2 6d

73. z ≤ 10 75. 7 ≤ y < 14 77.

or x 5 26x 5 6

|x| 5 6

79.

x 54

23 x 5 4

23x23

54

23

23x23

521223

23x 5 4 23x 5 212

4 2 4 2 3x 5 8 2 4 4 2 4 2 3x 5 28 2 4

4 2 3x 5 8 or 4 2 3x 5 28

|4 2 3x| 5 8 81.

x 5 0 x 5 285

5x5

505

5x5

5285

5x 5 0 5x 5 28

5x 1 4 2 4 5 4 2 4 5x 1 4 2 4 5 24 2 4

5x 1 4 5 4 or 5x 1 4 5 24

|5x 1 4| 5 4

|5x 1 4| 2 10 5 26

83.

x 524 5

12

x 5 3 4x 5 2

2x 5 6 3x 2 4 5 2x 2 2

3x 2 4 5 x 1 2 or 3x 2 4 5 2sx 1 2d|3x 2 4| 5 |x 1 2| 85.

x < 1 or x > 7

x 2 4 < 23 or x 2 4 > 3

|x 2 4| > 3

87.

x < 23 or x > 3

3x < 29 or 3x > 9

|3x| > 9 89.

24 < x < 11

28 < 2x < 22

215 < 2x 2 7 < 15

|2x 2 7| < 15


91.

b < 29 or b > 5

b 1 2 < 27 or b 1 2 > 7

|b 1 2| > 7

|b 1 2| 2 6 > 1 93.

Keystrokes: 2 5 1

or x ≥ 3x ≤ 2

|2x 2 5| ≥ 1

Y5 ABS x X,T, u 2

x

GRAPH≥

95.

|x 2 3| < 2

22 < x 2 3 < 2

1 2 3 < x 2 3 < 5 2 3

1 < x < 5

s1, 5d

97.

Labels:

Equation:

x 5 73, x 1 1 5 74

2x2

51462

2x 5 146

2x 1 1 2 1 5 147 2 1

2x 1 1 5 147

x 1 sx 1 1d 5 147

Sum 5 147

Second integer 5 x 1 1

First integer 5 x

Firstinteger

VerbalModel: 1 5 Sum

Secondinteger

99.

Labels:

Equation:

6% 5 x

9000

150,0005 x

9000 5 x ? 150,000

Sales 5 150,000

Percent rate 5 x

Commission 5 9000

CommissionVerbalModel:

5Percentrate

? Sales

101.

Labels:

Equation:

Labels:

Equation:

—CONTINUED—

x 5 $10.79

x 5 0.30 ? 35.95

List price 5 35.95

Discount rate 5 30%

Discount 5 x

x 5 $31.90

x 5 24.95 1 6.95

Shipping 5 6.95

List price 5 24.95

Total price 5 x

Listprice

VerbalModel:


Totalprice

VerbalModel:

5 1 ShippingListprice


101. —CONTINUED—

Labels:

Equation:

The department store price is the better buy.

x 5 $25.16

x 5 35.95 2 10.79

Discount 5 $10.79

List price 5 $35.95

Sale price 5 x

Saleprice

VerbalModel:


103.

Labels:

Equation:

x 5 $2.47

x 5 0.0725 ? 34

Cost 5 34

Percent rate 5 714%

Sales tax 5 x

Salestax

VerbalModel: 5

Percentrate ? Cost 105.

Proportion:

x 5 334 cups

x 5 112 ? 21

2

11

2

15

x

212

CupsBatches

VerbalModel:

5Cups

Batches

107.

Proportion:

x 5 25 pints

501

5x12

GasolineOil

VerbalModel:

5Gasoline

Oil109.

Proportion:

x 5143

3x 5 14

3

3.55

4x

BaseSide

VerbalModel:

5BaseSide

111.

Proportion:

x 5 80 feet

x 5120

112

x

205

6

112

Silo’s heightSilo’s shadow

VerbalModel:

5Your height

Your shadow113.

Labels:

Equation:

Labels:

Equation:

x 5 $27,166.25

x 5 25,750 1 1416.25

Increase 5 1416.25

Old price 5 25,750

New model 5 x

x 5 $1416.25

x 5 0.055 ? 25,750

Base number 5 25,750

Percent 5 512%

Increase 5 x

IncreaseVerbalModel:

5 1Newmodel

IncreaseVerbalModel:


Oldprice


115.

Labels:

Equation:

Labels:

Equation:

84.21% 5 x

80.0095.00

5 x

80.00 5 x ? 95.00

Cost 5 95.00

Markup rate 5 x

Markup 5 80.00

x 5 $80.00

x 5 175.00 2 95.00

Cost 5 95.00


Markup 5 x

Sellingprice

VerbalModel: 25 CostMarkup

Markuprate


117.

Labels:

Equation:

Labels:

Equation:

Labels:

Equation:

The department store price is the better buy.

x 5 $100.76

x 5 125.95 2 25.19

Discount 5 25.19

List price 5 125.95

Sale price 5 x

x 5 $25.19

x 5 0.20 ? 125.95

List price 5 125.95

Discount rate 5 20%

Discount 5 x

x 5 $104.47

x 5 99.97 1 4.50

Shipping 5 4.50

List price 5 99.97

Total price 5 x

Listprice

VerbalModel:


Totalprice

VerbalModel:

5 1 ShippingListprice

Saleprice

VerbalModel:


119.

Labels:

Equation:

10 2 x 5 623 liters of 60% solution

x 5 313 liters of 30% solution

20.30x 5 21

0.30x 1 6 2 0.60x 5 5

0.30x 1 0.60s10 2 xd 5 0.50s10d

Liters of final solution 5 10


Liters of solution 2 5 10 2 x


Liters of solution 1 5 x


Amount ofsolution 1

VerbalModel:



121.

Labels:

Equation:

d 5 2800 miles

d 5 1200 ? 2 13

Time 5 2 13 hours

Rate 5 1200 mph

Distance 5 d


5 ?Rate Time


123.

Labels:

Equation:

Labels:

Equation:

r < 43.6 mph

r 5 200 4 4.583

Total time 5 4.583 hours

Total distance 5 200 miles

Average speed 5 r

t 5 4.583 or 5512

t 510048

110040

t 5dr

d 5 rt

Time 5 t

Rates 5 48 mph and 40 mph

Distance 5 100 miles


5 ?Rate Time

Averagespeed

VerbalModel:

45Totaldistance

Totaltime

125.

Labels:

Equation:

2.57 hours 5 t

27

10.55 t

27 5 10.5t

27 5 6t 1 4.5t

1 51

4.5std 1

16

std

Time 5 t

Rate of person 2 516

Rate of person 1 51

4.5

Work done 5 1

Workdone

VerbalModel: 15

Work doneby person 1

Work doneby person 2

127.

Labels:

Equation:

i 5 $340

i 5 1000 ? 0.085 ? 4

Time 5 4

Rate 5 8.5%

Principal 5 $1000

Interest 5 i


Labels:

Equation:

$210,526.32 5 p

20,0000.095

5 p

20,000 5 p ? 0.095 ? 1

Time 5 1

Rate 5 9.5%

Principal 5 p

Interest 5 $20,000

InterestVerbalModel: 5 ?Principal Rate ? Time

131.

Labels:

Equation:

$30,000 5 50,000 2 p

$20,000 5 p

2300

20.0155 p

2300 5 20.015p

4700 5 0.085p 1 5000 2 0.10p

4700 5 0.085p 1 0.10s50,000 2 pd

Time 5 1

Rate 2 5 10%

Principal 2 5 50,000 2 p

Rate 1 5 8.5%

Principal 1 5 p

Interest 5 4700


Labels:

Equation:

8 inches 5 x

48 5 x ? 6

Width 5 6

Length 5 x

Area 5 48

AreaVerbalModel:

5 ?Length Width

Chapter Test for Chapter 1 135

135.

Label:

Inequality:

2 ≤ x ≤ 27

4 ≤ 2x ≤ 54

50 ≤ 2x 1 2s23d ≤ 100

Perimeter 5 2x 1 2s23d

137.

0 20 40 60 80 100 120

116.6t

40 < t < 116.6

238.3 < t 2 78.3 < 38.3

|t 2 78.3| < 38.3VerbalModel:

50 ≤ Perimeter ≤ 100

Chapter Test for Chapter 1

1.

x 5 4

6x6

5246

6x 5 24

6x 2 5 1 5 5 19 1 5

6x 2 5 5 19 2.

x 5 3

22x22

52622

22x 5 26

22x 2 6 1 6 5 212 1 6

22x 2 6 5 212

5x 2 7x 2 6 5 7x 2 7x 2 12

5x 2 6 5 7x 2 12

3.

x 5 4

4x4

5164

4x 5 16

8 2 8 1 4x 5 24 2 8

8 1 7x 2 3x 5 3x 1 24 2 3x

8 1 7x 5 3x 1 24

15 2 7 1 7x 5 3x 1 24

15 2 7s1 2 xd 5 3sx 1 8d 4.

x 5 24

4x 2 3x 5 3x 1 24 2 3x

4x 5 3x 1 24

612x3 2 5 1x

21 426

2x3

5x2

1 4

5.

Labels:

Equation:

x 5 864

x 5 0.27 ? 3200

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber


Basenumber

6.

Labels:

Equation:

150% 5 x

1.5 5 x

1200800 5 x

1200 5 x ? 800

a 5 p ? b

Base number 5 b

Percent 5 p

Compared number 5 a

Comparednumber


Basenumber


7.

Labels:

Equation:

$8000 5 x

64000.80

5 x

6400 5 0.80 ? x

List price 5 x

Percent 5 80%

Sale price 5 6400

Saleprice

VerbalModel:

5 ?Percent Listprice

8.

The 15-ounce can is the better buy.

Total priceTotal units

52.9915

5299

15005 0.1993 per ounce

Total priceTotal units

52.4912

5249

12005 0.2075 per ounce

9.

Proportion:

x < $1466.67

x 5s1200ds110,000d

90,000

1200

90,0005

x110,000

TaxAssessed value

VerbalModel:

5Tax

Assessed value 10.

Labels:

Equation:

x 5 212 hours

5 half hours 5 x

80 5 16x

165 5 85 1 6x

Cost of labor 5 6x


Cost of parts 5 85

Total bill 5 165

Totalbill

VerbalModel: 5

Cost ofparts 1

Cost oflabor

11.

Labels:

Equation:

100 2 x 5 6623 liters at 40%

x 5 3313 liters at 10%

20.30x 5 210

0.10x 1 40 2 0.40x 5 30

0.10x 1 0.40s100 2 xd 5 0.30s100d

Percent concentration of final solution 5 30%

Number of liters of final solution 5 100

Percent concentration of solution 2 5 40%

Number of liters of solution 2 5 100 2 x

Percent concentration of solution 1 5 10%

Number of liters of solution 1 5 x

Amount ofsolution 1

VerbalModel:


Amount offinal solution 12.

Labels:

Equation:

23 hour or 40 minutes 5 x

1015 5 x

10 5 15x

40x 1 10 5 55x

Distance of car 2 5 55x

Distance of car 1 5 40x

Time 5 x

Distance of car 1

VerbalModel:

1 10 miles 5Distance of car 2


13.

Labels:

Equation:

$2000 5 p

300 5 p ? 0.075 ? 2

Time 5 2

Rate 5 7.5%

Principal 5 p

Interest 5 300

InterestVerbalModel: 5 ?Principal Rate ? Time 14. (a)

(b)

(c)

x 5 212

x 5 224 x 5 5

24x 5 2 24x 5 220

9 2 4x 5 11 or 9 2 4x 5 211

|9 2 4x| 5 11

|9 2 4x| 2 10 5 1

x 569 5

23

x 5 243 9x 5 6

23x 5 4 9x 2 5 5 1

23x 2 5 5 21 3x 2 5 5 26x 1 1

3x 2 5 5 6x 2 1 or 3x 2 5 5 2s6x 2 1d|3x 2 5| 5 |6x 2 1|

x 5 5 x 5 211

2x 5 10 2x 5 222

2x 1 6 5 16 or 2x 1 6 5 216

|2x 1 6| 5 16

15. (a)

(d)

x

−2 −1 0 1 2

45

21 ≤ x <54

1512

> x ≥ 21

215 < 212x ≤ 12

27 < 8 2 12x ≤ 20

27 < 4s2 2 3xd ≤ 20

x

0−2−4−6−8

x ≥ 26

3x ≥ 218

3x 1 12 2 12 ≥ 26 2 12

3x 1 12 ≥ 26 (b)

x

4310 2

x > 2

3x > 6

1 1 2x > 7 2 x (c)

−8 −6 −4 −2 0 2 4

−7 1x

1 ≥ x > 27

21 ≤ 2x < 7

0 ≤ 1 2 x < 8

0 ≤1 2 x

4< 2


16. (a)

1 ≤ x ≤ 5

22 ≤ x 2 3 ≤ 2

|x 2 3| ≤ 2 (b)

x > 3 x < 295

5x > 15 5x < 29

5x 2 3 > 12 or 5x 2 3 < 212

|5x 2 3| > 12 (c)

2445

< x < 2365

28.8 < x < 27.2

20.8 < x 1 8 < 0.8

20.2 <x4

1 2 < 0.2

|x4 1 2| < 0.2

17. denotes the phrase is at least 8.”“tt ≥ 8

18.

Label:

Equation:

m ≤ 25,000 miles

0.37m ≤ 9250

0.37m 1 2700 ≤ 11,950

Number of miles 5 m

Operatingcost

VerbalModel:

≤ 11,950

Integrated Reviews 3

1. illustrates the Commutative Property ofAddition.5 1 x 5 x 1 5 2. illustrates the Multiplicative Inverse Property.10 ? 1

10 5 1

3. illustrates the DistributiveProperty.6sx 2 2d 5 6x 2 6 ? 2 4. illustrates the Associative

Property of Addition.3 1 s4 1 xd 5 s3 1 4d 1 x

5. 4 2 |23| 5 4 2 3 5 1 6.

5 4

5 210 1 14

210 2 s4 2 18d 5 210 2 s214d

7. 3 2 s5 2 20d

45

3 2 s215d4

53 1 15

45

184

592

8. |3 2 18|3

5 |215|3

5153

5 5

9. 61 2152 5

3 ? 2 ? 25 ? 3

545

10.

52815

57 ? 4 ? 44 ? 3 ? 5

712

45

165

712

?165

11. Money saved 5 $75s20ds12d 5 $18,000 12. Length of each piece 5135 feet

155 9 feet

1. An algebraic expression is a collection of letters (calledvariables) and real numbers (called constants) combined,using the operations of addition, subtraction, multiplica-tion, and division.

2. The terms of an algebraic expression are those parts sepa-rated by addition or subtraction.

3. am ? an 5 am1n 4. sabdm 5 ambm 5. 2360 1 120 5 2240

6. 5s57 2 33d 5 5s24d 5 120 7.

5 214

24

15?

1516

5 24 ? 5 ? 3

5 ? 3 ? 4 ? 48.

565

53 ? 8 ? 2

8 ? 5

38

45

165

38

?165

9. s12 2 15d3 5 s23d3 5 227 10. s58d2

5 s58ds5

8d 52564

CHAPTER 1 Linear Equations and Inequalities

SECTION 1.1 Linear Equations

SECTION 1.2 Linear Equations and Problem Solving

4 Integrated Reviews

12.

5 14x 1 2

5 s4x 1 2x 1 3x 1 x 1 x 1 3xd 1 s1 2 1 1 2d

Perimeter 5 s4x 1 1d 1 s2xd 1 s3x 2 1d 1 x 1 sx 1 2d 1 s3xd

1. The sign of is negative. The rule used is toadd two real numbers with like signs, add their absolutevalues and attach the common sign to the result.

s27d 1 s23d 2. The sign of the sum of is negative. The rule usedis to add two real numbers with unlike signs, subtract thesmaller absolute value from the greater absolute value andattach the sign of the number with the greater absolutevalue.

27 1 3

3. The sign of is positive. The rule used is tomultiply two real numbers with like signs, find theproduct of their absolute values.

s26ds22d 4. The sign of the product is negative. The rule usedis to multiply two real numbers with unlike signs, find theproduct of their absolute values. The product is negative.

6s22d

5.

x 5 14

x 2 5 1 5 5 9 1 5

x 2 5 5 9

2x 2 x 2 5 5 x 2 x 1 9

2x 2 5 5 x 1 9 6.

x 5 0

8x8

508

8x 5 0

8x 1 8 2 8 5 8 2 8

8x 1 8 5 8

6x 1 2x 1 8 5 8 2 2x 1 2x

6x 1 8 5 8 2 2x 7.

x 5 0

2x 5 0

2x 132 2

32 5

32 2

32

2x 132 5

32

8.

x 5 210,000

s210d ? 2x

105 1000s210d

2x

105 1000 9.

x 5 2200

20.35x20.35

570

20.35

20.35x 5 70 10.

x 5 40

0.60x0.60

524

0.60

0.60x 5 24

11.

Labels:

Equation:

0.7 mile 5 x

2.5 2 1.8 5 x

2.5 5 1.8 1 x

Length of last part 5 x

Length of first part 5 1.8

Length of race 5 2.5

Lengthof race

VerbalModel:

15Length offirst part

Length oflast part

11.

5 6x 1 1

5 sx 1 x 1 x 1 3xd 1 s3 2 2d

Perimeter 5 x 1 x 1 sx 1 3d 1 s3x 2 2d

SECTION 1.3 Business and Scientific Problems


12.

Equation:

x 5 774130 5 7811

30 tons

x 5 s34 1 18 1 25d 1 s1030 1

630 1

2530d

x 5 341030 1 18 6

30 1 252530

x 5 3413 1 181

5 1 2556

Total soybeans

VerbalModel: 15

Soybeansin January

Soybeansin February

1Soybeansin March

1. illustrates the Commu-tative Property of Multiplication.3yx 5 3xy 2. illustrates the

Additive Inverse Property.3xy 2 3xy 5 0 3. illustrates

the Distributive Property.6sx 2 2d 5 6x 2 6 ? 2

4. illustrates theAdditive Identity Property.3x 1 0 5 3x 5.

42 2 32 5 16 2 9 5 7

x2 2 y2, x 5 4, y 5 3 6.

4s3d 1 3s24d 5 12 1 212 5 0

4s 1 st, s 5 3, t 5 24

7.

002 1 32 5

09

5 0

xx2 1 y2 , x 5 0, y 5 3 8.

s21d2 1 222 2 1

51 1 24 2 1

533

5 1

z2 1 2x2 2 1

, x 5 2, z 5 21 9.

2

1 212

5212

5 4

a1 2 r

, a 5 2, r 512

10.

2s3d 1 2s1.5d 5 6 1 3 5 9

2l 1 2w, l 5 3, w 5 1.5 11.

A 5 19.8 square meters

A 512s7 1 4d3.6

A 512sb1 1 b2 dh 12.

A 5 104 square feet

A 512s16 1 10d8

A 512sb1 1 b2 dh

1. If is an integer, is an even integer and is anodd integer.

2n 1 12nn 2. and are not equal. By order of operationsand s22xd4 5 16x4.22x4 5 22x4

s22xd422x 4

3.

Divide the numerator and denominator by 7 to put thefraction in simplified form.

3514

57 ? 57 ? 2

552

4.

To divide fractions, multiply by the reciprocal of thedivisor.

45

4z3

545

?3z

5125z

5. because is to the left of 2 on the numberline.

23.223.2 < 2 6. because is to the right of on thenumber line.

24.123.223.2 > 24.1

7. because is to the right of on thenumber line.

252342

34 > 25 8. because is to the right of on

the number line.2

13s2 5

15d215s2 3

15d215 > 2

13

9. because is to the right of on the numberline.

23pp > 23 10. because 6 is to the left of on the numberline.

132 s61

2d6 < 132

SECTION 1.4 Linear Inequalities

SECTION 1.5 Absolute Value Equations and Inequalities


11.

Equation:

which is more than $500.x 5 $656

x 5 163,356 2 162,700

VerbalModel: 5Difference 2

Actualexpense

Budgetedamount

12.

Equation:

which is less than $500.x 5 |2305| 5 $305

x 5 |42,335 2 42,640|


Actualexpense

Budgetedamount

CHAPTER 2 Graphs and Functions

SECTION 2.1 The Rectangular Coordinate System

1. is a linear equation because it can be written in theform Since cannot be written inthe form it is not a linear equation.ax 1 b 5 0,

x2 1 3x 5 2ax 1 b 5 0.3x 5 7 2. To check is a solution of the equation

substitute 3 for x in the equation. If the result is true,is a solution.x 5 3

5x 2 4 5 11x 5 3

3. 6xs2x2d 5 s6 ? 2d ? sx ? x2d 5 12x3 4. 3t2 ? t4 5 3t214 5 3t6

5.

5 54x10

5 s21ds227ds2dsx6dsx4d

2s23x2d3s2x4d 5 s21ds23d3s2dsx2d3sx4d 6.

5 28x4y5

s4x3y2ds22xy3d 5 s4ds22dsx3dsxdsy2dsy3d

7. 4 2 3s2x 1 1d 5 4 2 6x 2 3 5 1 2 6x 8.

5 23x 1 22

5sx 1 2d 2 4s2x 2 3d 5 5x 1 10 2 8x 1 12

9. 241y3

1y62 5 8y 1 4y 5 12y 10.

5 0.02x 1 100

0.12x 1 0.05s2000 2 2xd 5 0.12x 1 100 2 0.1x

11. Your rate job per hour

Friend’s rate job per hour

Verbal model:

Labels: Work done

Your rate

Friend’s rate

Time

—CONTINUED—

5 t

515

514

5 1

Work doneby friend1

Work doneby you5

Workdone

515

514

C H A P T E R 2Graphs and Functions

Section 2.1 The Rectangular Coordinate System . . . . . . . . . . . .140

Section 2.2 Graphs of Equations . . . . . . . . . . . . . . . . . . . .146

Section 2.3 Slope and Graphs of Linear Equations . . . . . . . . . . .153

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159

Section 2.4 Equations of Lines . . . . . . . . . . . . . . . . . . . . .162

Section 2.5 Relations and Functions . . . . . . . . . . . . . . . . . .166

Section 2.6 Graphs of Functions . . . . . . . . . . . . . . . . . . . .171


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184

140

C H A P T E R 2Graphs and Functions

Section 2.1 The Rectangular Coordinate SystemSolutions to Odd-Numbered Exercises

1.

is 4 units to the right of the vertical axis and 3 unitsabove the horizontal axis.

is 5 units to the left of the vertical axis and 3 unitsabove the horizontal axis.

is 3 units to the right of the vertical axis and 5units below the horizontal axis.s3, 25d

s25, 3d

s4, 3d

)

6x

5)

y

,4(

66

43),5(

2 42

2

,(36

4

6 4

2

3

3.

is 8 units to the left of the vertical axis and 2units below the horizontal axis.

is 6 units to the right of the vertical axis and 2units below the horizontal axis.

is 6 units to the right of the vertical axis and 5 unitsabove the horizontal axis.s6, 5d

s6, 22d

s28, 22d

5)

6x

, 2)

y

,(6

8

4

66

2

2

2

4

2

6

(6

8 6

,( 28 )

4 4

5.

is units to the right of the vertical axis and 2units below the horizontal axis.

is 2 units to the left of the vertical axis and unitsabove the horizontal axis.

is units to the right of the vertical axis and units below the horizontal axis.

72

32s3

2, 272d

14s22, 14d

52s5

2, 22d

x3

2

y

2

14,21

2

5

1

,2

73

11

2

2,2

4

3

23

7.

is units to the right of the vertical axis and 1 unitabove the horizontal axis.

is 4 units to the right of the vertical axis and 3units below the horizontal axis.

is units to the left of the vertical axis and unitsabove the horizontal axis.

73

43s24

3, 73d

s4, 23d

32s3

2, 1d

x

y

(4, 3)−

( , 1)

( , )−

3

4 7

2

3 3

−2 −1 1 2 3 4

−3

−2

−1

1

3

2

9. Point Position Coordinates

A 2 left, 4 up

B 0 right or left, 2 down

C 4 right, 2 down s4, 22d

s0, 22d

s22, 4d

11. Point Position Coordinates

A 4 right, 2 down

B 3 left, down

C 3 right, up s3, 12d12

s23, 252d5

2

s4, 22d

Section 2.1 The Rectangular Coordinate System141

17.

x

, )2

6

0)4( ,

y

2

(

0( , 4)

)2,2

4

(

6

4

6

19.

x

y

(5, 5)

(2, 3)

(3, 2)

(0, 0) 2 3 4 5−1−1

1

2

3

4

5

21. Point 5 units left of y-axis and 2 units abovex-axis 5 s25, 2d

23. Point 3 units right of y-axis and 2 units belowx-axis 5 s3, 22d

25. The coordinates of the point are equal and located inQuadrant III, 10 units left of y-axis 5 s210, 210d.

27. Point on positive x-axis 10 units from theorigin 5 s10, 0d.

29. is in Quadrant III.s23, 25d 31. is in Quadrant IV.s3, 258d

33. is in Quadrant I.s200, 1365.6d 35. is in Quadrant IV.sx, yd, x > 0, y < 0

37. is in Quadrants I or II.sx, 4d 39. is in Quadrants II or III.s23, yd

41. is in Quadrants I or III.sx, yd, xy > 0 43.

8x

64

y

100

90

2

70

80

45. The relationshipbetween x and y is asx increases from 1 to7, y also increases, butas x increases from 7to 12, y decreases.

14x

y

50

70

60

8 1064

40

20

30

2

10

12

47. shifted 2 units right and 5 units up

shifted 2 units right and 5 units up

shifted 2 units right and 5 units up5 s3, 2ds1, 23d

5 s21, 1ds23, 24d

5 s0, 4ds22, 21d

13.

3,3 )

x4

y

6),(

4

0

),3 3((

)0,(0

2

2

24 2

15.)

4x

y

3,(

4

5

0,(2

321

),1 2(

121

1

3

)

5

142 Chapter 2 Graphs and Functions

49.

–3 –2 –1 4 5 6 7

5

10

15

20

25

30

x

y

(−2, −11)(0, −1)

(2, 9)

(4, 19)

(6, 29)

x 0 2 4 6

9 19 2921211y 5 5x 2 1

22

51.

x

y

( , 3)25

−6 −3 3 12

14

7

−7

−14

−21

−28

(−4, 14)

(4, −6)

(8, −16)

(12,−26)

x 4 8 12

14 3 22621626y 5 252x 1 4

2524

5 14

5 10 1 4

y 5252

s24d 1 4

5 226

5 230 1 4

y 5252

s12d 1 4

5 216

5 220 1 4

y 5252

s8d 1 4

5 26

5 210 1 4

y 5252

s4d 1 4

5 3

5 21 1 4

y 5252 12

52 1 4

53.

Keystrokes: 4 2

x 0 2 4 6

12 16 66 14822y 5 4x2 1 x 2 2

22

55.

(a)

Not a solution

3 Þ 25

9 2 6 5?

25

32 1 3s22d 5?

25

s3, 22d

x2 1 3y 5 25

(c)

Not a solution

26 Þ 25

9 2 15 5?

25

32 1 3s25d 5?

25

s3, 25d

(b)

Solution

25 5 25

4 2 9 5?

25

s22d2 1 3s23d 5?

25

s22, 23d

(d)

Solution

25 5 25

16 2 21 5?

25

42 1 3s27d 5?

25

s4, 27d

Y5 X,T,u X,T,ux2 2 GRAPH1


57.

(a)

Not a solution

1 Þ 0

4s0d 2 2s0d 1 1 5?

0

s0, 0d

4y 2 2x 1 1 5 0

(b)

Solution

0 5 0

0 2 1 1 1 5?

0

4s0d 2 2s12d 1 1 5

?0

s12, 0d

(c)

Solution

0 5 0

27 1 6 1 1 5?

0

4s274d 2 2s23d 1 1 5

?0

s23, 274d (d)

Not a solution

24 Þ 0

23 2 2 1 1 5?

0

4s234d 2 2s1d 1 1 5

?0

s1, 234d

59.

(a)

Solution

4 5 4

4 5?

1 1 3

4 5? 7

8s87d 1 3

s87, 4d

y 578x 1 3

(b)

Solution

10 5 10

10 5?

7 1 3

10 5? 7

8s8d 1 3

s8, 10d

(c)

Not a solution

0 Þ 3

0 5?

0 1 3

0 5? 7

8s0d 1 3

s0, 0d (d)

Not a solution

14 Þ 211

14 5?

214 1 3

14 5? 7

8s216d 1 3

s216, 14d

61.

Vertical line

6x

y

)3( , 566

4

4

2

2

2),(4

32

24

5 7

5 |7| d 5 |5 2 s22d| 63.

Horizontal line

x

y

3

3

2

, 2)( 01

8

2

1

4

( ),

12

5 7

5 |7| d 5 |10 2 3|


65.

Vertical line

x

y

9

34

2

−3 −2 −1

1

2

3

−3,( )−3,( )

5 34

5 |64 2

94|

d 5 |32 2

94| 67.

Horizontal line

x

y

−4( , )13 ( , )5 1

2 3

2 31

2

1

4

3

6

5

−2 −1−1

−4 −3

5 132

5 |132 |

5 |52 1

82|

d 5 |52 2 s24d|

69.

5 !s21d2 1 s2d2 5 !1 1 4 5 !5

d 5 !s3 2 4d2 1 s7 2 5d2 71.

5 !16 1 9 5 !25 5 5

d 5 !s1 2 5d2 1 s3 2 6d2

73.

5 !s27d2 1 s3d2 5 !49 1 9 5 !58

d 5 !s23 2 4d2 1 f0 2 s23dg2 75.

5 !36 1 25 5 !61

d 5 !s22 2 4d2 1 s23 2 2d2

77. d 5 !s1 2 3d2 1 f3 2 s22dg2 5 !4 1 25 5 !29

79.

By the Pythagorean Theorem, it is a right triangle.

26 5 26

13 1 13 5?

26

s!13d21 s!13d2

5? s!26d2

d3 5 !s3 2 4d2 1 s6 2 1d2 5 !1 1 25 5 !26

d2 5 !s1 2 4d2 1 s3 2 1d2 5 !9 1 4 5 !13

d1 5 !s1 2 3d2 1 s3 2 6d2 5 !4 1 9 5 !13

(1, 3)(4, 1)

(3, 6)

x

y

d1

d2

d3

1 2

1

−1

2

3

4

5

6

7

3 4 5 6 7

81.

By the Pythagorean Theorem, it is a right triangle.

40 5 40

20 1 20 5?

40

s!20d21 s!20d2

5? s!40d2

d3 5 !s1 2 3d2 1 s23 2 3d2 5 !4 1 36 5 !40

d2 5 !s21 2 1d2 1 f1 2 s23dg2 5 !4 1 16 5 !20

d1 5 !s21 2 3d2 1 s1 2 3d2 5 !16 1 4 5 !20

(9, 4)

(1, 3)−

( 1, 1)−

x

y

d1

d2 d3−2

−2

2

3

31

4

4−3

−3

−4

−4


83. Not collinear

d 5 !s2 2 6d2 1 s6 2 3d2 5 !16 1 9 5 5

d 5 !s2 2 6d2 1 s3 2 3d2 5 !16 1 0 5 4

3 1 4 Þ 5 d 5 !s2 2 2d2 1 s3 2 6d2 5 !0 1 9 5 3

85. Collinear

d 5 !s5 2 2d2 1 s2 2 1d2 5 !9 1 1 5 !10

d 5 !s8 2 2d2 1 s3 2 1d2 5 !36 1 4 5 !40 5 2!10

!10 1 !10 5 2!10 d 5 !s8 2 5d2 1 s3 2 2d2 5 !9 1 1 5 !10

87.

8x

y

8,(48

10

)4,(1

642

),2 0(

242

4

66

)

M 5 122 1 42

, 0 1 8

2 2 5 s1, 4d 89.

)3

7x

2,27 9

6)

y

,1(7

5

66

6(

53 4

4

3

2

1

1

2 6

,

M 5 11 1 62

, 6 1 3

2 2 5 172

, 922

91.

For each additional 50 units produced, costs increase by $1400.

x 100 150 200 250 300

5800 7200 8600 10,000 11,400c 5 28x 1 3000

5 5800

5 2800 1 3000

y 5 28s100d 1 3000

5 7200

5 4200 1 3000

y 5 28s150d 1 3000

5 8600

5 5600 1 3000

y 5 28s200d 1 3000

5 10,000

5 7000 1 3000

y 5 28s250d 1 3000

5 11,400

5 8400 1 3000

y 5 28s300d 1 3000

93.

Rafter feet5 2 1 x < 18.55294536 < 18.55

x 5 !274 < 16.55294536

x2 5 49 1 225

x2 5 72 1 152 95.

P 5 !29 1 !26 1 3 < 13.48

d 5 !s22 2 1d2 1 s0 2 0d2 5 !9 5 3

d 5 !s0 2 1d2 1 s5 2 0d2 5 !1 1 25 5 !26

d 5 !s22 2 0d2 1 s0 2 5d2 5 !4 1 25 5 !29

97. The word orderedis significant because each number in the pair has a particular interpretation. The first measures horizontaldistance and the second measures vertical distance.

99. The x-coordinate of any point on the y-axis is 0.

The y-coordinate of any point on the x-axis is 0.


101. No. The scales on the x and y-axes are determined by the magnitudes of the quantities being measured by x and y.

103.

When the sign of the x-coordinate is changed, the point is on the opposite side of the y-axis as the original point.

x

y

(−7, −3) (7,−3)

(2, 1)(−2, 1)

(−3, 5) (3, 5)

−8 −6

−6

−4 −2 42

2

86

6

8

4

−4

−8

Section 2.2 Graphs of Equations

1. matches graph (e).y 5 2 3. matched graph (f)y 5 2 2 x 5. matches graph (d).y 5 x2 2 4

7.

x

y

9

6

3

−3−6−9

−9

3 6 9

9.

5x

y

4

4321

2

1

11

3

x 0 1 2

0 3 6

Solution s2, 6ds1, 3ds0, 0ds21, 23ds22, 26d

2326y 5 3x

2122

x 0 1 2

6 5 4 3 2


y 5 4 2 x

2122

11.

x

y

1 2 3−2−3

1

2

−2

−1

3

−3

−1

x 0 1 2

1


21232527y 5 2x 2 3

2122

Section 2.2 Graphs of Equations 147

15.

–3 –2 2 3

–4

–3

–2

–1

1

2

x

y

x 0 1 2

0


24212124y 5 2x2

2122

17.

3x

y

1

1

13

2

3

5

1

x 0 1 2

0 0


232423y 5 x2 2 4

2122

19.

x

y

1 2−2−4

1

2

3

−2

−3

−1

x 0 1 2

0 4 10


2222y 5 x2 1 3x

2122

21.

x

y

1 32 4−2

−2

1

2

4

3

−1−1

x 0 1 2

7 2


212221y 5 x2 2 2x 2 1

2122

13.

x

y

2 3−2−3

1

−2

−1

3

−3

−1

x 0 1 2

4 1


22212

52y 5 2

32x 1 1

2122


23.

3x

y

5

4

2

1

21111

23

3

25.

x

y

21 3−2 −1−3

2

3

4

5

1

−1

x 0 1 2

5 4 3 4 5


y 5 |x| 1 3

2122

x 0 1 2

2 1 0 1 2


y 5 |x|2122

27.

–7 –6 –5 –4 –3 –2 –1 1

–2

1

3

4

5

6

x

yx 0 1 2

1 2 3 4 5


y 5 |x 1 3|2122

29.

x

y

21 3−2 −1−3

2

3

1

−1

−2

−3

x 0 1 2

8 1 0


2821y 5 2x3

2122

31.

y-intercept:

x-intercept:

s12, 0d 12 5 x

36 5 x

3 5 6x

0 5 6x 2 3

s0, 23d y 5 23

y 5 6s0d 2 3

y 5 6x 2 3 33.

y-intercept:

x-intercept:

s220, 0d 220 5 x

215 534x

0 534x 1 15

s0, 15d y 5 15

y 534s0d 1 15

y 534x 1 15


39.

y-intercept:

x-intercept:

s1, 0d, s21, 0d ±1 5 x

1 5 |x| 0 5 |x| 2 1

s0, 21d y 5 21

y 5 |0| 2 1

y 5 |x| 2 1 41.

y-intercept:

x-intercept:

s22, 0d 22 5 x

0 5 x 1 2

0 5 |x 1 2|s0, 2d y 5 2

y 5 |0 1 2|y 5 |x 1 2|

43.

y-intercept:

x-intercept:

s4, 0d, s22, 0d 4 5 x 22 5 x

3 5 x 2 1 or 23 5 x 2 1

3 5 |x 2 1| 0 5 |x 2 1| 2 3

s0, 22d y 5 22

y 5 1 2 3

y 5 |0 2 1| 2 3

y 5 |x 2 1| 2 3

45.

Estimate: y-intercept

x-intercept< 3

< 2

2x 1 3y 5 6

Check:

s3, 0d x 5 3

2x 5 6

2x 1 3s0d 5 6

s0, 2d y 5 2

3y 5 6

2s0d 1 3y 5 6

47.


no x-intercepts

< 3

y 5 x2 1 3

Check:

no real solution 23 5 x2

0 5 x2 1 3

s0, 3d y 5 02 1 3 5 3

35.

y-intercept:

x-intercept:

s10, 0d x 5 10

x 1 2s0d 5 10

s0, 5d y 5 5

0 1 2y 5 0

x 1 2y 5 10 37.

y-intercept:

x-intercept:

s234, 0d x 5 2

34

4x 5 23

4x 2 0 1 3 5 0

s0, 3d 3 5 y

4s0d 2 y 1 3 5 0

4x 2 y 1 3 5 0


51.

Keystrokes: 1 6


x-intercepts< 1, 6

−6

−8

14

12< 6

y 5 sx 2 1dsx 2 6d

Check:

s0, 6d y 5 6

y 5 s0 2 1ds0 2 6d Check:

s1, 0d, s6, 0d x 5 1 x 5 6

0 5 sx 2 1dsx 2 6d

53.

Keystrokes: 4 6 2


x-intercepts< 21, 22

−12

−4

8

16< 4

y 5 |4x 1 6| 2 2

Check:

s0, 4d y 5 4

y 5 6 2 2

y 5 |4s0d 1 6| 2 2 Check:

or

s21, 0d, s22, 0d

21 5 x

24 5 4x

2 5 4x 1 6

2 5 |4x 1 6| 0 5 |4x 1 6| 2 2

x 5 22

4x 5 28

4x 1 6 5 22

55.

s1, 2d y 5 2

y 5 3 2 1

s3, 0d x 5 3

0 5 3 2 x

s0, 3d y 5 3

y 5 3 2 0

)0

4x

)30,

y

4

3 (

)

,3(

,(1 2

31 2

2

1

1

1

57.

s3, 3d y 5 3

y 5 2s3d 2 3

s32, 0d 32 5 x

3 5 2x

0 5 2x 2 3

s0, 23d y 5 23

y 5 2s0d 2 3

4x

y

),3( 3

2

3

2 3

0,2

),0( 3

12

3

2

1

1 3

Y5 x xX,T,u X,T,u2 2d d GRAPH

Y5 x X,T,u 2d GRAPHABS 1

49.

Keystrokes: 4 6


x-intercept< 32

< 26−3

−7

7

3y 5 4x 2 6

Check:

s0, 26d y 5 26

y 5 4s0d 2 6 Check:

s32, 0d 32 5 x

64 5 x

6 5 4x

0 5 4x 2 6

Y5 X,T,u 2 GRAPH


59.

s1, 21d y 5 21

4s1d 1 y 5 3

s34, 0d x 5

34

4x 5 3

4x 1 0 5 3

s0, 3d y 5 3

4s0d 1 y 5 3

4x 1 y 5 3

x

)1

2

3, )(3

2

0

y

, 043

1

,1(1

12

61.

s1, 243d y 5 2

43

23y 5 4

2s1d 2 3y 5 6

s3, 0d x 5 3

2x 5 6

2x 2 3s0d 5 6

s0, 22d y 5 22

23y 5 6

2s0d 2 3y 5 6

2x 2 3y 5 6

x4

)3, 0(

y

1

32

, 312)

41

,0(

3

1

63.

s5, 1d y 5 1

5y 5 5

5 1 5y 5 10

s10, 0d x 5 10

x 1 5s0d 5 10

s0, 2d y 5 2

0 1 5y 5 10

x 1 5y 5 10

),0 0

10x

y

66

4

(, 1)

86

(

4

5,

2

4

2

0 2)(

1

65.

)0x

6

y

,(3

3

0),3(

9),(9 0

6

s0, 29d y 5 29

y 5 02 2 9

s3, 0ds23, 0d 3 5 x x 5 23

0 5 sx 2 3dsx 1 3d

0 5 x2 2 9

y 5 x2 2 9

67.

s21, 0d y 5 0

y 5 1 2 1

y 5 1 2 s21d2

s1, 0d y 5 0

y 5 1 2 1

y 5 1 2 s1d2

s0, 1d y 5 1

y 5 1 2 0

y 5 1 2 x2 69.

0)x

3

y

3

2

,(2),(0 0

), 1(1

1

1

1

s0, 0d, s2, 0d 0 5 x x 5 2

0 5 xsx 2 2d

0 5 x2 2 2x

s0, 0d y 5 0

y 5 02 2 2s0d

y 5 xsx 2 2d


71.

, 0)x

3

y

(

2

10),3(

2111

2

,

4

30 )(3

23

3

s3, 0d, s23, 0d 3 5 x x 5 23

3 5 |x| 0 5 |x| 2 3

s0, 23d y 5 23

y 5 |x| 2 3 73.

–5 –4 1

3

4

5

x

y

(−2, 0)

(0, 2)

(1, 3)

s24, 2d 5 2

y 5 |24 1 2|s22, 0d 5 0

y 5 |22 1 2|s0, 2d 5 2

y 5 |x 1 2|

75.

or

x

y

4321

1

2

3

4

−2

−2

−3

−3

−4

−4

(0, 1)), 0

(3, 1)12(−

0 Þ 1

x 5 x 1 1

|x| 5 |x 1 1| 0 5 2|x| 1 |x 1 1|

s0, 1d y 5 1

y 5 2|0| 1 |0 1 1| y 5 2|x| 1 |x 1 1| 77.

87t

225,000

175,000200,000

y

5 64321

100,00075,000

125,000

1

25,00050,000

150,000

s8, 65,000d 5 65,000

5 225,000 2 160,000

y 5 225,000 2 20,000s8d

s0, 225,000d 5 225,000

y 5 225,000 2 20,000s0d

y 5 225,000 2 20,000t

s3, 1d y 5 1

y 5 23 1 4

y 5 2|3| 1 |3 1 1|

s212, 0d x 5 2

12

22x 5 1

2x 5 x 1 1

79.

y 5 25000x 1 40,000

m 540,000 2 5,000

0 2 75

35,00027

5 25000

s0, 40,000d, s7, 5000d

t

y

1 2 3 4 5 6 7

40,000

20,000

30,000

10,000

0 ≤ x ≤ 7

Section 2.3 Slope and Graphs of Linear Equations153

81. (a) (b)

(c) F doubles.3 6 9 12 15

3

6

9

12

15

x

Fx 0 3 6 9 12

0 4 8 12 1643x

83. The scales on the y-axes are different. From graph (a) it appears that sales have not increased. From graph (b) it appears thatsales have increased dramatically.

85. The graph of an equation is the set of all solutions of the equation plotted on a rectangular coordinate system.

87. To find the x-intercepts, let and solve the equation for x. To find the y-intercepts, let and solve the equation for y.

Example:

x 5 0y 5 0

x-intercept

s2, 0d x 5 2

2x 5 4

2x 2 0 5 4

2x 2 y 5 4

y-intercept

s0, 24d y 5 24

2y 5 4

2s0d 2 y 5 4

89. (a) It is 6 miles from the person’s home.

(b) For time the person is stopped since the graph is a constant line.

(c) The person’s speed was greatest during because the graph is steepest there.6 ≤ t ≤ 10

4 < t < 6

Section 2.3 Slope and Graphs of Linear Equations

1.

m 56 2 26 2 0

546

523

s0, 2d and s6, 6d 3.

m 50 2 84 2 0

5284

5 22

s0, 8d and s4, 0d

5.

undefinedm 58 2 03 2 3

580

5

s3, 0d and s3, 8d 7. (a) (b)

(c) m 5 23 ⇒ L1

m 5 0 ⇒ L2m 534

⇒ L3


9. Line rises.

x8

5,(7

y

8

66

642

4

, 0)2

(0

2

)

m 55 2 07 2 0

557

11. Line falls.

1 2 3 4 5

–5

–4

–3

–2

–1

x

y

(0, 0)

(5, −4)

m 524 2 05 2 0

5245

13. Line rises.

y

x1−2−3−5 −4

1

2

3

5

4

−1−1

(−2, 5)

(−4, 3)

m 53 2 5

24 2 s22d 52222

5 1 15. undefined

Line is vertical.

2x

y

4

2

4, )( 5

2

2

4

)3,

46

( 5

m 54 2 s23d

25 2 s25d 570

5

17. Line is horizontal.

8x

)5

62 4

y

2

2

(7,5),2

2

4

8

6

(

m 525 2 s25d

7 2 25

05

5 019. Line falls.

5x

3,4 2

y

2

421

52,5

11

3

2

1

m 5

2 2252

34

2 5?

44

58 1 103 2 20

518

217


21. Line rises.

x

y

−2 −1 1 2

1

2

−1

−2

18,3

2( (−14,3

4( (

m 5

14

218

34

2232

?88

52 2 1

6 1 125

118

23. Line rises.

25)5.5,4.( 7

x

56

2, )8

y

66

2

4

2.2

(2

2

4

4

m 55.25 2 s22d4.75 2 2.5

57.252.25

5725225

5299

25.

m 51 2 s21d

1 2 05

21

5 2

x

y

−2 −1 1 2

1

2

−1 (0, 1)−

(1, 1)

27.

5 212

m 5

72

2 4

1 2 05

72

282

x

y

−4

−2

2

6

2−2 4 6 8

(8, 0)

(0, 4)

x 0 1

1

Solution s1, 1ds0, 21ds21, 23d

2123y 5 2x 2 1

21 x 0 1

4

Solution s1, 72ds0, 4ds21, 92d

72

92y 5 2

12x 1 4

21

29.

x

y

−1

−2

3

4

−1−2 21 4

(0, 2)

1

4, − 65( (

5 245

m 5

65

2 2

1 2 05

65

2105

y 5 245

x 1 2

5y 5 24x 1 10

4x 1 5y 5 10

x 0 1

2

Solution s1, 65ds0, 2ds21, 145 d

65

145y 5 2

45x 1 10

21


31.

x 5 1

22x 5 22

22x 1 8 5 6

22sx 2 4d 5 6

223

57 2 5x 2 4

33.

215 5 y

30 5 22y

36 5 6 2 2y

3s12d 5 2s3 2 yd

32

53 2 y

9 2 s23d 35.

Horizontal line:

Any points with a y-coordinate of 2

s1, 2d, s0, 2d, s3, 2d

0 5y 2 2x 2 5

37.

let solve for y:

let solve for y:

2 5 y

6 5 y 1 4

3 5y 1 45 2 3

x 5 5,

21 5 y

3 5 y 1 4

3 5y 1 44 2 3

x 5 4,

s4, 21d, s5, 2d

3 5y 1 4x 2 3

39.

let solve for y:

let solve for y:

1 5 y

22 5 y 2 3

21 5y 2 32 2 0

x 5 2,

2 5 y

21 5 y 2 3

21 5y 2 31 2 0

x 5 1,

s1, 2d, s2, 1d

21 5y 2 3x 2 0

41.

let solve for y:

let solve for y:

8 5 y

43

5y

1 1 5

x 5 1,

4 5 y

43

5y

22 1 5

x 5 22,

s22, 4d, s1, 8d

43

5y 2 0x 1 5

43.

y 5 2x 2 3

23y23

526x23

19

23

23y 5 26x 1 9

6x 2 3y 5 9 45.

y 514

x 2 1

4y4

5x4

244

4y 5 x 2 4

4y 2 x 5 24 47.

y 5225

x 135

5y5

522x

51

35

5y 5 22x 1 3

2x 1 5y 2 3 5 0

49. y 512x 1 2 51.

m 5 3; s0, 22d

y 5 3x 2 2 53.

m 523; s0, 1d

y 523x 1 1

3y 5 2x 1 3

3y 2 2x 5 3

55.

m 5253

; 10, 232

y 5253

x 123

3y 5 25x 1 2

5x 1 3y 2 2 5 0 57.

slope

y-intercept5 22

5 3

y 5 3x 2 2

2y 5 23x 1 2

y

3

x32

2)

1

1

,0(

1

2

23 1

2

3x 2 y 2 2 5 0


59.

slope

y-intercept5 0

5 21

y 5 2x 1 0

y

2

x2

0),0

1

(

1

1

2

12

x 1 y 5 0 61.

slope

y-intercept5 1

5232

y 5232

x 1 1

2y 5 23x 1 2

y

x

)

2

, 1(01

1

2

2 1

2

3x 1 2y 2 2 5 0

63.

slope

y-intercept512

514

y 514

x 112

24y 5 2x 2 2

x

y

,01

1

2

2

2

1

12 1

2

x 2 4y 1 2 5 0 65.

Locate a second point with the slope of 3.

m 531

5Change in yChange in x

1 3 4

1

2

3

4

x

y

(3, 2)

67.

Locate a second point with the slope of

m 5213

5Change in yChange in x

213

.

1 2 3 4

1

2

4

x

y

(3, 2)

69.

m is undefined so the line is vertical.

y

1

(3, 2)

−1

−2

x1−1 2 4

2

3

71.

s22, 0d x 5 22

2x 5 24

2x 2 0 1 4 5 0

s0, 4d 4 5 y

2s0d 2 y 1 4 5 0

x4

y

4),

66

(4 0

2

2

2

, 0)( 2

4

2x 2 y 1 4 5 0


73.

s24, 0d x 5 24

25x 5 20

25x 1 2s0d 2 20 5 0

s0, 10d y 5 10

2y 5 20

25s0d 1 2y 2 20 5 0

25x 1 2y 2 20 5 075.

so the lines are parallel.L1 5 m2

m1 512

and m2 512

L2: y 512

x 1 3

L1: y 512

x 2 2

83.

(a)

(b)

(c) On the average, tuition and fees increased $192.64 each year from 1990 to 1996. The increase is the slope of the graph.

(d) for 2005, so

5 $4905.39

y 5 192.64s15d 1 2015.79t 5 15

2000

2500

y

3000

2t

4 61 3 5

y 5 192.64t 1 2015.79

t 0 1 2 3 4 5 6

y $2015.79 $2208.43 $2401.07 $2593.71 $2786.35 $2978.99 $3171.63

)10

4x

y

12

,(010

66

2

4

4 0

22

6

),

810

(

2

77.

so the linesare perpendicular.

m1 ? m2 5 21

m1 534

and m2 5243

L2: y 5243

x 1 1

L1: y 534

x 2 3 79.

The change in horizontalposition is 16,667 feet.

x < 16,667

212x 5 2200,000

212

1005

22000x

81.

The maximum height in

the attic is 454

feet 5 11.25 feet.

454

5 h

45 5 4h

34

5h

15

85. (a)

5,000

15,000

y

20,000

(0, 15,900)

(3, 10,200)

2x

1 3

10,000

(b)

(c) The slope is the annual depreciation.

m 515,900 2 10,200

0 2 35 21900


87. Negative slope: line falls to the right.

Zero slope: line is horizontal

Positive slope: line rises to the right

89. In the form m represents the slope and b represents the y-intercept.y 5 mx 1 b,

91. No, it is not possible for two lines with positive slopes to be perpendicular to each other. Their slopes must be negativereciprocals of each other.


1. Quadrants I or II. Since x can be any real number and y is4, the point can only be located in quadrants inwhich the y coordinate is positive.

sx, 4d2. s10, 23d

3.

(a)

not a solution

(c)

solution 10 5 10

10 2 0 5?

10

4s2.5d 2 3s0d 5?

10s2.5, 0d

5 Þ 10

8 2 3 5?

10

4s2d 2 3s1d 5?

10s2, 1d

4x 2 3y 5 10

(b)

solution

(d)

solution 10 5 10

8 1 2 5?

10

4s2d 2 3s223d 5

?10s2, 22

3d 10 5 10

4 1 6 5?

10

4s1d 2 3s22d 5?

10s1, 22d

4.

5 5

5 !25

5 !16 1 9

d 5 !s21 2 3d2 1 s5 2 2d2

x

y

−3 −2 1 2 3−1

4

1

2

3

( 1, 5)−

(3, 2)

5.

5 13

5 !169

5 !25 1 144

d 5 !s23 2 2d2 1 s22 2 10d2

x

y

−6 −4 2 4 6

8

10

4

6

( 3, 2)− −

(2, 10)

6.

x-intercept:

y-intercept:

s0, 6d y 5 6

28y 5 248

6s0d 2 8y 1 48 5 0

s28, 0d x 5 28

6x 5 248

6x 2 8s0d 1 48 5 0

6x 2 8y 1 48 5 0 7.

s2, 1d 5 1

y 5 2s2d 2 3

s32, 0d 32 5 x

3 5 2x

0 5 2x 2 3

s0, 23d 5 23

y 5 2s0d 2 3

y 5 2x 2 3

x

y

( , 0)32

−1−2−3

−3

−2

−1

1

2

(2, 1)

(0, −3)


8.

s1, 3d y 5 3

3s1d 1 y 2 6 5 0

s2, 0d x 5 2

3x 5 6

3x 1 0 2 6 5 0

s0, 26d y 5 6

3s0d 1 y 2 6 5 0

–4 –2 4 6 8–2

2

4

6

x

y

(0, 6)

(1, 3)

(2, 0)

3x 1 y 2 6 5 0

9.

s3, 9d 5 9

5 18 2 9

y 5 6s3d 2 32

s6, 0d 5 0

y 5 6s6d 2 62

s0, 0d 5 0

y 5 6s0d 2 02

–6 –3 3 9 12–3

3

6

9

x

y

(0, 0) (6, 0)

(3, 9)

y 5 6x 2 x2

10.

s22, 0d 5 0

y 5 s22d2 2 4

s2, 0d 5 0

y 5 22 2 4

s0, 24d 5 24

y 5 02 2 4

–1 1

–5

–3

–2

–1

1

x

y

(−2, 0) (2, 0)

(0, −4)

y 5 x2 2 4

11.

s21, 2d 5 2

y 5 |21| 1 1

s1, 2d 5 2

y 5 |1| 1 1

s0, 1d 5 1

y 5 |0| 1 1

y

x−2 −1 1 2

3

−1

( 2, 3)−

(0, 1)

2

(2, 3)

y 5 |x| 1 1

12.

s2, 23d 5 23

y 5 |2 2 2| 2 3

s5, 0d 5 0

y 5 |5 2 2| 2 3

s0, 21d 5 21

y 5 |0 2 2| 2 3

x

y

1 52 63

1

−2

−3

−4

3

4

( 1, 0)−(5, 0)

(0, 1)−

(2, 3)−

y 5 |x 2 2| 2 3


13. undefined

Line is vertical.

m 522 2 35 2 5

5250

5 14. Line is horizontal.m 58 2 8

7 2 s23d 50

105 0

15. Line rises.m 55 2 06 2 3

553

16. Line falls.m 526 2 4

5 2 s21d 5210

65

253

17.

x

y

−2

−1

3

1−1 2 4

2

(0, 1)

m 5212

; s0, 1d

y 5212

x 1 1

6y 5 23x 1 6

3x 1 6y 5 6 18.

−2x

−8 2

y

−2

2

4

6

8

−6

(0, 8)

m 5 2; s0, 8d

y 5 2x 1 8

22x 1 y 5 8 19.

y

−4

1

x1−1 2 4

(0, 2)−

−1

−3

m 512

; s0, 22d

y 512

x 2 2

22y 5 2x 1 4

x 2 2y 5 4

20.

Lines are perpendicular.

m1 ? m2 5 21

m1 5 3 m2 5213

y 5 3x 1 2; y 5213

x 2 4 21.

Lines are neither.

m1 ? m2 Þ 21

m1 Þ m2

m1 5 2 m2 5 22

y 5 2x 1 3; y 5 22x 2 3 22.

Lines are parallel.

m1 5 m2

m1 5 4 m2 5 4

y 5 4x 1 3; y 512

s8x 1 5d

23.

20,000

80,000

V

100,000

6t

2 8

60,000

40,000

4 10

V 5 28100t 1 85,000, 0 ≤ t ≤ 10

m 54000 2 85,000

10 2 05

281,00010

5 28100

s0, $85,000d, s10, $4000d


Section 2.4 Equations of Lines

1. matches graph (b).y 523x 1 2 3. matches graph (a).y 5 2

32x 1 2

5.

3x 2 y 5 1

3x 2 6 5 y 2 5

3sx 2 2d 5 y 2 5

3 5y 2 5x 2 2

7.

x 1 2y 5 21

x 1 3 5 22y 1 2

x 1 3 5 22sy 2 1d

212

5y 2 1

x 2 s23d 9.

4x 2 5y 5 8

4x 2 3 5 5y 1 5

41x 2342 5 5sy 1 1d

45

5y 2 s21d

x 234

11.

y 5 212x

y 2 0 5 212sx 2 0d 13.

y 1 4 5 3x

y 1 4 5 3sx 2 0d 15.

y 2 6 5 234x

y 2 6 5 234sx 2 0d

17.

y 2 8 5 22sx 1 2d

y 2 8 5 22fx 2 s22dg 19.

y 1 7 554sx 1 4d

y 2 s27d 554fx 2 s24dg 21.

y 272 5 24sx 1 2d

y 272 5 24fx 2 s22dg

23. y 252 5

43sx 2

34d 25.

y 1 1 5 0

y 2 s21d 5 0sx 2 2d

27.

3x 2 2y 5 0

2y 5 3x

y 532

x

y 2 0 532

sx 2 0d

m 53 2 02 2 0

532

29.

x 1 y 2 4 5 0

y 2 4 5 2x

y 2 4 5 21sx 2 0d

m 50 2 44 2 0

5244

5 21

31.

x 1 2y 2 4 5 0

2y 5 2x 1 4

y 5212

x 1 2

y 2 0 5212

sx 2 4d

m 50 2 3

4 2 s22d 5236

5212

33.

2x 1 5y 5 0

5y 5 22x

y 5 225

x

y 2 2 5 225

x 2 2

y 2 2 5 225

sx 1 5d

m 522 2 25 1 5

5 24

105 2

25

Section 2.4 Equations of Lines 163

35.

2x 2 6y 1 15 5 0

6y 2 18 5 2x 2 3

y 2 3 513

x 212

y 2 3 5131x 2

322

m 54 2 392

232

5162

513

37.

5x 1 34y 2 67 5 0

34y 2 17 5 25x 1 50

y 212

5 2534

x 15034

y 212

5 2534

sx 2 10d

m 5

74

212

1322 2 10

?44

57 2 2

6 2 405

5234

39.

52x 1 15y 2 395 5 0

15y 2 135 5 252x 1 260

y 2 9 525215

x 1523

y 2 9 525215

sx 2 5d

m 521.4 2 9

8 2 55

210.43

52104

305

25215

41.

4x 1 5y 2 11 5 0

8x 1 10y 2 22 5 0

0.8x 1 y 2 2.2 5 0

y 2 0.6 5 20.8x 1 1.6

y 2 0.6 5 20.8sx 2 2d

m 524.2 2 0.6

8 2 25 2

4.86

5 20.8

43.

y 512

x 1 3

y 2 5 512

x 2 2

y 2 5 512

sx 2 4d

m 55 2 24 1 2

536

512

45.

y 5 3

y 2 3 5 0

y 2 3 5 0sx 2 4d

m 53 2 34 1 2

506

5 0 47. because everyx-coordinate is 21.x 5 21

49. because every y-coordinateis 6.y 5 6 51. because both points have

an x-coordinate of 27.x 5 27

53. slope

(a)

(b)

y 5 213x 1

53

y 5 213x 1

23 1

33

y 2 1 5 213x 1

23

y 2 1 5 213sx 2 2d

y 5 3x 2 5

y 2 1 5 3x 2 6

y 2 1 5 3sx 2 2d

y 5 3x 232

22y 5 26x 1 3

5 3 6x 2 2y 5 3 55.

slope

(a)

(b)

y 545x 1 8

y 2 4 545x 1 4

y 2 4 545sx 1 5d

y 2 4 545fx 2 s25dg

y 5 254x 2

94

y 5 254x 2

254 1

164

y 2 4 5 254x 2

254

y 2 4 5 254sx 1 5d

y 2 4 5 254fx 2 s25dg

5 254 y 5 2

54x 1 6

4y 5 25x 1 24

5x 1 4y 5 24


57. slope

(a)

(b)

y 5 214x 1

314

y 5 214x 1

34 1

284

y 2 7 5 214x 1

34

y 2 7 5 214sx 2 3d

y 5 4x 2 5

y 2 7 5 4x 2 12

y 2 7 5 4sx 2 3d

4x 2 3 5 y

5 4 4x 2 y 2 3 5 0 59.

The slope is undefined.

(a)

(b)

3y 2 4 5 0

y 243 5 0 or

y 543

3x 2 2 5 0

x 223 5 0 or

x 523

x 5 5

x 2 5 5 0

61.

The slope is zero.

(a)

(b)

x 1 1 5 0

x 5 21

y 2 2 5 0

y 2 2 5 0sx 1 1d

y 5 25

y 1 5 5 0 63.

x3

1y2

5 1

xa

1yb

5 1, a Þ 0, b Þ 0

65.

26x5

23y7

5 1

x

256

1y

273

5 1

xa

1yb

5 1, a Þ 0, b Þ 0 67.

5 $13,000

C 5 20s400d 1 5000

C 5 20x 1 5000

C 2 5000 5 20sx 2 0d

M 56000 2 5000

50 2 05

100050

5 20

69.

S 5 100,000s6d 5 $600,000

S 5 100,000t

S 2 500,000 5 100,000t 2 500,000

S 2 500,000 5 100,000st 2 5d

5 100,000

52300,000

23

M 5200,000 2 500,000

2 2 571.

0.03 5 3%

S 5 0.03M 1 1500

S 53

100M 1 1500 or

S 2 1500 53

100sM 2 0d

53

100

530

1000

M 51530 2 1500

1000 2 0

Section 2.4 Equations of Lines 165

73. (a)

(b)

S 5 $94.50

S 5 0.70s135d

S 5 0.70L 75. (a)

(b)

Thus, after 2 years, the photocopier has avalue of $4450.

V 5 4450

V 5 22950 1 7400

V 5 21475s2d 1 7400

V 5 21475t 1 7400

V 2 7400 5 21475t

V 2 7400 5 21475st 2 0d

m 57400 2 1500

0 2 45

590024

5 21475

s4, 1500ds0, 7400d

77. (a) N 5 1500 1 60t (b)

5 2400

5 1500 1 900

N 5 1500 1 60s15d (c)

5 1800

5 1500 1 300

N 5 1500 1 60s5d

79. (a) & (b)

(d)

years E < 48.8

E 5 225.8 1 74.56

E 5 20.86s30d 1 74.56

A

E

20

40

60

100

20 40 60 80

(c) Two points taken from the “best-fitting” line sketched inpart (b) are

E 5 20.86A 1 74.56

E 5 20.86A 1 72.7

E 2 38.3 5 20.86sA 2 40d

m 538.3 2 21.1

40 2 605

17.2220

5 20.86

0 and 10.

81.

—CONTINUED—

x 2 8y 5 0

8y 5 x

y 518x

y 2 0 518sx 2 0d

m 55 2 0

40 2 05

540

518

s0, 0d, s40, 5dDistance from deep end 0 8 16 24 32 40

Depth of water 9 8 7 6 5 4


81. —CONTINUED—

Depth of water5 9 2 y

(a)

x 5 0

x 2 8s0d 5 0

0 5 y

9 5 9 2 y

Depth 5 9 2 y (b)

x 5 8

x 2 8s1d 5 0

1 5 y

21 5 2y

8 5 9 2 y

Depth 5 9 2 y (c)

x 5 16

x 2 8s2d 5 0

2 5 y

22 5 2y

7 5 9 2 y

Depth 5 9 2 y

(d)

x 5 24

x 2 8s3d 5 0

3 5 y

23 5 2y

6 5 9 2 y

Depth 5 9 2 y (e)

x 5 32

x 2 8s4d 5 0

4 5 y

24 5 2y

5 5 9 2 y

Depth 5 9 2 y (f)

x 5 40

x 2 8s5d 5 0

5 5 y

25 5 2y

4 5 9 2 y

Depth 5 9 2 y

83. Yes. When different pairs of points are selected, the change in y and the change in x are the lengths of the sides of similartriangles. Corresponding sides of similar triangles are proportional.

85. In the equation 3 is the slope and 5 is the y-intercept.y 5 3x 1 5,

Section 2.5 Relations and Functions

1. Domain

Range

−3 −2 1 2

1

2

3

4

y

x(−2, 0)

(0, 1)

(1, 4)

(0, −1)

5 H21, 0, 1, 4J

5 H22, 0, 1J 3. Domain

Range

−4 −2−2

2 4 6 8

−4

2

4

6

8

y

x(0, 0)

(5, 5) (6, 5)

(2, 8)

(4, −3)

5 H23, 0, 5, 8J

5 H0, 2, 4, 5, 6J

5. s3, 150d, s2, 100d, s8, 400d, s6, 300d, s12, 25d 7. s1, 1d, s2, 8d, s3, 27d, s4, 64d, s5, 125d, s6, 216d, s7, 343d

9. (1995, Atlanta Braves), (1996, New York Yankees), (1997, Florida Marlins), (1998, New York Yankees)

11. No, this relation is not a function because in thedomain is paired to 2 numbers in the range.s6 and 7d

21

13. Yes, this relation is a function as each number in thedomain is paired with exactly one number in the range.

15. No, this relation is not a function as 0 in the domain ispaired with 2 numbers in the range .s5 and 9d

Section 2.5 Relations and Functions 167

17. No, this relation is not a function because both CBS andABC in the domain are each paired to 3 different TVshows in the range.

19. Yes, this relation is a function as each number in thedomain is paired with exactly one number in the range.

21. No, this relation is not a function as the 4 and the 7 in the domain are each paired with 2 different numbers in the range.

23. (a) Yes, this relation is a function as each number in the domain is paired with exactly one number in the range.

(b) No, this relation is not a function as the 1 in the domain is paired with 2 different numbers in the range.

(c) Yes, this relation is a function as each number in the domain is paired with exactly one number in the range.

(d) No, this relation is not a function as each number in the domain is not paired with a number.

25.

Both are solutions of which implies y is not a function of x.

x2 1 y2 5 25s0, 5d and s0, 25d

25 5 25

02 1 52 5?

25

x2 1 y2 5 25 27.

Both are solutions of which implies y is not a function of x.

|y| 5 x 1 2s1, 3d and s1, 23d

3 5 3

|3| 5?

1 1 2

|y| 5 x 1 2

25 5 25

02 1 s25d2 5?

25

3 5 3

|23| 5?

1 1 2

29. represents y as a function of x because there is one value of y associated with one value of x.y 5 10x 1 12

31. represents y as a function of x because there is one value of y associated with one value of x.3x 1 7y 2 2 5 0

33. represents y as a function of x because there is one value of y associated with one value of x.y 5 xsx 2 10d

35.

(a)

(b)

(c)

(d) f sk 1 1d 5 3sk 1 1d 1 5 5 3k 1 3 1 5 5 3k 1 8

f skd 5 3skd 1 5 5 3k 1 5

f s22d 5 3s22d 1 5 5 21

f s2d 5 3s2d 1 5 5 11

f sxd 5 3x 1 5 37.

(a)

(b)

(c)

(d) f s2td 5 3 2 s2td2 5 3 2 4t2

f smd 5 3 2 m2

f s23d 5 3 2 s23d2 5 3 2 9 5 26

f s0d 5 3 2 02 5 3

f sxd 5 3 2 x2

39.

(a)

(b)

(c)

(d) f ss 2 2d 5s 2 2

ss 2 2d 1 25

s 2 2s

f ssd 5s

s 1 2

f s24d 524

24 1 25

2422

5 2

f s3d 53

3 1 25

35

f sxd 5x

x 1 2


41.

(a)

(b)

(c)

(d) 5 12a 1 5f sa 1 1d 5 12sa 1 1d 2 7 5 12a 1 12 2 7

f sad 1 f s1d 5 f12sad 2 7g 1 f12s1d 2 7g 5 12a 2 7 1 12 2 7 5 12a 2 2

f s32d 5 12s3

2d 2 7 5 11

f s3d 5 12s3d 2 7 5 29

f sxd 5 12x 2 7

43.

(a)

(b)

(c)

(d) 5 165 2 1 145 s2 2 16 1 16d 1 s2 2 24 1 36d gs4d 1 gs6d 5 f2 2 4s4d 1 42g 1 f2 2 4s6d 1 62g

5 2 2 8y 1 4y2 gs2yd 5 2 2 4s2yd 1 s2yd2

5 2 gs0d 5 2 2 4s0d 1 02

5 25 2 2 16 1 16 gs4d 5 2 2 4s4d 1 42

gsxd 5 2 2 4x 1 x2

45.

(a) (b)

(c) (d) f s5zd 5 !5z 1 5f sz 2 5d 5 !z 2 5 1 5 5 !z

f s4d 5 !4 1 5 5 3f s21d 5 !21 1 5 5 2

f sxd 5 !x 1 5

47.

(a)

(b)

(c)

(d) 5 8 2 |x 2 6| gsx 2 2d 5 8 2 |x 2 2 2 4|5 275 24 2 35 s8 2 12d 2 s8 2 5d gs16d 2 gs21d 5 s8 2 |16 2 4|d 2 s8 2 |21 2 4|d

5 45 8 2 4 gs8d 5 8 2 |8 2 4|5 45 8 2 4 gs0d 5 8 2 |0 2 4|

gsxd 5 8 2 |x 2 4|

49.

(a)

(b)

(c)

(d) f sx 1 4d 53sx 1 4d

x 1 4 2 55

3x 1 12x 2 1

56

235

2326

5 22 212

5252

f s2d 2 f s21d 5 3 3s2d2 2 54 2 3 3s21d

21 2 54

f1532 5

31532

53

2 5?

33

515

5 2 155

15210

53

22

f s0d 53s0d

0 2 55 0

f sxd 53x

x 2 5

51.

(a)

(b)

(c)

(d) 5 10 2 12 2 6 5 28 f s6d 2 f s22d 5 f10 2 2s6dg 2 f22 1 8g

f s0d 5 10 2 2s0d 5 10

f s210d 5 210 1 8 5 22

f s4d 5 10 2 2s4d 5 10 2 8 5 2

f sxd 5 5x 1 8,10 2 2x,

if if

x < 0x ≥ 0

Section 2.5 Relations and Functions 169

53.

(a)

(b)

(c)

(d) 5 4 2 9 1 5 5 0 hs23d 1 hs7d 5 f4 2 s23d2g 1 f7 2 2g

hs5d 5 5 2 2 5 3

5 4 294 5

164 2

94 5

74 hs23

2d 5 4 2 s232d2

hs2d 5 4 2 22 5 0

hsxd 5 54 2 x2,x 2 2,

if if

x ≤ 2x > 2

55.

(a)

(b)f sx 2 3d 2 f s3d

x5

f2sx 2 3d 1 5g 2 f2s3d 1 5gx

52x 2 6 1 5 2 6 2 5

x5

2x 2 12x

f sx 1 2d 2 f s2dx

5f2sx 1 2d 1 5g 2 f2s2d 1 5g

x5

2x 1 4 1 5 2 4 2 5x

52xx

5 2

f sxd 5 2x 1 5

57. Domain of is all real numbers x.f sxd 5 5 2 2x

59. Domain of is all real numbers x such that because means x Þ 3.x 2 3 Þ 0x Þ 3f sxd 52x

x 2 3

61. Domain of is all real numbers t such that because means t Þ 0 and t Þ 22.tst 1 2d Þ 0t Þ 0, 22f std 5t 1 3

tst 1 2d

63. Domain of is all real numbers x such that x ≥ 24 because x 1 4 ≥ 0 means x ≥ 24.gsxd 5 !x 1 4

65. Domain of is all real numbers x such that because 2x 2 1 ≥ 0 means x ≥ 12.x ≥ 1

2f sxd 5 !2x 2 1

67. Domain of is all real numbers t.f std 5 |t 2 4|

69. Domain

Range5 H0, 1, 8, 27J

5 H0, 2, 4, 6J 71. Domain

Range5 52172

, 252

, 2, 1165 H23, 21, 4, 10J 73. Domain

Range5 C > 0

5 r > 0 75. Domain

Range5 A > 0

5 r > 0

77. Verbal model: 4

Labels: Perimeter

Length of side

Function: P sxd 5 4x

5 x

5 P sxd

Length of side?5Perimeter

79. Verbal model:

Labels: Volume

Length of side

Function: V sxd 5 x3

5 x

5 V sxd

3Length of side5Volume


81. Verbal model:

Labels: Distance

Rate

Time

Function: d std 5 230t

5 t

5 230

5 d std

Time?Rate5Distance

83. Verbal model:

Labels: Volume

Length

Width

Height

Function: V sxd 5 xs24 2 2xd2 or 4xs12 2 xd2

5 x

5 s24 2 2xd

5 s24 2 2xd

5 V sxd

Height?Width?Length5Volume

85. Verbal model:

Labels: Area

Length

Width

Function:

Asxd 5 s32 2 xd2

A sxd 5 s32 2 xds32 2 xd

5 s32 2 xd

5 s32 2 xd

5 A sxd

Width?Length5Area

87.

(a) pounds (b) poundsSs16d 5128,160

165 8010Ss12d 5

128,16012

5 10,680

SsLd 5128,160

L

89. Yes to both questions. For each year there is associated one public school enrollment and one private school enrollment.

91. (g)

(i) Domain: all real numbers x such that

Range: all real numbers y such that 0 < y ≤ 15,900

0 < x ≤ 8.37

5 $2600

5 15,900 2 13,300

y s7d 5 15,900 2 1900s7d

y sxd 5 15,900 2 1900x (h) Straight-line depreciation might not be a fair model forautomobile depreciation because the car depreciates moreslowly as the car ages.

x

y

16,000

12,000

8,000

4,000

2 4 6 8 10

93. (a) This is not a correct mathematical use of the word function.

(b) This is a correct mathematical use of the word function.

Section 2.6 Graphs of Functions 171

95. No, every relation is not a function because some relations have more than one y value paired with each x value.For example, is a relation, but not a function.Hs4, 3d, s4, 22dJ

97. You can name the function That is convenient when there is more than one function used in solving a problem.The values of the independent and the dependent variables are easily seen in function notation.

sf, g, etc.d.

Section 2.6 Graphs of Functions

1.

Domain:

Range:2` < y < `

2` < x < `

6x

y

4

2

224

2

4

6

3.

Domain:

Range: f0, `d or 0 ≤ y < `

2` < x < `

3x

y

5

4

2

1

2111

23

3

5.

Domain:

Range: s2`, 0g or 2` < y ≤ 0

2` < x < `

–1 1 2 3

–3

–1

1

x

y 7.

Domain:

Range: f21, `d or 21 ≤ y < `

2` < x < `

8x

y

10

8

64224

4

2

2

66

9.

Domain:

Range:21 ≤ y < `

0 ≤ x < `

x

y

−2

−1

1

2

3

1−1 2 3 4

11.

Domain:

Range: f0, `d or 0 ≤ y < `

f2, `d or 2 ≤ t < `

t6

y

66

4

42

2

2

2


13.

Domain:

Range:y 5 8

2` < x < `

x

y

−2

2

10

4

6

−2−4−6 2 64

15.

Domain:

Range:2` < y < `

2` < s < `

s

y

−1

−2

2

3

4

−2−3 1 32

17.

Domain:

Range: f0, `d or 0 ≤ y < `

2` < x < `

2x

y

8

66

4

2

4 268

19.

Domain:

Range: f1, `d or 1 ≤ y < `

2` < x < `

2 4 6 8

2

4

8

y

s

21.

Domain:

Range:0 ≤ y ≤ 6 or f0, 6g

0 ≤ x ≤ 2 or f0, 2g

6x

y

8

66

4

2

4

2

224

23.

Domain:

Range:28 ≤ y ≤ 8 or f28, 8g

22 ≤ x ≤ 2 or f22, 2g

2 4 6 8−2−4−6−8

2

4

6

8

x

y

25.

Domain:

Range: s2`, 3g or ` < y ≤ 3

2` < x < `

−1 1 2 3

1

3

4

5

y

x

h x xx

( ) 2 + 3< 0

=

h x xx

( ) 30

= −≥

27.

Domain:

Range:24 ≤ y < ` or f24, `d

2` < x < `

−2−3 −1 2 3 51

−4

−2

−3

2

3

4

y

x

f x xx

( )0

= −≤

f x x xx

( ) 40

= −>

2


29. Keystrokes:

1

Domain Range5 s2`, 1g or 2` < y ≤ 15 2` < x < ` −10

−10

10

10

31. Keystrokes:

2

Domain Range

2 ≤ x < `

5 f0, `d or 0 ≤ y < `5 f2, `d or x ≥ 2 −10

−10

10

10

33. Yes, passes the Vertical Line Test and is afunction of x.

y 513x3 35. Yes, y is a function of x by the Vertical Line Test.

37. No, y is not a function of x by the Vertical Line Test. 39. No, does not pass the Vertical Line Test and y isnot a function of x.

y2 5 x

41.

y is a function of x.

5x

321

y

1

1

2

3

5

4

1

43.

y is not a function of x.

4x

y

3

2

1

32112

1

2

3

45. (b) graph matches f sxd 5 x2 2 1. 47. (a) graph matches f sxd 5 2 2 |x|.

49. (b) shows the most complete graph.

(a)

00

10

30

(b)

0

−10

20

60

(c)

15

−10

30

60

2 X,T,u x2Y5 GRAPH

Y5 GRAPH! x dX,T,u 2


51. (a) Vertical shift 2 units upward

4x

y

7

5

66

3

1

2111

234

4

3

(b) Vertical shift 4 units downward

–3 –1 1 3

–2

–1

1

2

x

y

(c) Horizontal shift 2 units to the left

1x

y

5

4

1

112345

(d) Horizontal shift 4 units to the right

2 4 6 8

2

4

6

8

x

y

(e) Reflection in the x-axis.

3x

y

2

1

23

2

3

5

4

1

(f) Reflection in the x-axis and a vertical shift 4 units upward

–3 –1 1 3

–2

–1

1

2

3

x

y

(g) Horizontal shift 3 units to the right and a vertical shift 1unit upward

x

y

−1

2

1

3

4

5

6

−1 21 43 5 6

(h) Reflection in the x-axis, a horizontal shift 2 units to theleft, and a vertical shift 3 units downward

x

y

−4

−3

−2

−1

1

−5 −4 −3 −2 −1 1 2


53. Keystrokes:

5

Horizontal shift 5 units to the right

−10 10

−10

10

55. Keystrokes:

5

Vertical shift 5 units downward

−10 10

−10

10

57. Keystrokes:

Reflection in the x-axis

−10 10

−10

10

59. Graph is shifted 3 units left

hsxd 5 sx 1 3d2

61. Graph is reflected in the x-axis

hsxd 5 2x2

63. Graph is shifted 3 units left and reflected in the x-axis

hsxd 5 2sx 1 3d2

65. Graph is reflected inthe x-axis and shiftedup 2 units

hsxd 5 2x2 1 2

67. f sxd 5 2!x 69. f sxd 5 !x 1 2 71. f sxd 5 !2x

73. (a)

—CONTINUED—

yy

x21 3 4 5

4

3

2

1

5

(0, 1)

(1, 2)

(3, 3)

(4, 4)

y 5 f sxd 1 2 (b)

yy

x1 3 4 5

2

1

−1

−2

−3

(0, 1)

(1, 0)

(3, 1)−

(4, 2)−

y 5 2f sxd

Y5 GRAPHx dX,T,u 2ABS

Y5 GRAPHX,T,u

Y5 GRAPHX,T,u 2ABS

x2c ABS


(c)

y

x2 3 5 61 4

(2, 1)−

(3, 0) (5, 1)

(6, 2)

−1

−2

1

2

3

4

y 5 f sx 2 2d (d)

y

x1−3 −2 2

2

1

3

−1

−2

(1, 1)

(2, 2)

( 1, 0)−

( 2, 1)− −

y 5 f sx 1 2d73. —CONTINUED—

(e)

yy

x1 3 4 5

2

1

−1

−2

−3

(4, 1)

(3, 0)

(1, 1)−

(0, 2)−

y 5 f sxd 2 1 (f)

yy

x−1−3 −2−4−5

2

3

−2

(0, 1)−

( 1, 0)−( 3, 1)−

( 4, 2)−

y 5 f s2xd

75. (a) Keystrokes:

26 .0242 20 60

(b) 46%<

20 60

−50

20

Y5 2 X,T,u X,T,u X,T,ux2 4 x ≥ ≤AND d GRAPH

77. (a)

Keystrokes:

100

(b) (c) When the largest value of A is 2500. isthe highest point on the graph of A giving the largest valueof the function.

s50, 2500dx 5 50,

0 1000

3000

A 5 xs100 2 xd

A 5 l ? w

100− x

x

100 2 x 5 width

Let x 5 length

100 2 l 5 w

100 5 l 1 w

200 5 2l 1 2w

P 5 2l 1 2w

Y5 X,T,u X,T,ux d GRAPH2


79. (a) Keystrokes:

5.46 2665.56 153,363

(b) corresponds to the year 1970 (20 years later).

(c) Keystrokes:

5.46 20 2665.56 20 153,363

−20 28150,000

280,000

t 5 0

0 48150,000

280,000

Y5 X,T,u X,T,ux2 GRAPH1 1

Y5 X,T,u X,T,u GRAPH1 1 1 1x d dx2 x

81. If the domain of the function changes from then the range changed from f0, 4g to f0, 8g.f0, 2g to f0, 4g,f sxd 5 2x

83. The four types of shifts of the graph of a function are vertical shift upward, vertical shift downward, horizontal shift to the left,horizontal shift to the right.

85. is a reflection in the y-axis of the graph of f sxd.gsxd 5 f s2xd


1.5

4 5x

y

5,25

4

3

2

32

1

11

), 30(3

4

2

234

)4,2(

3.

x

y

(4, 20)

(12, 9)

(1, 1)

4 8 12 16 20

4

8

12

16

20

5. Quadrant IV

–6 –4 –2 2 4 6

–6

–4

–2

2

4

6

x

y

(2, −6)

7. Quadrant I, IV

y

x−2 2

2

−2−6 −4 4 6

4

6

−4

−6

(4, y)

(4, y)

x2c

x2c


23.

s1, 0d, s21, 0d x 5 1 x 5 21

0 5 sx 2 1dsx 1 1d

0 5 x2 2 1

s0, 21d 5 21

y 5 02 2 1

−3 −2 1 2 3

−2

1

2

3

4

y

x−1

( 1, 0)− (1, 0)

(0, 1)−

y 5 x2 2 1

21.

s232, 0d x 5 2

32

22x 5 3

3s0d 2 2x 2 3 5 0

s0, 1d y 5 1

3y 5 3

3y 2 2s0d 2 3 5 0

3y 2 2x 2 3 5 0

1)

1x

y

2

,(01

1

0,

1

2

3

23

9. (a)

yes

(c)

no 0 Þ 6

0 5?

4 1 2

0 5?

4 212s24ds24, 0d

2 5 2

2 5?

4 2 2

2 5?

4 212s4ds4, 2d (b)

no

(d)

yes 0 5 0

0 5?

4 2 4

0 5?

4 212s8ds8, 0d

5 Þ 412

5 5?

4 112

5 5?

4 212s21ds21, 5d

11.

5 5

5 !25

5 !0 1 25

d 5 !s4 2 4d2 1 s3 2 8d2 13.

5 3!5

5 !45

5 !36 1 9

d 5 !s25 2 1d2 1 s21 2 2d2

15. matches graph (c).y 5 5 232x 17. matched graph (a).y 5 |x| 1 4

19.

s18, 0d x 5 18

13x 5 6

0 5 6 213x

s0, 6d y 5 6

y 5 6 213s0d

y 5 6 213x

),8 0

20x

y

20

16

(

), 6

161284

(08

4

44

12

1


25.

s2, 0d, s22, 0d ±2 5 x

2 5 |x| 0 5 |x| 2 2

5 22

y 5 |0| 2 2

−2

1

2

3

y

−3 −2 1 2 3x

−1

( 2, 0)− (2, 0)

(0, 2)−−3

y 5 |x| 2 2

27.

y-intercept

x-intercept

s32, 0d 32 5 x

64 5 x

6 5 4x

0 5 4x 2 6

s0, 26d 5 26

y 5 4s0d 2 6

y 5 4x 2 6 29.

y-intercept

x-intercept

s22, 0d x 5 22

7x 5 214

7x 2 2s0d 5 214

s0, 7d y 5 7

22y 5 214

7s0d 2 2y 5 214

7x 2 2y 5 214

31.

y-intercept

x-intercept

s5, 0d 5 5 x

0 5 x 2 5

0 5 |x 2 5|

s0, 5d 5 5

y 5 |0 2 5|

y 5 |x 2 5| 33.

y-intercept

x-intercepts

or

s2, 0d, s23, 0d

2 5 x

4 5 2x

5 5 2x 1 1

5 5 |2x 1 1| 0 5 |2x 1 1| 2 5

s0, 24d 5 24

5 25

y 5 |2s0d 1 1| 2 5

y 5 |2x 1 1| 2 5

23 5 x

26 5 2x

25 5 2x 1 1

35. Keystrokes:

3 3

s1.27, 0d, s4.73, 0d, s0, 6d

− 4 10

− 4

10

Y5 x X,T,u 2 2d x2 GRAPH

37. Keystrokes:

4 7

no x-intercepts

s0, 211d

0 8

−12

0

Y5 x X,T,u 22 d GRAPHx2c ABS


39. Keystrokes:

3

s3, 0d, s0, 1.73d

−6 4

−1

4

41. m 53 2 1

6 2 s21d 527

Y5 x X,T,u2 d GRAPH!

43. m 53 2 3

4 2 s21d 505

5 0 45. m 50 2 68 2 0

5268

5234

5 234 47.

32

5 t

23 5 22t

3 5 6 2 2t

32

53 2 t1 2 0

m 53 2 s23d1 2 s23d 5

64

532

49.

s0, 2d, s1, 21d

23 5y 1 4x 2 2

51.

s7, 6d, s11, 11d

54

5y 2 1x 2 3

53. Since m is undefined the line isa vertical line so points such as

are onthis line.s3, 0d, s3, 1d, and s3, 22d

55.

x

y

1

1

−1

−1

−2

−2

2

3

2 3 4

y 552x 2 2

22y 5 25x 1 4

5x 2 2y 2 4 5 0 57.

1−1

−1

2

2

3

y

x

y 5 212x 1 1

2y 5 2x 1 2

x 1 2y 2 2 5 0


59.

So lines are neither

m1 Þ m2, m1 ? m2 Þ 21

m1 532, m2 5

23

L2: y 523x 2 1

L1: y 532x 1 1 61.

So lines are perpendicular

m1 ? m2 5 21

m1 532, m2 5 2

23

L2: y 5 223x 1 1

L1: y 532x 2 2 63.

So lines are neither

m1 Þ m2, m1 ? m2 Þ 21

m2 5 212

y 5 212x 1 3

L2: 2y 5 2x 1 6

m1 523

y 523x 2

53

L1: 23y 5 22x 1 5

L2: x 1 2y 2 6 5 0

L1: 2x 2 3y 2 5 5 0

65.

2x 2 y 2 6 5 0

y 1 4 5 2x 2 2

y 1 4 5 2sx 2 1d 67.

4x 1 y 5 0

y 2 4 5 24x 2 4

y 2 4 5 24sx 1 1d 69.

2x 1 3y 2 17 5 0

3y 2 12 5 22x 1 5

y 2 4 5 223x 1

53

y 2 4 5 223sx 2

52d

71.

y 2 5 5 0

y 2 5 5 0fx 2 s26dg 73.

x 1 2y 1 6 5 0

2y 5 2x 2 6

y 5 212

x 2 3

y 2 0 5 212

sx 1 6d

m 50 1 3

26 2 05

326

5 212 75.

3x 2 2y 5 0

2y 2 12 5 3x 2 12

2sy 2 6d 5 2132

x 2 62

y 2 6 532

x 2 6

y 2 6 532

sx 2 4d

m 56 2 s23d4 2 s22d 5

6 1 34 1 2

596

532

77.

9x 2 24y 2 8 5 0

18x 2 48y 2 16 5 0

48y 2 56 5 18x 2 72

y 276

538

x 2128

y 276

538

sx 2 4d

m 5

76

216

4 243

?66

57 2 1

24 2 85

616

538

79.

(a)

(b) y 145

5131x 2

352 or x 2 3y 2 3 5 0

y 145

5 231x 2352 or 3x 1 y 2 1 5 0

y 5 23x 1 2

3x 1 y 5 2

81.

undefined

(a)

(b) y 5 1 or y 2 1 5 0

x 5 12 or x 2 12 5 0

m 5 x 535

5x 5 3 83. No, this relation is not a functionbecause the 8 in the domain ispaired to two numbers (1 and 2)in the range.

85. Yes, this relation is a functionbecause each number in thedomain is paired to only onenumber in the range.


99.

–2 2 4 6 8

–6

–4

–2

2

4

x

y 101.

1

2

3

4

y

−1 1 2 3x

4−1

5

5

103.

2

4

6

y

−4

−6 −4 2 4 6x

−2

8

8

105.

−1−2 1 2 3

−8

−4

4

12

y

x4

107.

y x= 2 + ( 1)− 2

x ≥ 1

2y x= 2 ( 1)− −x < 1

x

y

−1 1 2 3

1

2

3

4

109. No, y is not a function of x.

87.

(a)

(b)

(c)

(d) f sx 1 hd 5 4 252sx 1 hd 5 4 2

52x 2

52h

5 4 252t 1 4 1 10 5 18 2

52t f std 1 f s24d 5 s4 2

52td 1 f4 2

52s24dg

f s25d 5 4 2

52s2

5d 5 4 2 1 5 3

f s210d 5 4 252s210d 5 4 1 25 5 29

f sxd 5 4 252x

89.

(a)

(b)

(c)

(d) f s5zd 5 !5 2 5z

5 !2 f s3d 5 !5 2 3

5 0 f s5d 5 !5 2 5

5 35 !9 f s24d 5 !5 2 s24d

f std 5 !5 2 t

91.

(a)

(b)

(c)

(d) 5 275 1 2 16 2 1 1 9 f s4d 2 f s3d 5 s1 2 42d 2 s1 2 32d

5 0 f s1d 5 1 2 12

5 2 f s223d 5 23s22

3d5 23 f s2d 5 1 2 22

523x,1 2 x2,

if if

x ≤ 0x > 06

93. (a)

(b) 522x 1 12

x5

3 2 2x 1 6 2 3 1 6x

f sx 2 3d 2 f s3d

x5

f3 2 2sx 2 3dg 2 f3 2 2s3dgx

522x

x5 225

3 2 2x 2 4 2 3 1 4x

f sx 1 2d 2 f s2d

x5

f3 2 2sx 1 2dg 2 f3 2 2s2dgx

95. Find the domain of

Domain: or s2`, `d2` < x < `

hsxd 5 4x2 2 7. 97. Find the domain of

Domain: or 2` < x ≤ 52s2`. 52g

f sxd 5 !5 2 2x.


111. Yes, y is a function of x. 113. is a reflection in the x-axis of

y

−1 1 2 3x

4 5

1

−2

−1

−3

−4

−5

f sxd 5 !xhsxd 5 2!x

115. is a horizontalshift 1 unit to the rightof f sxd 5 !x

hsxd 5 !x 2 1 117.


y 5 x2 2 2 119.

Reflection in the x-axis and ahorizontal shift 3 units to the left

y 5 2sx 1 3d2

121. Verbal model:

Proportion:

Verbal model:

Labels: Leg 1

Leg 2

Hypotenuse

Equation:

feet 3!145 5 x < 36.12

!1305 5 x

9 1 1296 5 x2

32 1 362 5 x2

5 x

5 36

5 3

2Hypotenuse5

2Leg21

2Leg1

x 5 36

1

125

3x

5RiseRun

RiseRun

123.

V 5 22,000t 1 20,000, 0 ≤ t ≤ 7

V 2 20,000 5 22,000t

V 2 20,000 5 22,000st 2 0d

m 56,000 2 20,000

7 2 05

214,0007

5 22,000

t

V

2 4 6 8 10

10,000

20,000

5,000

15,000

25,000

s0, $20,000d, s7, $6000d

125.

y 5 2x 1 3.87

y 5 2x 1 4.75 2 0.88

y 1 0.88 5 21sx 2 4.75d

m 520.88 2 4.75

4.75 2 s20.88d 525.635.63

5 21


127.

Verbal model:

Labels: Area

Length

Width

Function:

Domain: 0 < x <752

Asxd 5 s75 2 xdx

5 x

5 75 2 x

5 Asxd

Width?Length5Area

150 22− x

150 22− x

x x

Verbal model: 2 2

75 2 x 5 Length

150 2 2x

25 Length

150 5 2Length 1 2x

Width1Length5Perimeter

129. (a)

feet per second v 5 16

v 5 264 1 80

v 5 232s2d 1 80 (b)

seconds t 552

t 58032

32t 5 80

0 5 232t 1 80 (c)

feet per second v 5 216

v 5 296 1 80

v 5 232s3d 1 80


1. lies in Quadrant IV if x > 0 and y < 0.sx, yd 2.

1 2 3 4 5

1

2

3

4

5

y

x

(0, 5)

(3, 1)

d 5 !s0 2 3d2 1 s5 2 1d2 5 !9 1 16 5 !25 5 5

3. (a) y-intercept

(b)

x-intercepts21, 0d; x 5 21,

0 5 23sx 1 1d

s0, 23d;y 5 23s0 1 1d 5 23 4.

1 2 3 4

1

3

4

y

x


5. (a)

(b) undefinedm 56 1 23 2 3

580

5

m 53 2 72 1 4

5 246

5 223

6.

2−2 6

−6

−8

−2

−4

y

x

(0, 6)−

7.

s5, 0d x 5 5

2x 5 10

2x 1 5s0d 5 10

s0, 2d y 5 2

5y 5 10

2s0d 1 5y 5 10

2x 1 5y 5 10 8.

m 535

y 5253

x 1 3

3y 5 25x 1 9

5x 1 3y 2 9 5 0

1 2 3 4 5

1

−1

−2

3

4

y

x

(0, 2)

(5, 0)

9.

x 2 2y 2 55 5 0

2y 2 20 5 x 2 75

y 2 10 512

x 2752

y 2 10 512

sx 2 75d

m 510 1 1575 2 25

52550

512

10.

2x 1 y 5 0

y 1 4 5 22x 1 4

y 2 s24d 5 22sx 2 2d

11.

x 1 2 5 0

x 5 22 12. No, is not a function of x, because thegraph does not pass the Vertical Line Test.

y2s4 2 xd 5 x3

13. (a) The relation is a function because each x number is paired with exactly one y number.

(b) The relation is not a function because 0 is paired with two numbers, 0 and 24.

14. (a)

(b)

(c) gsx 1 2d 5x 1 2

sx 1 2d 2 35

x 1 2x 2 1

57

7 2 65 7 g17

22 5

72

72

2 3

5 22 gs2d 52

2 2 3 15. (a) (b)

Domain:

Domain: t ≤ 9 or s2`, 9g

t ≤ 9

2t ≥ 29

x Þ 4 9 2 t ≥ 0

f sxd 5x 1 1x 2 4

hstd 5 !9 2 t


16.

−3 −2 1 2 3

−2

1

2

3

4

y

x

17. is a reflection in the x-axis,horizontal shift 2 units to the right and a vertical shift1 unit upward.

gsxd 5 2sx 2 2d2 1 1

18.

2.5 552

5 t

210,00024000

5 t

210,000 5 24000t

16,000 5 24000t 1 26,000

V 5 24000t 1 26,000

V 2 26,000 5 24000st 2 0d

m 510,000 2 26,000

4 2 05

216,0004

5 24000

s0, $26,000d, s4, $10,000d 19. (a)

(b)

(c) y 5 2|x| 1 2 or 2 2 |x|y 5 |x| 2 2

y 5 |x 2 2|


11.

Equation:

which is more than $500.x 5 $656

x 5 163,356 2 162,700


Actualexpense

Budgetedamount

12.

Equation:

which is less than $500.x 5 |2305| 5 $305

x 5 |42,335 2 42,640|


Actualexpense

Budgetedamount

CHAPTER 2 Graphs and Functions

SECTION 2.1 The Rectangular Coordinate System

1. is a linear equation because it can be written in theform Since cannot be written inthe form it is not a linear equation.ax 1 b 5 0,

x2 1 3x 5 2ax 1 b 5 0.3x 5 7 2. To check is a solution of the equation

substitute 3 for x in the equation. If the result is true,is a solution.x 5 3

5x 2 4 5 11x 5 3

3. 6xs2x2d 5 s6 ? 2d ? sx ? x2d 5 12x3 4. 3t2 ? t4 5 3t214 5 3t6

5.

5 54x10

5 s21ds227ds2dsx6dsx4d

2s23x2d3s2x4d 5 s21ds23d3s2dsx2d3sx4d 6.

5 28x4y5

s4x3y2ds22xy3d 5 s4ds22dsx3dsxdsy2dsy3d

7. 4 2 3s2x 1 1d 5 4 2 6x 2 3 5 1 2 6x 8.

5 23x 1 22

5sx 1 2d 2 4s2x 2 3d 5 5x 1 10 2 8x 1 12

9. 241y3

1y62 5 8y 1 4y 5 12y 10.

5 0.02x 1 100

0.12x 1 0.05s2000 2 2xd 5 0.12x 1 100 2 0.1x

11. Your rate job per hour

Friend’s rate job per hour

Verbal model:

Labels: Work done

Your rate

Friend’s rate

Time

—CONTINUED—

5 t

515

514

5 1

Work doneby friend1

Work doneby you5

Workdone

515

514


11. —CONTINUED—

Equation:

hours 209

5 t < 2.2

1920

5 t

1 5 1 9202t

1 5 114

1152t

1 514

t 115

t

SECTION 2.2 Graphs of Equations

12. Verbal model:

Labels: Distance 200 miles at 50 mph

200 miles at 42 mph

400 miles at x mph

Rate

Time

Equation:

mph x 521s400d

184< 45.65

18421

5400

x

4 110021

5400

x

20050

120042

5400

x

520050

120042

or 400

x

5 50, 42, x

5

Time?Rate5Distance

1. If and c is an algebraic expression, then t 2 3 1 c > 7 1 c.

t 2 3 > 7 2. If t 2 3 < 7 and c < 0, then st 2 3dc > 7c.

3. Multiplicative Inverse Property:

y11y2 5 1

4. illustrates the Commutative Property ofAddition.u 1 v 5 v 1 u


5.

x ≥ 1

2x2

≥22

2x ≥ 2

2x 1 3 2 3 ≥ 5 2 3

2x 1 3 ≥ 5 6.

x < 23

23x23

<9

23

23x > 9

5 2 5 2 3x > 14 2 5

5 2 3x > 14

7.

212

< x <12

2510

<10x10

<510

25 < 10x < 5

24 2 1 < 10x 1 1 2 1 < 6 2 1

24 < 10x 1 1 < 6 8.

212

≤ x ≤32

32

≥ x ≥212

2322

≥22x22

≥1

22

23 ≤ 22x ≤ 1

22 2 1 ≤ 1 2 1 2 2x ≤ 2 2 1

22 ≤ 1 2 2x ≤ 2

9.

26 ≤ x ≤ 6

6 ≥ x ≥ 26

2621

≥2x21

≥6

21

26 ≤ 2x ≤ 6

2 ? 23 ≤ 2 ? 2x2

≤ 3 ? 2

23 ≤ 2x2

≤ 3 10.

20 < x < 30

25 1 25 < x 2 25 1 25 < 5 1 25

25 < x 2 25 < 5

11. Verbal model:

Labels: Compared number

Percent

Base number

Equation:

$29,018 < b

32,5001.12 5 b

32,500 5 1.12b

a 5 pb

5 b

5 p

5 a

Basenumber?Percent5

Comparednumber


12. Verbal model:


Percent

Base number

Equation:

a 5 $108.50

a 5 s0.035ds3100d

a 5 pb

5 b

5 p

5 a

Basenumber?Percent5

Comparednumber

1. Two equations having the same set of solutions are calledequivalent.

2.

12x 5 13 1 5

12x 2 5 5 13

3.

x 583

3x3

583

3x 5 8

2x 1 x 5 8

21x 1x22 5 s4d2

x 1x2

5 4 4.

x 5 27

x 1 3 5 30

3113

x 1 12 5 s10d3

13

x 1 1 5 10

SECTION 2.3 Slope and Graphs of Linear Equations

5.

x 5 5

24x24

522024

24x 5 220

24x 1 20 5 0

24sx 2 5d 5 0 6.

x 5103

3x 5 10

3x 1 6 5 16

8138

x 1342 5 s2d8

38

x 134

5 2

7.

x 5 18

8x8

51448

8x 5 144

8x 2 112 5 32

8sx 2 14d 5 32 8.

x 5 219

2x 5 238

36 1 2x 5 22

36 2 12x 5 214x 2 2

36 2 12x 5 5 2 14x 2 7

12s3 2 xd 5 5 2 7s2x 1 1d


9.

No solution

219

3Þ 0

224

31

53

5 0

22x 2 8 1 2x 153

5 0

2s2x 1 8d 113

s6x 1 5d 5 0 10.

r 5 0.1

r 5 1.1 2 1

1 1 r 5 1.1

1 1 r 5550500

s1 1 rd500 5 550

11. Verbal model:0.45

Labels: Total cost

Cost of first minute

Number of additional minutes

Inequality:

(with first minute) 0 < t ≤ 23

0 < t ≤ 22

22 ≥ t > 0

9.9 ≥ 0.45t > 0

11 ≥ 1.10 1 0.45t > 1.10

5 x

5 $1.10

5 $11

Number ofadditionalminutes

1Cost of

first minute5

Totalcost

12.

m < 23,846

0.65m < 15,500

0.65m 1 4500 < 20,000

1. The ratio of the real number a to the real number b is ab

. 2. is a proportion.45

512u

3. Verbal model:


Percent

Base number

Equation:

a 5 1.875

a 5 0.075 ? 25

a 5 p ? b

5 b

5 p

5 a

Base number?Percent5Comparednumber

SECTION 2.4 Equations of Lines


4. Verbal model:


Percent

Base number

Equation:

a 5 9000

a 5 1.50s6000d

a 5 pb

5 b

5 p

5 a


5. Verbal model:


Percent

Base number

Equation:

150% 5 p

1.5 5 p

225150 5 p

225 5 p ? 150

a 5 p ? b

5 b

5 p

5 a


6. Verbal model:


Percent

Base number

Equation:

15.5% 5 p

0.155 5 p

93600 5 p

93 5 p ? 600

a 5 pb

5 b

5 p

5 a



7. Verbal model:


Percent

Base number

Equation:

6623% 5 p

0.6623 5 p

160240 5 p

160 5 p ? 240

a 5 p ? b

5 b

5 p

5 a


8. Verbal model:


Percent

Base number

Equation:

350 5 b

420.12 5 b

42 5 0.12b

a 5 pb

5 b

5 p

5 a


9. Verbal model:


Percent

Base number

Equation:

80,000 5 b

4000.005 5 b

400 5 0.005b

a 5 pb

5 b

5 p

5 a


10. Verbal model:


Percent

Base number

Equation:

275 5 b

1320.48 5 b

132 5 0.48b

a 5 pb

5 b

5 p

5 a



11. Verbal model:

Proportion:

pounds 72 5 x

360 5 5x

360 2 4x 5 x

4s90 2 xd 5 x

14

590 2 x

x

5Cement

SandCement

Sand

12.

seconds 3 5 t

9632 5 t

96 5 32t

96 2 32t 5 0

1. If by the Transitive Property.a < b and b < c, then a < c 2.

x 5 4

9x 519s36d

9x 5 36

3. “y is no more than 45” translates into y ≤ 45. 4. “x is at least 15” can be expressed in inequality notationas x ≥ 15.

5.

5 24y

6y 2 3x 1 3x 2 10y 5 s6y 2 10yd 1 s23x 1 3xd 6.

5 5sx 2 2d

5 5x 2 10

8sx 2 2d 2 3sx 2 2d 5 8x 2 16 2 3x 1 6

SECTION 2.5 Relations and Functions

7.

5 32t 2

58

5 96t 2

58

5 s46 1

56dt 2

58

23t 258 1

56t 5 s2

3 156dt 2

58 8.

5 724x 1 8

5 s 924 2

224dx 1 8

38x 21

12x 1 8 5 s38 2

112dx 1 8

9.

5 230x2 1 23x 1 3

3x2 2 5x 1 3 1 28x 2 33x2 5 s3x2 2 33x2d 1 s25x 1 28xd 1 3

10.

5 4x3 1 12x2y 1 4xy2 1 y3

4x3 2 3x2y 1 4xy2 1 15x2y 1 y3 5 4x3 1 s23 1 15dsx2yd 1 4xy2 1 y3


11. Verbal model:

Proportion:

cups x 5354

5 834

x 552

?72

x 5 212

? 312

2

121

5x

312

5Cups flour

Batches cookiesCups flour

Batches cookies

12. Verbal model:

Proportion:

pints or 4 gallons x 5 16

x 5 32 ?12

321

5x12

5Gasoline

OilGasoline

Oil

1. illustrates the Multiplicative Inverse Property.8x ?18x

5 1 2. illustrates the Additive Identity Property.3x 1 0 5 3x

3. illustrates theDistributive Property24sx 1 10d 5 24 ? x 1 s24ds10d 4. illustrates the Associative

Property of Addition.5 1 s23 1 xd 5 s5 2 3d 1 x

5. 5x4sx2d 5 5x412 5 5x6 6. 3sx 1 1d2sx 1 1d3 5 3sx 1 1d213 5 3sx 1 1d5

7. s24t3d 5 s24d3std3 5 264t3 8. 2s22xd4 5 2s22d4x4 5 2s116dx4 5 216x4

9. su2vd4 5 su2d4v4 5 u8v4 10.

5 18a4b5

5 s9 ? 2dsa4dsb213d

s3a2bd2s2b3d 5 32sa2d2b2s2dsb3d

SECTION 2.6 Graphs of Functions


11. Verbal model:

Labels: Discount

Discount rate

List price

Equation:

Verbal model:

Labels: Total cost

List price

Shipping

Equation:

Verbal model:

Labels: Sale price

List price

Discount

Equation:

The department store price is a better bargain.

x 5 $191.96

x 5 239.95 2 47.99

5 $47.99

5 $239.95

5 x

Discount2List price5Sale price

x 5 $193.27

x 5 188.95 1 4.32

5 $4.32

5 $188.95

5 x

Shipping1List price5Total cost

x 5 $47.99

x 5 0.20s239.95d

5 $239.95

5 20%

5 x

List price?Discount rate5Discount

12. Verbal model:


Percent

Base number

Equation:

a 5 $960.70

a 5 1.30s739d

a 5 pb

5 b

5 p

5 a


C H A P T E R 3Polynomials and Factoring

Section 3.1 Adding and Subtracting Polynomials . . . . . . . . . . .188

Section 3.2 Multiplying Polynomials . . . . . . . . . . . . . . . . . .192

Section 3.3 Factoring Polynomials . . . . . . . . . . . . . . . . . . .199

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202

Section 3.4 Factoring Trinomials . . . . . . . . . . . . . . . . . . . .203

Section 3.5 Solving Polynomial Equations . . . . . . . . . . . . . . .207


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218

Cumulative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .220

188

C H A P T E R 3Polynomials and Factoring

Section 3.1 Adding and Subtracting PolynomialsSolutions to Odd-Numbered Exercises

1. Standard form:

Degree: 1

Leading coefficient: 10

10x 2 4 3. Standard form:

Degree: 2


3x2 2 x 1 2

5. Standard form:

Degree: 5


y5 2 3y4 2 2y3 1 5 7. Standard form:

Degree: 3

Leading coefficient:23

23x3 2 2x2 2 3

9. Standard form:

Degree: 0


24 11. Standard form:

Degree: 2


216t2 1 v0t

13. is a binomial.12 2 5y2 15. is a trinomial.x3 1 2x2 2 4 17. is a monomial.5

19. A monomial of degree 3 is any term of form where ais any real number.

ax3 21. A binomial of degree 2 and leading coefficient of 8 is anybinomial beginning and containing one other term ofdegree less than 2 such as 8x2 1 4 or 8x2 1 x.

8x2

23. is not a polynomial because the first term is not of the form (k must be nonnegative).axky23 2 2

25. is not a polynomial because the term is not of the form (k must be nonnegative).axk8x

27. 5 1 s2 1 3xd 5 s5 1 2d 1 3x 5 7 1 3x 29. s2x2 2 3d 1 s5x2 1 6d 5 s2x2 1 5x2d 1 s23 1 6d 5 7x2 1 3

31. s5y 1 6d 1 s4y2 2 6y 2 3d 5 4y2 1 s5y 2 6yd 1 s6 2 3d 5 4y2 2 y 1 3

33. s2 2 8yd 1 s22y4 1 3y 1 2d 5 s22y4d 1 s28y 1 3yd 1 s2 1 2d 5 22y4 2 5y 1 4

35. s8 2 t 4d 1 s5 1 t 4d 5 s8 1 5d 1 s2t 4 1 t 4d 5 13

37. sx2 2 3x 1 8d 1 s2x2 2 4xd 1 3x2 5 sx2 1 2x2 1 3x2d 1 s23x 2 4xd 1 s8d 5 6x2 2 7x 1 8

39. 5 s46x3 2

36x3d 1 3x 1 s5

5 235d 5

16x3 1 3x 1

25 s2

3x3 2 4x 1 1d 1 s235 1 7x 2

12x3d 5 s2

3x3 212x3d 1 s24x 1 7xd 1 s1 2

35d

41. 5 2.69t2 1 7.35t 2 4.2 s6.32t 2 4.51t2d 1 s7.2t2 1 1.03t 2 4.2d 5 s24.51t2 1 7.2t2d 1 s6.32t 1 1.03td 2 4.2

Section 3.1 Adding and Subtracting Polynomials189

43.

2x2 2 3x

23x2 2 4

5x2 2 3x 1 4 45.

4x3 1 2x2 1 9x 2 6

4x2 1 x 2 6

4x3 2 2x2 1 8x 47.

2p2 2 2p 2 5

23p2 1 2p 2 7

5p2 2 4p 1 2

49.

0.6b2 2 0.6b 1 7.1

6.6b2

22.4b2 2 3.1b 1 7.1

23.6b2 1 2.5b 51.

5 22y3

5 s4 2 4d 1 s2y3 2 y3d

s4 2 y3d 2 s4 1 y3d 5 s4 2 y3d 1 s24 2 y3d

53.

5 x2 2 3x 1 2

5 s3x2 2 2x2d 1 s22x 2 xd 1 s1 1 1d

s3x2 2 2x 1 1d 2 s2x2 1 x 2 1d 5 s3x2 2 2x 1 1d 1 s22x2 2 x 1 1d

55.

5 7t3 2 t 2 10

5 s6t3 1 t3d 2 t 1 s212 1 2d

s6t3 2 12d 2 s2t3 1 t 2 2d 5 s6t3 2 12d 1 st3 2 t 1 2d

57.

5 74y2 2 9y 2 12

5 s14y2 1

32y2d 1 s25y 2 4yd 2 12

s14y2 2 5yd 2 s12 1 4y 2

32y2d 5 s1

4y2 2 5yd 1 s212 2 4y 132y2d

59.

5 9.37t5 1 10.4t 4 2 5.4t2 1 7.35t 2 2.6

5 s20.23t5 1 9.6t5d 1 10.4t 4 1 s1.3t2 2 6.7t2d 1 7.35t 2 2.6

s10.4t4 2 0.23t5 1 1.3t2d 2 s2.6 2 7.35t 1 6.7t2 2 9.6t5d 5 s10.4t4 2 0.23t5 1 1.3t2d 1 s22.6 1 7.35t 2 6.7t2 1 9.6t5d

61.

5 22x3 1 x2 1 2x

5 x3 2 3x 2 3x3 1 x2 1 5x

sx3 2 3xd 2 f3x3 2 sx2 1 5xdg 5 sx3 2 3xd 2 f3x3 2 x2 2 5xg 63.

x2 2 2x 1 5

2 sx 2 2d ⇒ 2 x 1 2

x2 2 x 1 3 ⇒ x2 2 x 1 3

65.

24x3 2 2x 1 13

2s12 2 13x 1 2x3d ⇒ 22x3 1 13x 2 12

25 2 15x 2 2x3 ⇒ 22x3 2 15x 1 25 67.

211x7 2 10x5 1 8x4 1 16

2s8x7 1 10x5 2 2x4 2 12d ⇒ 28x7 2 10x5 1 2x4 1 12

23x7 1 6x4 1 4 ⇒ 23x7 1 6x4 1 4

69.

5 2x3 2 2x 1 3

5 s22x3 1 4x3d 1 s22xd 1 s3d

2s2x3 2 3d 1 s4x3 2 2xd 5 22x3 1 3 1 4x3 2 2x

71.

5 22x3 2 x2 1 6x 2 11

5 s4x5 2 8x5 1 4x5d 1 s210x3 1 3x3 1 5x3d 2 x2 1 6x 2 11

s4x5 2 10x3 1 6xd 2 s8x5 2 3x3 1 11d 1 s4x5 1 5x3 2 x2d 5 s4x5 2 10x3 1 6xd 1 s28x5 1 3x3 2 11d 1 s4x5 1 5x3 2 x2d

190 Chapter 3 Polynomials and Factoring

73.

5 7y2 2 9y 1 2

5 s5y2 1 2y2d 1 s22y 2 7yd 1 2

5 s5y2 2 2yd 1 s2y2 2 7y 1 2d

5 s5y2 2 2yd 2 f22y2 1 7y 2 2g

5 s5y2 2 2yd 2 fsy2 2 3y2d 1 sy 1 6yd 2 2g

s5y2 2 2yd 2 fsy2 1 yd 2 s3y2 2 6y 1 2dg 5 s5y2 2 2yd 2 fsy2 1 yd 1 s23y2 1 6y 2 2dg

75.

5 7x3 1 2x

5 s8x3 2 x3d 1 s24x2 1 4x2d 1 s3x 2 xd

5 s8x3 2 4x2 1 3xd 1 s2x3 1 4x2 2 xd

s8x3 2 4x2 1 3xd 2 fsx3 2 4x2 1 5d 1 sx 2 5dg 5 s8x3 2 4x2 1 3xd 2 fx3 2 4x2 1 xg

77.

5 3x3 1 5x2 1 2

3s4x2 2 1d 1 s3x3 2 7x2 1 5d 5 12x2 2 3 1 3x3 2 7x2 1 5

79.

5 3t2 1 29

5 s2t2 2 5t2 1 6t2d 1 s24 2 25 1 30d

2st2 1 12d 2 5st2 1 5d 1 6st2 1 5d 5 2t2 1 24 2 5t2 2 25 1 6t2 1 30

81.

5 3v2 1 78v 1 27

5 s3v2d 1 s15v 2 9v 1 72vd 1 27

15v 2 3s3v 2 v2d 1 9s8v 1 3d 5 15v 2 9v 1 3v2 1 72v 1 27

83.

5 29s 1 8

5 s5s 2 6s 1 30sd 1 s8d

5s 2 f6s 2 s30s 1 8dg 5 5s 2 f6s 2 30s 2 8g

85. Keystrokes:

3 3 2 1

3 4 3

represent equivalent expressions since the graphs of are identical.y1 and y2y1 and y2

−10

−13

11

1

y2

y1 Y5 x xX,T,u X,T,u X,T,u> 2 22x2 x2d d1 ENTER

X,T,u > 2 X,T,u x2 2 GRAPH

87.

5 2x3 2 4x2 2 x 1 16

5 s4x3 2 5x3d 1 s23x2 2 x2d 2 x 1 s7 1 9d

5 s4x3 2 3x2 1 7d 1 s9 2 x 2 x2 2 5x3d

hsxd 5 f sxd 1 gsxd

Section 3.1 Adding and Subtracting Polynomials191

89. Polynomial Value Substitute Simplify

(a) 64 feet

(b) 60 feet

(c) 48 feet

(d) 0 feet

At time the object is at 64 feet and continues to fall, reaching the ground at time .t 5 2t 5 0,

216s2d2 1 64t 5 2

216s1d2 1 64t 5 1

216s12d2

1 64t 512

216s0d2 1 64t 5 0hstd 5 216t2 1 64

91. Polynomial Value Substitute Simplify

(a) 50 feet

(b) 146 feet

(c) 114 feet

(d) 50 feet

At time the object is at a height of 50 feet. The object moves upward, reaches a maximum height and returns down-ward. At time , object is again at a height of 50 feet.t 5 5

t 5 0,

216s5d2 1 80s5d 1 50t 5 5

216s4d2 1 80s4d 1 50t 5 4

216s2d2 1 80s2d 1 50t 5 2

216s0d2 1 80s0d 1 50t 5 0hstd 5 216t2 1 80t 1 50

93. The free-falling object was dropped.

feet216s0d2 1 100 5 100

95. The free-falling object was thrown downward.

feet216s0d2 2 24s0d 1 50 5 50

97. feet

feet

feet h 5 216s3d2 1 40s3d 1 200 5 176

h 5 216s2d2 1 40s2d 1 200 5 216

h 5 216s1d2 1 40s1d 1 200 5 224

99. Verbal model:

Equation:

P 5 $15,000

P 5 6s5000d 2 15,000

P 5 6x 2 15,000

P 5 14x 2 s8x 1 15,000d

P 5 R 2 C

Cost2Revenu5Profit

101.

5 14x 1 8

5 4x 1 8 1 4x 1 6x

Perimeter of region 5 2s2x 1 4d 1 4x 1 2s3xd 103.

5 36x or 36x

5 9x 1 27x or 6f122 xg

Area of region 5 s6 ? 32xd 1 s6 ? 9

2xd or 6 ? f32x 1

92xg

105.

5 5x 1 72

5 12x 1 72 2 7x

Area 5 12sx 1 6d 2 7x


107. (a) Verbal model:

Equation:

(b) Keystrokes:

59.89 11.4141 .42

No, this model was increasing over the interval 6 ≤ t ≤ 13.6.

10016

150

6

5 20.42t2 1 11.424t 1 59.89

5 59.89 1 11.424t 2 0.42t2

5 s231.06 2 171.17d 1 s0.009t 1 11.415td 1 s20.095t2 2 0.325t2d

y 5 231.06 1 0.009t 2 0.095t2 1 s2171.17 1 11.415t 2 0.325t2d

y 5 s231.06 1 0.009t 2 0.095t2d 2 s171.17 2 11.415t 1 0.325t2d

Per capitaconsumption

of wholemilk

2

Per capitaconsumption

of allbeverage

milks

5

Per capitaconsumptionof all bever-

agemilks otherthan whole

Y5 X,T,u 2 X,T,u x2 GRAPH

109. The degree of the term is k. The term of highest degree in a polynomial has the same degree as the polynomial.axk

111. 8x2 2 3x2 5 s8 2 3dx2 5 5x2

113. No, not every trinomial is a second-degree polynomial. For example, is a trinomial of third-degree.x3 1 2x 1 3

Section 3.2 Multiplying Polynomials

1. t3 ? t 4 5 st ? t ? tdst ? t ? t ? td 5 t314 5 t7 3.

5 s25d5x5 5 23125x5

5 25 ? 25 ? 25 ? 25 ? 25 ? x ? x ? x ? x ? x

s25xd5 5 25x ? 25x ? 25x ? 25x ? 25x

5.

5 u8

5 u414

su4d2 5 u4 ? u4 7.x6

x4 5x ? x ? x ? x ? x ? x

x ? x ? x ? x5 x624 5 x2

9. 1y52

4

5y5

?y5

?y5

?y5

5y ? y ? y ? y5 ? 5 ? 5 ? 5

5y4

54 5y4

625

1

Section 3.2 Multiplying Polynomials 193

11. (a) 23x3 ? x5 5 23sx3 ? x5d 5 23x315 5 23x8 (b) s23xd2 ? x5 5 9x2 ? x5 5 9x215 5 9x7

13. (a) s25z2d3 5 s25d3 ? sz2d3 5 2125z2?3 5 2125z6 (b) s25z4d2 5 s25d2sz4d2 5 25z4?2 5 25z8

15. (a) su3vds2v2d 5 2 ? u3 ? v112 5 2u3v3 (b) s24u4dsu5vd 5 24 ? u415 ? v 5 24u9v

17. (a) 5u2 ? s23u6d 5 5 ? 23 ? u2 ? u6 5 215u216 5 215u8 (b) s2ud4s4ud 5 24u4 ? 4u 5 16 ? 4 ? u411 5 64u5

19. (a)

5 2m1514 ? n314 5 2m19n7

5 2m15n3 ? m4n4

2sm5nd3s2m2n2d2 5 2m5?3n3 ? m2?2n2?2 (b) s2m5ndsm2n2d 5 2m512n112 5 2m7n3

21. (a)

5 3m4n3

5 3 ? m521 ? n623

27m5n6

9mn3 5279

?m5

m?

n6

n3 (b)

5 3m2n3

5 3 ? m321 ? n623

218m3n6

26mn3 521826

?m3

m?

n6

n3

23. (a)

59x2

16y2

13x4y2

2

532 ? x2

42 ? y2 (b)

5125u3

27v3

15u3v2

3

553 ? u3

33 ? v3

25. (a)

5 3x4y

527x622 y322

9

5 2s227dx6y3

9x2y2

2s23x2yd3

9x2y2 5 2s23d3sx2d3y3

9x2y2 (b)

5 22x2y4

3

5 22x2 y622

3

5 24x2y6

6y2

2s22xy3d2

6y2 5 2s22d2x2sy3d2

6y2

27. (a)

5254

u8v2

5 352

u4v42

5 352

? u622 ? v22142

5 32510

?u6

u2 ?v2

v 42

5 325u6v2

10u2v 42

3s25u3vd2

10u2v 42

5 3s25d2 ? su3d2 ? svd2

10u2v 42

(a)

514

u8v2

5 3212

u4v42

5 3212

? u622 ? v22142

5 32510

?u6

u2 ?v2

v 42

5 325u6v2

10u2v 42

325su3vd2

10u2v 42

5 325 ? su3d2 ? svd2

10u2v 42


29. (a)

(b) x6nyn27

x4n12y5 5 x6n2s4n12dyn2725 5 x6n24n22yn212 5 x2n22yn212

x2n14y4n

x5y2n11 5 x2n1425y4n2s2n11d 5 x2n21y4n22n21 5 x2n21y2n21

31. s22a2ds28ad 5 s22ds28da2 ? a 5 16a211 5 16a3 33. 2ys5 2 yd 5 s2yds5d 2 s2ydsyd 5 10y 2 2y2

35. 4xs2x2 2 3x 1 5d 5 s4xds2x2d 2 s4xds3xd 1 s4xds5d 5 8x3 2 12x2 1 20x

37. 22x2s5 1 3x2 2 7x3d 5 s22x2ds5d 1 s22x2ds3x2d 2 s22x2ds7x3d 5 210x2 2 6x4 1 14x5

39. 5 2x7 1 2x6 2 5x4 1 6x3 2x3sx4 2 2x3 1 5x 2 6d 5 2x3sx4d 2 x3s22x3d 2 x3s5xd 2 x3s26d

41. 23xs25xds5x 1 2d 5 s23xds25xds5x 1 2d 5 15x2s5x 1 2d 5 15x2s5xd 1 15x2s2d 5 75x3 1 30x2

43. u2vs3u4 2 5u2 1 6uv3d 5 u2vs3u4d 1 u2vs25u2d 1 u2vs6uv3d 5 3u6v 2 5u4v 1 6u3v4

45. sx 1 2dsx 1 4d 5 x2 1 4x 1 2x 1 8 5 x2 1 6x 1 8

47. sx 2 6dsx 1 5d 5 x2 1 5x 2 6x 2 30 5 x2 2 x 2 30

49. sx 2 4dsx 2 4d 5 x2 2 4x 2 4x 1 16 5 x2 2 8x 1 16

51. s2x 2 3dsx 1 5d 5 2x2 1 10x 2 3x 2 15 5 2x2 1 7x 2 15

53. s5x 2 2ds2x 2 6d 5 10x2 2 30x 2 4x 1 12 5 10x2 2 34x 1 12

55. s8 2 3x2ds4x 1 1d 5 32x 1 8 2 12x3 2 3x2 5 212x3 2 3x2 1 32x 1 8

57. s4y 213ds12y 1 9d 5 48y2 1 36y 2 4y 2 3 5 48y2 1 32y 2 3

59. s2x 1 yds3x 1 2yd 5 6x2 1 4xy 1 3xy 1 2y2 5 6x2 1 7xy 1 2y2

61. s2t 2 1dst 1 1d 1 1s2t 2 5dst 2 1d 5 2t2 1 2t 2 t 2 1 1 2t2 2 2t 2 5t 1 5 5 4t2 2 6t 1 4

63.

5 x3 2 5x2 1 10x 2 6

5 x3 2 x2 2 4x2 1 4x 1 6x 2 6

sx 2 1dsx2 2 4x 1 6d 5 sx 2 1dsx2d 1 sx 2 1ds24xd 1 sx 2 1ds6d

65.

5 3a3 1 11a2 1 9a 1 2

5 3a3 1 2a2 1 9a2 1 6a 1 3a 1 2

s3a 1 2dsa2 1 3a 1 1d 5 s3a 1 2dsa2d 1 s3a 1 2ds3ad 1 s3a 1 2ds1d


67.

5 8u3 1 22u2 2 u 2 20

5 8u3 1 10u2 1 12u2 1 15u 2 16u 2 20

s2u2 1 3u 2 4ds4u 1 5d 5 s4u 1 5ds2u2d 1 s4u 1 5ds3ud 1 s4u 1 5ds24d

69.

5 x4 2 2x3 2 3x2 1 8x 2 4

5 x4 2 2x3 2 3x2 1 6x 1 2x 2 4

sx3 2 3x 1 2dsx 2 2d 5 x3sx 2 2d 1 s23xdsx 2 2d 1 2sx 2 2d

71.

5 5x4 1 20x3 2 3x2 1 8x 2 2

5 5x4 1 2x2 1 20x3 1 8x 2 5x2 2 2

s5x2 1 2dsx2 1 4x 2 1d 5 s5x2 1 2dsx2d 1 s5x2 1 2ds4xd 1 s5x2 1 2ds21d

73.

5 t 4 2 t2 1 4t 2 4

5 t 4 2 t3 1 2t2 1 t3 2 t2 1 2t 2 2t2 1 2t 2 4

st2 1 t 2 2dst2 2 t 1 2d 5 t2st2 2 t 1 2d 1 tst2 2 t 1 2d 2 2st2 2 t 1 2d

75.

28x5 2 56x4 1 36x3 1 21x2 2 42x 1 27

28x5 2 56x4 1 36x3

1 21x2 2 42x 1 27

4x3 1 3

7x2 2 14x 1 9 77.

2u3 1 u2 2 7u 2 6

2u3 1 5u2 1 3u

24u2 2 10u 2 6

u 2 2

2u2 1 5u 1 3

79.

22x3 1 3x2 2 1

22x3 1 4x2 2 2x

2x2 1 2x 2 1

2x 1 1

2x2 1 2x 2 1 81.

t 4 2 t2 1 4t 2 4

t 4 1 t3 2 2t2

2 t3 2 t2 1 2t

1 2t2 1 2t 2 4

t2 2 t 1 2

t2 1 t 2 2

83.

5 x2 2 4

sx 1 2dsx 2 2d 5 sxd2 2 s2d2 85.

5 x2 2 49

sx 2 7dsx 1 7d 5 sxd2 2 s7d2

87. s2 1 7yds2 2 7yd 5 s2d2 2 s7yd2 5 4 2 49y2 89. s6 2 4xds6 1 4xd 5 s6d2 2 s4xd2 5 36 2 16x2

91. s2a 1 5bds2a 2 5bd 5 s2ad2 2 s5bd2 5 4a2 2 25b2 93. s6x 2 9yds6x 1 9yd 5 s6xd2 2 s9yd2 5 36x2 2 81y2

95. s2x 214ds2x 1

14d 5 s2xd2 2 s1

4d25 4x2 2

116 97.

5 0.04t2 2 0.25

s0.2t 1 0.5ds0.2t 2 0.5d 5 s0.2td2 2 s0.5d2

99. sx 1 5d2 5 sxd2 1 2sxds5d 1 s5d2 5 x2 1 10x 1 25 101. sx 2 10d2 5 sxd2 2 2sxds10d 1 102 5 x2 2 20x 1 100


103. s2x 1 5d2 5 s2xd2 1 2s2xds5d 1 s5d2 5 4x2 1 20x 1 25 105. s6x 2 1d2 5 s6xd2 2 2s6xds1d 1 s1d2 5 36x2 2 12x 1 1

107. s2x 2 7yd2 5 s2xd2 2 2s2xds7yd 1 s7yd2 5 4x2 2 28xy 1 49y2

109. fsx 1 2d 1 yg2 5 sx 1 2d2 1 2sx 1 2dy 1 y2 5 sxd2 1 2sxds2d 1 s2d2 1 2xy 1 4y 1 y2 5 x2 1 4x 1 4 1 2xy 1 4y 1 y2

111. fu 2 sv 2 3dgfu 1 sv 2 3dg 5 sud2 2 sv 2 3d2 5 u2 2 fv2 2 2svds3d 1 s3d2g 5 u2 2 sv2 2 6v 1 9d 5 u2 2 v2 1 6v 2 9

113.

x3 1 9x2 1 27x 1 27

x3 1 6x2 1 9x

3x2 1 18x 1 27

x 1 3

x2 1 6x 1 9

5 sx2 1 6x 1 9dsx 1 3d

5 sx2 1 3x 1 3x 1 9dsx 1 3d

sx 1 3d3 5 sx 1 3dsx 1 3dsx 1 3d 115.

u3 1 3u2v 1 3uv2 1 v3

u3 1 2u2v 1 uv2

u2v 1 2uv2 1 v3

u 1 v

u2 1 2uv 1 v2

5 su2 1 2uv 1 v2dsu 1 vd

5 su2 1 uv 1 uv 1 v2dsu 1 vd

su 1 vd3 5 su 1 vdsu 1 vdsu 1 vd

117. Keystrokes:

1 2

3 2

y1 5 y2 because sx 1 1dsx2 2 x 1 2d 5 x3 2 x2 1 2x 1 x2 2 x 1 2 5 x3 1 x 1 2

−9

−4

9

8

y2

y1 Y5 x xX,T,u X,T,u X,T,u1 d x2 2 1 d ENTER

X,T,u X,T,u> 1 1 GRAPH

119. Keystrokes:

2 3 2

2 6

y1 5 y2 because s2x 2 3dsx 1 2d 5 2x2 1 4x 2 3x 2 6 5 2x2 1 x 2 6

−12

−8

12

8

y2

y1 Y5 x X,T,u X,T,u2 d x 1 d ENTER

X,T,u X,T,ux2 1 2 GRAPH

121. (a)

5 t2 2 8t 1 15

5 t2 2 6t 1 9 2 2t 1 6

f st 2 3d 5 st 2 3d2 2 2st 2 3d (b)

5 2h 1 h2

5 s4 1 4h 1 h2 2 4 2 2hd 2 s0d

f s2 1 hd 2 f s2d 5 fs2 1 hd2 2 2s2 1 hdg 2 f22 2 2s2dg



Function:

(b)

cubic inches

(c) Verbal model:

Function:

(d) Function:

5 n2 1 10n 1 24

5 sn 1 4dsn 1 6d

Asn 1 4d 5 sn 1 4dsn 1 4 1 2d

5 n2 1 10n 1 24 5 Asn 1 4d

5 n2 1 6n 1 4n 1 24

5 sn 1 4dsn 1 6d

Area 5 sn 1 4dsn 1 2 1 4d

5 n2 1 2n

Asnd 5 n ? sn 1 2d

Width?Length5Area

5 48

5 2s4ds6d

Vs2d 5 2 ? s2 1 2d ? s2 1 4d

5 n3 1 6n2 1 8n

5 nsn2 1 6n 1 8d

Vsnd 5 n ? sn 1 2d ? sn 1 4d


125. Verbal model:

Function:

5 8x2 1 26x

5 9x2 1 30x 2 x2 2 4x

Asxd 5 3xs3x 1 10d 2 xsx 1 4d

Area ofInside

Rectangle2

Area ofOutside

Rectangle5

Area ofShadedRegion

127. Verbal model:

Function:

5 1.2x2

5 1.6x2 2 0.4x2

Asxd 512s2xds1.6xd 2

12sxds0.8xd

Area ofSmallerTriangle

2

Area ofLarger

Triangle5

Area ofShadedRegion

129. (a) Verbal model: 2 2 (b) Verbal model: Width?Length5AreaWidth1Length5Perimeter

w

w32

P 5 5w

5 3w 1 2w

P 5 2s32wd 1 2w

A 532w2

A 5 s32wdswd


131.

5 1000 1 2000r 1 1000r2

5 1000s1 1 2r 1 r2d

5 1000s1 1 rds1 1 rd

Interest 5 1000s1 1 rd2 133.


Formula: sx 1 adsx 1 bd 5 x2 1 ax 1 bx 1 ab.

5 x2 1 ax 1 bx 1 ab

Area 5 sx ? xd 1 sx ? ad 1 sx ? bd 1 sa ? bd

5 x2 1 ax 1 bx 1 ab

5 sx 1 adsx 1 bd

Area 5 l ? w

135. (a)

(b)

(c)

sx 2 1dsx4 1 x3 1 x2 1 x 1 1d 5 x5 2 1

sx 2 1dsx3 1 x2 1 x 1 1d 5 x4 1 x3 1 x2 1 x 2 x3 2 x2 2 x 2 1 5 x4 2 1

sx 2 1dsx2 1 x 1 1d 5 x3 1 x2 1 x 2 x2 2 x 2 1 5 x3 2 1

sx 2 1dsx 1 1d 5 x2 2 1


Labels: Volume

Length

Width

Height

Function:

(b) Verbal model:

Labels: Volume

Area of base

Height

Function:

(c) Verbal model:

Function:

5 73x3 1

533 x2 2 20x 1 6

5 3x3 1 13x2 2 10x 223x3 1

143 x2 2 10x 1 6

VSsxd 5 s3x3 1 13x2 2 10xd 2 s23x3 2

143 x2 1 10x 2 6d

Volumeof pyramid2

Volumeof bin5

Volumeof grain

5 23x3 2

143 x2 1 10x 2 6

5 13s2x3 2 14x2 1 30x 2 18d

5 13s2x3 2 8x2 1 6x 2 6x2 1 24x 2 18d

5 13s2x2 2 8x 1 6dsx 2 3d

VPsxd 513s2x 2 6dsx 2 1dsx 2 3d

5 x 2 3

5 s2x 2 6dsx 2 1d

5 VPsxd

Height?Area of base13

?5Volume

5 3x3 1 13x2 2 10x

5 s3x2 1 13x 2 10dx

5 s3x2 1 15x 2 2x 2 10dx

VBsxd 5 s3x 2 2dsx 1 5dx

5 x

5 x 1 5

5 3x 2 2

5 VBsxd


Section 3.3 Factoring Polynomials 199

139. s2xd3 5 23 ? x3 5 8x3 Þ 2x3 141.

L 5 Last

I 5 Inner

O 5 Outer

F 5 First

143. (a) True, the product of two monomials is a monomial.

(b) False, the product of two binomials can be a trinomial. For example, sx 1 2dsx 2 3d 5 x2 2 x 2 6.

Section 3.3 Factoring Polynomials

1.

GCF 5 2 ? 3 5 6

96 5 25 ? 3

90 5 5 ? 2 ? 32

48 5 24 ? 3 3.

GCF 5 3x

12x 5 22 ? 3 ? x

3x2 5 3 ? x ? x 5.

5 6z2

GCF 5 2 ? 3 ? z ? z

212z3 5 21 ? 22 ? 3 ? z ? z ? z

30z2 5 2 ? 3 ? 5 ? z ? z

7.

5 14b2

GCF 5 7 ? 2 ? b ? b

42b5 5 7 ? 2 ? 3 ? b ? b ? b ? b ? b

14b3 5 7 ? 2 ? b ? b ? b

28b2 5 7 ? 22 ? b ? b 9.

5 21sx 1 8d2

GCF 5 7 ? 3sx 1 8d2

63sx 1 8d3 5 7 ? 32 ? sx 1 8d3

42sx 1 8d2 5 7 ? 3 ? 2 ? sx 1 8d2

11. 8z 2 8 5 8sz 2 1d 13. 4u 1 10 5 2s2u 1 5d 15. 24x2 2 18 5 6s4x2 2 3d

17. 2x2 1 x 5 xs2x 1 1d 19. 21u2 2 14u 5 7us3u 2 2d 21. is prime (No commonfactor other than 1.)11u2 1 9

23. 28x2 1 16x 2 8 5 4s7x2 1 4x 2 2d 25. 3x2y2 2 15y 5 3ysx2y 2 5d

27. 15xy2 2 3x2y 1 9xy 5 3xys5y 2 x 1 3d 29. 14x4y3 1 21x3y2 1 9x2 5 x2s14x2y3 1 21xy2 1 9d

31. 10 2 x 5 21s210 1 xd 5 21sx 2 10d 33. 7 2 14x 5 27s21 1 2xd 5 27s2x 2 1d

35. 16 1 4x 2 6x2 5 21s216 2 4x 1 6x2d 5 21s6x2 2 4x 2 16d 5 22s3x2 2 2x 2 8d

37. y 2 3y3 2 2y2 5 21s2y 1 3y3 1 2y2d 5 21s3y3 1 2y2 2 yd 5 2ys3y2 1 2y 2 1d

39. 2y 235 5

15s10y 2 3d 41. 3

2x 154 5

14s6x 1 5d

43. 2ysy 2 3d 1 5sy 2 3d 5 sy 2 3ds2y 1 5d 45. 5xs3x 1 2d 2 3s3x 1 2d 5 s3x 1 2ds5x 2 3d

47. 2s7a 1 6d 2 3a2s7a 1 6d 5 s7a 1 6ds2 2 3a2d 49. 8t3s4t 2 1d2 1 3s4t 2 1d2 5 s4t 2 1d2s8t3 1 3d


51. sx 2 5ds4x 1 9d 2 s3x 1 4ds4x 1 9d 5 s4x 1 9dsx 2 5 2 3x 2 4d 5 s4x 1 9ds22x 2 9d

53. x2 1 25x 1 x 1 25 5 sx2 1 25xd 1 sx 1 25d 5 xsx 1 25d 1 1sx 1 25d 5 sx 1 25dsx 1 1d

55.

5 sy 2 6dsy 1 2d

5 ysy 2 6d 1 2sy 2 6d

y2 2 6y 1 2y 2 12 5 sy2 2 6yd 1 s2y 2 12d 57.

5 sx 1 2dsx2 1 1d

5 x2sx 1 2d 1 1sx 1 2d

x3 1 2x2 1 x 1 2 5 sx3 1 2x2d 1 sx 1 2d

59.

5 sa 2 4ds3a2 2 2d

5 3a2sa 2 4d 2 2sa 2 4d

3a3 2 12a2 2 2a 1 8 5 s3a3 2 12a2d 1 s22a 1 8d 61.

5 sz3 2 2dsz 1 3d

5 zsz3 2 2d 1 3sz3 2 2d

z422z 1 3z3 2 6 5 sz4 2 2zd 1 s3z3 2 6d

63.

5 sx 2 2yds5x2 1 7y2d

5 5x2sx 2 2yd 1 7y2sx 2 2yd

5x3 2 10x2y 1 7xy2 2 14y3 5 s5x3 2 10x2yd 1 s7xy2 2 14y3d

65.

5 sx 2 8dsx 1 8d

x2 2 64 5 x2 2 82 67.

5 s1 2 ads1 1 ad

1 2 a2 5 12 2 a2 69.

5 s4y 2 3ds4y 1 3d

16y2 2 9 5 s4yd2 2 32

71.

5 s9 2 2xds9 1 2xd

81 2 4x2 5 92 2 s2xd2 73. 4z2 2 y2 5 s2z 2 yds2z 1 yd 75.

5 s6x 2 5yds6x 1 5yd

36x2 2 25y2 5 s6xd2 2 s5yd2

77.

5 su 214dsu 1

14d

u2 21

16 5 u2 2 s14d2

79.

5 s23x 2

45yds2

3x 145yd

49x2 21625y2 5 s2

3xd22 s4

5yd2

81. sx 2 1d2 2 16 5 fsx 2 1d 2 4gfsx 2 1d 1 4g 5 sx 2 5dsx 1 3d

83. 81 2 sz 1 5d2 5 92 2 sz 1 5d2 5 f9 2 sz 1 5dgf9 1 sz 1 5dg 5 f9 2 z 2 5gf9 1 z 1 5g 5 s4 2 zds14 1 zd

85. s2x 1 5d2 2 sx 2 4d2 5 fs2x 1 5d 2 sx 2 4dgfs2x 1 5d 1 sx 2 4dg 5 f2x 1 5 2 x 1 4gf2x 1 5 1 x 2 4g 5 sx 1 9ds3x 1 1d

87.

5 sx 2 2dsx2 1 2x 1 4d

x3 2 8 5 x3 2 23 89.

5 sy 1 4dsy2 2 4y 1 16d

y3 1 64 5 y3 1 43 91.

5 s2t 2 3ds4t2 1 6t 1 9d

8t3 2 27 5 s2td3 2 33

93.

5 s3u 1 1ds9u2 2 3u 1 1d

27u3 1 1 5 s3ud3 1 13 95.

5 s4a 1 bds16a2 2 4ab 1 b2d

64a3 1 b3 5 s4ad3 1 b3

97.

5 sx 1 3ydsx2 2 3xy 1 9y2d

x3 1 27y3 5 x3 1 s3yd3 99.

5 2f2 2 5xgf2 1 5xg

5 2f22 2 s5xd2g

8 2 50x2 5 2s4 2 25x2d

Section 3.3 Factoring Polynomials 201

101.

5 8sx 1 2dsx2 2 2x 1 4d

5 8sx3 1 23d

8x3 1 64 5 8sx3 1 8d 103.

5 sy 2 3dsy 1 3dsy2 1 9d

5 sy2 2 9dsy2 1 9d

y4 2 81 5 sy2d2 2 92

105.

5 3x2sx 2 10dsx 1 10d

3x4 2 300x2 5 3x2sx2 2 100d 107.

5 6sx2 2 2y2dsx4 1 2x2y2 1 4y4d

5 6fsx2d3 2 s2y2d3g

6x6 2 48y6 5 6sx6 2 8y6d

109.

5 s2xn 2 5ds2xn 1 5d

4x2n 2 25 5 s2xnd2 2 52

111. Keystrokes:

3 6

3 2

y1 5 y2

y2

y1 Y5 X,T,u 2 ENTER

x X,T,u 2 d GRAPH

−6

−8

12

4

113. Keystrokes:

4

2 2

y1 5 y2

y2

y1

x xX,T,u X,T,u1 d

Y5 X,T,u x2 2 ENTER

2 d GRAPH

−6

−5

6

3

115.

or

5 sx 2 1dsx 1 1ds3x 1 4d

5 sx2 2 1ds3x 1 4d

5 x2s3x 1 4d 2 1s3x 1 4d

3x3 1 4x2 2 3x 2 4 5 s3x3 1 4x2d 1 s23x 2 4d

5 sx 2 1dsx 1 1ds3x 1 4d

5 sx2 2 1ds3x 1 4d

5 3xsx2 2 1d 1 4sx2 2 1d

5 s3x3 2 3xd 1 s4x2 2 4d

117.

p 5 800 2 0.25x

R 5 xp

5 xs800 2 0.25xd

R 5 800x 2 0.25x2 119. P 1 Prt 5 Ps1 1 rtd 121.

so

w 5 45 2 l

5 ls45 2 ld

A 5 45l 2 l2

123.

S 5 2xsx 1 2hd

S 5 2x2 1 4xh 125.

5 psR 2 rdsR 1 rd

5 psR2 2 r2d

A 5 pR2 2 pr2 127. A polynomial is in factored formwhen the polynomial is written asa product of polynomials.

129. The method of finding the greatest common factor of two or more integers is first determine the prime factorization of eachinteger. Then the greatest common factor is the product of each common prime factor raised to its lowest power in either oneof the integers.

131. The Distributive Property is used to factor a polynomial in this example x2 1 2x 5 xsx 1 2d.



1. degree

leading coefficient5 22

5 4 2. is not a polynomial because the termhas degree The degree of the variable x is not

an integer.

12.2 3x1/2

2x 2 3x1/2 1 5

3. s2t3 1 3t2 2 2d 1 st3 1 9d 5 3t3 1 3t2 1 7 4. s3 2 7yd 1 s7y2 1 2y 2 3d 5 7y2 2 5y

5.

5 9x3 2 4x2 1 1

s7x3 2 3x2 1 1d 2 sx2 2 2x3d 5 7x3 2 3x2 1 1 2 x2 1 2x3 6.

5 2u2 2 u 1 1

5 5 2 u 2 4 1 2u2

5 s5 2 ud 2 2f2 2 u2g

s5 2 ud 2 2f3 2 su2 1 1dg 5 s5 2 ud 2 2f3 2 u2 2 1g

7. s25n2ds22n3d 5 10n5 8.

5 28x10

5 28x6 ? x4

s22x2d3sx4d 5 s22d3sx2d3sx4d 9.

5 23x4

6x7

s22x2d3 56x7

28x6

10.

516y4

25x2

14y2

5x 2 5 14y2

5x 214y2

5x 211. 7ys4 2 3yd 5 28y 2 21y2

12.

5 x2 2 4x 2 21

sx 2 7dsx 1 3d 5 x2 1 3x 2 7x 2 21 13.

5 24x2 2 26xy 1 5y2

s4x 2 yds6x 2 5yd 5 24x2 2 20xy 2 6xy 1 5y2

14.

5 2z2 1 3z 2 35

2zsz 1 5d 2 7sz 1 5d 5 2z2 1 10z 2 7z 2 35 15. s6r 1 5ds6r 2 5d 5 36r2 2 25

16.

5 4x2 2 12x 1 9

s2x 2 3d2 5 s2x 2 3ds2x 2 3d 17.

5 x3 1 1

sx 1 1dsx2 2 x 1 1d 5 x3 2 x2 1 x 1 x2 2 x 1 1

18.

5 x4 1 2x3 2 23x2 1 40x 2 20

5 x4 1 5x3 2 10x2 2 3x3 2 15x2 1 30x 1 2x2 1 10x 2 20

sx2 2 3x 1 2dsx2 1 5x 2 10d 5 x2sx2 1 5x 2 10d 2 3xsx2 1 5x 2 10d 1 2sx2 1 5x 2 10d

19. 28a2 2 21a 5 7as4a 2 3d 20. 25 2 4x2 5 s5 2 2xds5 1 2xd

21.

5 sz 1 3d2sz 2 3d

5 sz 1 3dsz 1 3dsz 2 3d

5 sz 1 3dsz2 2 9d

z3 1 3z2 2 9z 2 27 5 z2sz 1 3d 2 9sz 1 3d 22.

5 4sy 2 2xdsy2 1 2xy 1 4x2d

5 4fy3 2 s2xd3g

4y3 2 32x3 5 4sy3 2 8x3d

Section 3.4 Factoring Trinomials 203

23.

s5x 1 5ds2x 1 2d

s5x 1 2ds2x 1 5d

s5x 1 1ds2x 1 10d

s5x 1 10ds2x 1 1d

s5x 2 5ds2x 2 2d

s5x 2 2ds2x 2 5d

s5x 2 1ds2x 2 10d

s5x 2 10ds2x 2 1d

24. Verbal model:

Equation:

5 2x 1 2

5 12x2 1 2x 1 2 2

12x2

5 12sx2 1 4x 1 4d 2

12x2

A 512sx 1 2d2 2

12x2

Area ofsmall

triangle2

Area oflarge

triangle5

Area ofshadedregion

25.

5 26 feet

5 264 1 90

5 216s4d 2 10 1 100

h s2d 5 216s2d2 2 5s2d 1 100

5 79 feet

5 216 2 5 1 100

h s1d 5 216s1d2 2 5s1d 1 100 26.

Ps1000d 5 14s1000d 2 2000 5 $12,000

5 14x 2 2000

5 19x 2 5x 2 2000

5 19x 2 s5x 1 2000d

Psxd 5 Rsxd 2 Csxd

Section 3.4 Factoring Trinomials

1. x2 1 4x 1 4 5 x2 1 2s2xd 1 22 5 sx 1 2d2 3. a2 2 12a 1 36 5 a2 2 2s6ad 1 62 5 sa 2 6d2

5. 25y2 2 10y 1 1 5 s5yd2 2 2s5yd 1 1 5 s5y 2 1d2 7. 9b2 1 12b 1 4 5 s3bd2 1 2s3bds2d 1 22 5 s3b 1 2d2

9. u2 1 8uv 1 16v2 5 u2 1 2s4uvd 1 s4vd2 5 su 1 4vd2

11. 36x2 2 60xy 1 25y2 5 s6xd2 2 2s6xds5yd 1 s5yd2 5 s6x 2 5yd2

13. 5x2 1 30x 1 45 5 5sx2 1 6x 1 9d 5 5fx2 1 2s3dsxd 1 32g 5 5sx 1 3d2

15. 2x2 1 24x2 1 72x 5 2xsx2 1 12x 1 36d 5 2xfx2 1 2s6dx 1 62g 5 2xsx 1 6d2

17. 20v4 2 60v3 1 45v2 5 5v2s4v2 2 12v 1 9d 5 5v2fs2vd2 2 2s2vds3d 1 32g 5 5v2s2v 2 3d2


19. or

5 s12x 2

23d2

14x2 223x 1

49 5 s1

2xd22 2s1

2xds23d 1 s2

3d2

5 136s3x 2 4d2

5 136fs3xd2 2 2s3xds4d 1 42g

5 136s9x2 2 24x 1 16d

5 936x2 2

2436x 1

1636

21.

(a) or

5 sx 1 9d2

x2 1 18x 1 92 5 x2 1 2s9xd 1 92

b 5 18

x2 1 bx 1 81 5 x2 1 bx 1 92

(b)

5 sx 2 9d2

x2 2 18x 1 92 5 x2 2 2s9xd 1 92

b 5 218

23.

(a) or

5 s2x 1 3d2

s2xd2 1 12x 1 32 5 s2xd2 1 2s2xds3d 1 32

b 5 12

4x2 1 bx 1 9 5 s2xd2 1 bx 1 32

(b)

5 s2x 2 3d2

s2xd2 2 12x 1 32 5 s2xd2 2 2s2xds3d 1 32

b 5 212

25.

5 sx 1 4d2

5 x2 1 2s4xd 1 42

x2 1 8x 1 c 5 x2 1 2s4xd 1 c

c 5 16 27.

5 sy 2 3d2

5 y2 2 2s3yd 1 32

y2 2 6y 1 c 5 y2 2 2s3yd 1 c

c 5 9

29. x2 1 5x 1 4 5 sx 1 4dsx 1 1d 31. y2 2 y 2 20 5 sy 1 4dsy 2 5d 33. x2 2 2x 2 24 5 sx 1 4dsx 2 6d

35. z2 2 6z 1 8 5 sz 2 4dsz 2 2d 37. x2 1 4x 1 3 5 sx 1 3dsx 1 1d 39. x2 2 5x 1 6 5 sx 2 3dsx 2 2d

41. y2 1 7y 2 30 5 sy 1 10dsy 2 3d 43. t2 2 4t 2 21 5 st 2 7dst 1 3d 45. x2 2 20x 1 96 5 sx 2 12dsx 2 8d

47. x2 2 2xy 2 35y2 5 sx 2 7ydsx 1 5yd 49. x2 1 30xy 1 216y2 5 sx 1 12ydsx 1 18yd

51.

x2 2 11x 1 18 5 sx 2 9dsx 2 2db 5 211:

x2 1 11x 1 18 5 sx 1 9dsx 1 2db 5 11:

x2 2 9x 1 18 5 sx 2 6dsx 2 3db 5 29:

x2 1 9x 1 18 5 sx 1 6dsx 1 3db 5 9:

x2 2 19x 1 18 5 sx 2 18dsx 2 1db 5 219:

x2 1 19x 1 18 5 sx 1 18dsx 1 1db 5 19: 53.

x2 2 4x 2 21 5 sx 2 7dsx 1 3db 5 24:

x2 1 4x 2 21 5 sx 1 7dsx 2 3db 5 4:

x2 2 20x 2 21 5 sx 2 21dsx 1 1db 5 220:

x2 1 20x 2 21 5 sx 1 21dsx 2 1db 5 20:

55.

x2 2 12x 1 35 5 sx 2 7dsx 2 5db 5 212:

x2 1 12x 1 35 5 sx 1 7dsx 1 5db 5 12:

x2 2 36x 1 35 5 sx 2 35dsx 2 1db 5 236:

x2 1 36x 1 35 5 sx 1 35dsx 1 1db 5 36:

Section 3.4 Factoring Trinomials 205

57. There are many possibilities, such as:

Also note that if a negative number, there are manypossibilities for c such as the following.

x2 1 6x 2 27 5 sx 1 9dsx 2 3dc 5 227

x2 1 6x 2 16 5 sx 1 8dsx 2 2dc 5 216

x2 1 6x 2 7 5 sx 1 7dsx 2 1dc 5 27

c 5

x2 1 6x 1 9 5 sx 1 3dsx 1 3dc 5 9

x2 1 6x 1 8 5 sx 1 4dsx 1 2dc 5 8

x2 1 6x 1 5 5 sx 1 5dsx 1 1dc 5 5

59. There are many possibilities, such as:

There are more possibilities.

x2 2 3x 2 18 5 sx 2 6dsx 1 3dc 5 218

x2 2 3x 2 10 5 sx 2 5dsx 1 2dc 5 210

x2 2 3x 2 4 5 sx 2 4dsx 1 1dc 5 24

x2 2 3x 1 2 5 sx 2 2dsx 2 1dc 5 2

61. 5x2 1 18x 1 9 5 sx 1 3ds5x 1 3d 63. 5a2 1 12a 2 9 5 sa 1 3ds5a 2 3d 65. 2y2 2 3y 2 27 5 sy 1 3ds2y 2 9d

67. 3x2 1 4x 1 1 5 s3x 1 1dsx 1 1d 69. 7x2 1 15x 1 2 5 s7x 1 1dsx 1 2d 71. 2x2 2 9x 1 9 5 s2x 2 3dsx 2 3d

73. 6x2 2 11x 1 3 5 s3x 2 1ds2x 2 3d 75. 3t2 2 4t 2 10 5 prime 77. 6b2 1 19b 2 7 5 s3b 2 1ds2b 1 7d

79. 18y2 1 35y 1 12 5 s2y 1 3ds9y 1 4d 81. 22x2 2 x 1 6 5 21s2x2 1 x 2 6d 5 21s2x 2 3dsx 1 2d

83.

5 21s15x 2 1ds4x 1 1d

5 21s60x2 1 11x 2 1d

1 2 11x 2 60x2 5 260x2 2 11x 1 1 85.

5 3s2x 1 7dsx 2 4d

6x2 2 3x 2 84 5 3s2x2 2 x 2 28d

87.

5 5ys3y 2 2ds4y 1 5d

60y3 1 35y2 2 50y 5 5ys12y2 1 7y 2 10d 89. 10a2 1 23ab 1 6b2 5 sa 1 2bds10a 1 3bd

91. 24x2 2 14xy 2 3y2 5 s6x 1 yds4x 2 3yd 93.

5 s3x 1 4dsx 1 2d

5 3xsx 1 2d 1 4sx 1 2d

5 s3x2 1 6xd 1 s4x 1 8d

3x2 1 10x 1 8 5 3x2 1 6x 1 4x 1 8

95.

5 s2x 2 1ds3x 1 2d

5 2xs3x 1 2d 2 1s3x 1 2d

5 s6x2 1 4xd 1 s23x 2 2d

6x2 1 x 2 2 5 6x2 1 4x 2 3x 2 2 97.

5 s3x 2 1ds5x 2 2d

5 3xs5x 2 2d 2 1s5x 2 2d

5 s15x2 2 6xd 1 s25x 1 2d

15x2 2 11x 1 2 5 15x2 2 6x 2 5x 1 2

99. 3x4 2 12x3 5 3x3sx 2 4d 101.

5 2ts5t 2 9dst 1 2d

10t3 1 2t2 2 36t 5 2ts5t2 1 t 2 18d

103.

5 2s3x 2 1ds9x2 1 3x 1 1d

54x3 2 2 5 2s27x3 2 1d 105.

5 9ab2s3ab 1 2dsab 2 1d

27a3b4 2 9a2b3 2 18ab2 5 9ab2s3a2b2 2 ab 2 2d


107.

5 sx 1 2dsx 2 4dsx 1 4d

5 sx 1 2dsx2 2 16d

5 x2sx 1 2d 2 16sx 1 2d

x3 1 2x2 2 16x 2 32 5 sx3 1 2x2d 1 s216x 2 32d 109.

5 s3 2 zds9 1 zd

5 f6 2 z 2 3gf6 1 z 1 3g

36 2 sz 1 3d2 5 f6 2 sz 1 3dgf6 1 sz 1 3dg

111.

5 sx 2 5 1 ydsx 2 5 2 yd

5 fsx 2 5d 1 ygfsx 2 5d 2 yg

x2 2 10x 1 25 2 y2 5 sx 2 5d2 2 y2 113.

5 sx 2 1dsx 1 1dsx2 1 1dsx4 1 1d

5 sx2 2 1dsx2 1 1dsx4 1 1d

5 fsx2d2 2 12gsx4 1 1d

x8 2 1 5 sx4d2 2 12 5 sx4 2 1dsx4 1 1d

115. Keystrokes:

6 9

3

y1 5 y2

y2

y1 Y5 X,T,u X,T,ux2 1 1 ENTER

x X,T,u 1 d x2 GRAPH

−12

−2

6

10

117. Keystrokes:

2 3

1 3

y1 5 y2

y2

y1 Y5 X,T,u X,T,ux2 1 ENTER

x X,T,u d −10

−6

8

6

2

2 x X,T,u 1 d GRAPH

119. matches graph (c).a2 2 b2 5 sa 1 bdsa 2 bd 121. matches graph (b).a2 1 2ab 1 b2 5 sa 1 bd2

123. Verbal model:

Equation:

5 4s6 1 xds6 2 xd

5 4s36 2 x2d

5 144 2 4x2

Area 5 s8 ? 18d 2 4 ? x2

Area ofsquares2

Area ofrectangle5

Area ofshadedregion

125. (a)

(b)

2n 1 2 5 2s10d 1 2 5 22

2n 2 2 5 2s10d 2 2 5 18

If n 5 10, 2n 5 2s10d 5 20

5 2ns2n 2 2ds2n 1 2d

5 2nfs2nd2 2 22g

8n3 2 8n 5 2ns4n2 2 4d

127. To factor begin by finding the factors of 6 whose sum is They are The factorizationis sx 2 2dsx 2 3d.

22 and 23.25.x2 2 5x 1 6

129. Check the factors of a trinomial by multiplication. The factors of

sx 2 2dsx 2 3d 5 x2 2 5x 1 6.

x2 2 5x 1 6 are x 2 2 and x 2 3 because

131. No, is not in factored form. It is not yet a product. xsx 1 2d 2 2sx 1 2d 5 sx 1 2dsx 2 2dxsx 1 2d 2 2sx 1 2d

Section 3.5 Solving Polynomial Equations 207

Section 3.5 Solving Polynomial Equations

1.

x 5 8 x 5 0

x 2 8 5 0 2x 5 0

2xsx 2 8d 5 0 3.

y 5 210 y 5 3

y 1 10 5 0 y 2 3 5 0

sy 2 3dsy 1 10d 5 0 5.

a 5 2 a 5 24

a 2 2 5 0 a 1 4 5 0

25sa 1 4dsa 2 2d 5 0

7.

t 5 213 t 5 2

52

3t 1 1 5 0 2t 1 5 5 0

s2t 1 5ds3t 1 1d 5 0 9.

x 5 2252 x 5

32 x 5 0

2x 1 25 5 0 2x 2 3 5 0 4x 5 0

4xs2x 2 3ds2x 1 25d 5 0

11.

x 5 24 x 5 212 x 5 3

x 1 4 5 0 2x 1 1 5 0 x 2 3 5 0

sx 2 3ds2x 1 1dsx 1 4d 5 0 13.

5 5 y

5 2 y 5 0y 5 0

ys5 2 yd 5 0

5y 2 y2 5 0

15.

x 5 253 x 5 0

3x 1 5 5 0 3x 5 0

3xs3x 1 5d 5 0

9x2 1 15x 5 0 17.

x 5 10 x 5 22

x 2 10 5 0 x 1 2 5 0

sx 1 2dsx 2 10d 5 0

xsx 1 2d 2 10sx 1 2d 5 0

19.

u 5 23 u 5 3

u 1 3 5 0 u 2 3 5 0

su 2 3dsu 1 3d 5 0

usu 2 3d 1 3su 2 3d 5 0 21.

x 5 5 x 5 25

x 2 5 5 0 x 1 5 5 0

sx 1 5dsx 2 5d 5 0

x2 2 25 5 0

23.

y 5 4 y 5 24

y 2 4 5 0 y 1 4 5 0

3sy 1 4dsy 2 4d 5 0

3sy2 2 16d 5 0

3y2 2 48 5 0 25.

x 5 22 x 5 5

x 1 2 5 0 x 2 5 5 0

sx 2 5dsx 1 2d 5 0

x2 2 3x 2 10 5 0

27.

x 5 4 x 5 6

x 2 4 5 0 x 2 6 5 0

sx 2 6dsx 2 4d 5 0

x2 2 10x 1 24 5 0 29.

x 554

x 5 25 4x 5 5

x 1 5 5 0 4x 2 5 5 0

s4x 2 5dsx 1 5d 5 0

4x2 1 15x 2 25 5 0


31.

212 5 x 7 5 x

1 1 2x 5 0 7 2 x 5 0

s7 2 xds1 1 2xd 5 0

7 1 13x 2 2x2 5 0 33.

m 5 4

m 2 4 5 0

sm 2 4d2 5 0

m2 2 8m 1 16 5 0

m2 2 8m 1 18 5 2

35.

x 5 28

x 1 8 5 0

sx 1 8d2 5 0

x2 1 16x 1 64 5 0

x2 1 16x 1 57 5 27 37.

z 532

2z 2 3 5 0

s2z 2 3d2 5 0

4z2 2 12z 1 9 5 0

4z2 2 12z 1 15 5 6 39.

x 5 24 x 5 9

x 1 4 5 0 x 2 9 5 0

sx 2 9dsx 1 4d 5 0

x2 2 5x 2 36 5 0

x2 2 5x 5 36

xsx 2 5d 5 36

41.

y 5 6 y 5 212

y 2 6 5 0 y 1 12 5 0

sy 1 12dsy 2 6d 5 0

y2 1 6y 2 72 5 0

ysy 1 6d 5 72 43.

t 5 5 t 5 272

t 2 5 5 0 2t 1 7 5 0

s2t 1 7dst 2 5d 5 0

2t2 2 3t 2 35 5 0

ts2t 2 3d 5 35

45.

a 5 27 a 5 0

a 1 7 5 0 a 5 0

asa 1 7d 5 0

a2 1 7a 5 0

a2 1 7a 1 10 2 10 5 0

sa 1 2dsa 1 5d 5 10 47.

x 5 5 x 5 26

x 2 5 5 0 x 1 6 5 0

sx 1 6dsx 2 5d 5 0

x2 1 x 2 30 5 0

x2 1 x 2 20 2 10 5 0

sx 2 4dsx 1 5d 5 10

49.

t 5 6 t 5 22

t 2 6 5 0 t 1 2 5 0

st 1 2dst 2 6d 5 0

st 2 2 1 4dst 2 2 2 4d 5 0

st 2 2d2 2 16 5 0 51.

x 5 25 x 5 1

sx 1 5d 5 0 sx 2 1d 5 0

sx 2 1dsx 1 5d 5 0

fsx 1 2d 2 3gfsx 1 2d 1 3g 5 0

sx 1 2d2 5 9

53.

x 5 7 x 5 12 x 5 0

x 2 7 5 0 x 2 12 5 0 x 5 0

xsx 2 12dsx 2 7d 5 0

xsx2 2 19x 1 84d 5 0

x3 2 19x2 1 84x 5 0 55.

t 512 t 5 2

13 t 5 0

2t 2 1 5 0 3t 1 1 5 0 t 5 0

ts3t 1 1ds2t 2 1d 5 0

ts6t2 2 t 2 1d 5 0

6t3 2 t2 2 t 5 0


57.

z 5 22 z 5 2 z 5 22

z 1 2 5 0 z 2 2 5 0 z 1 2 5 0

sz 1 2dsz 2 2dsz 1 2d 5 0

sz 1 2dsz2 2 4d 5 0

z2sz 1 2d 2 4sz 1 2d 5 0 59.

a 5 23 a 5 3 a 5 22

a 1 3 5 0 a 2 3 5 0 a 1 2 5 0

sa 1 2dsa 2 3dsa 1 3d 5 0

sa 1 2dsa2 2 9d 5 0

a2sa 1 2d 2 9sa 1 2d 5 0

sa3 1 2a2d 1 s29a 2 18d 5 0

a3 1 2a2 2 9a 2 18 5 0

61.

c 5 23 c 5 3 c 5 3

c 1 3 5 0 c 2 3 5 0 c 2 3 5 0

sc 2 3dsc 2 3dsc 1 3d 5 0

sc 2 3dsc2 2 9d 5 0

c2sc 2 3d 2 9sc 2 3d 5 0

c3 2 3c2 2 9c 1 27 5 0 63.

x 5 21 x 5 1 x 5 3

x 1 1 5 0 x 2 1 5 0x 5 0 x 2 3 5 0

sx 2 3d xsx 2 1dsx 1 1d 5 0

sx 2 3d xsx2 2 1d 5 0

sx 2 3dsx3 2 xd 5 0

x3sx 2 3d 2 xsx 2 3d 5 0

x4 2 3x3 2 x2 1 3x 5 0

65.

x 5 22 x 5 2 x 5 0 x 5 232

x 1 2 5 0 x 2 2 5 0 4x 5 0 2x 1 3 5 0

s2x 1 3ds4xdsx 2 2dsx 1 2d 5 0

s2x 1 3d 4xsx2 2 4d 5 0

s2x 1 3ds4x3 2 16xd 5 0

4x3s2x 1 3d 2 16xs2x 1 3d 5 0

8x4 1 12x3 2 32x2 2 48x 5 0 67. From the graph, the x-intercepts are The solutions of the equation are 3 and

23 5 x 3 5 x

0 5 x 1 3 0 5 x 2 3

0 5 sx 2 3dsx 1 3d

23.0 5 x2 2 9s23, 0d and s3, 0d.

69. From the graph, the x-intercepts are The solutions of the equation

x 5 21 3 5 x

x 1 1 5 0 0 5 x 2 3

0 5 sx 2 3dsx 1 1d

0 5 x2 2 2x 2 3

0 5 x2 2 2x 2 3 are 21 and 3.s21, 0d and s3, 0d.

71. Keystrokes:

6

The x-intercepts are 0 and 6, so the solutions are 0 and 6.

Y5 X,T,u X,T,ux2 2 GRAPH

10

−1 7

−10

73. Keystrokes:

8 12


Y5 X,T,u X,T,ux2 2 GRAPH1

8

0 8

−5


75. Keystrokes:

2 5 12

The x-intercepts are so the solutions are 24 and 32.24 and 32,

Y5 X,T,u X,T,ux2 1 2 GRAPH

10

−5 3

−20

77. Keystrokes:

2 3 5 12

The x-intercepts are so the solutions are 232, 0, and 4.2

32, 0, and 4,

Y5 X,T,u X,T,u X,T,u> 2 x2 2 GRAPH

8

−2 5

−32

79.

x 5 2ba

ax 5 2b

ax 1 b 5 0x 5 0

xsax 1 bd 5 0

ax2 1 bx 5 0 81.

x2 2 2x 2 15 5 0

sx 1 3dsx 2 5d 5 0

fx 2 s23dgsx 2 5d 5 0

x 5 23, x 5 5

83. Verbal model:

Labels: Number

Its square

Equation:

reject

x 5 15 x 5 216

x 2 15 5 0 x 1 16 5 0

sx 1 16dsx 2 15d 5 0

x2 1 x 2 240 5 0

x 1 x2 5 240

5 x2

5 x

2405Its Square1Number

85. Verbal model:

Labels: First integer

Second integer

Equation:

1st integer

reject 2nd integer x 1 1 5 12

x 5 11 x 5 212

x 2 11 5 0 x 1 12 5 0

sx 1 12dsx 2 11d 5 0

x2 1 x 2 132 5 0

x ? sx 1 1d 5 132

x 1 1

5 x

1325SecondInteger?

FirstInteger


87. Verbal model:

Labels: Length

Width

Equation:

width

reject length x 1 7 5 22 feet

x 5 15 feet x 5 222

x 2 15 5 0 x 1 22 5 0

sx 1 22dsx 2 15d 5 0

x2 1 7x 2 330 5 0

x2 1 7x 5 330

sx 1 7d ? x 5 330

5 x

5 x 1 7

Area5Width?Length

89. Verbal model:

Labels: Base

Height

Equation:

inches base

reject inches height 32x 5 12

x 5 8 x 5 28

x 2 8 5 0 x 1 8 5 0

sx 1 8dsx 2 8d 5 0

3sx2 2 64d 5 0

3x2 2 192 5 0

34x2 2 48 5 0

12 ? x ? 32x 5 48

532x

5 x

Area5Height?Base?12

91. (a)

(b) Domain: Each side must be positive.

so

(c)

(e) Keystrokes:

5 2 4 2

yields the box of greatest volume.x 5 0.74

x < 2x < 52 x 5 0 x 5 2 x 5

52

0 < x < 2 4 2 2x > 0 5 2 2x > 0x > 0 x 5 0 4 2 2x 5 0 5 2 2x 5 0

0 5 s5 2 2xds4 2 2xdx

V 5 s5 2 2xds4 2 2xdx

Volume V 5 Length ? Width ? Height

x 0.25 0.50 0.75 1.00 1.25 1.50 1.75

V 3.94 6 6.56 6 4.69 3 1.31

(d) If

3 5 3

3 5 s2ds1ds1.5d

3 5 s5 2 3ds4 2 3ds1.5d

3 5 f5 2 2s1.5dgf4 2 2s1.5dgs1.5d

V 5 3, then x 5 1.5.

Y5 x x2 2X,T,u X,T,u X,T,ud d GRAPH

10

0 20


93.

reject

The object reaches the ground after 20 seconds.

t 5 220 t 5 20

t 1 20 5 0 t 2 20 5 0

216st 2 20dst 1 20d 5 0

216st2 2 400d 5 0

216t2 1 6400 5 0 95. Verbal model:

Equation:

units units

x 5 10 x 5 20

x 2 10 5 0 x 2 20 5 0

0 5 sx 2 20dsx 2 10d

0 5 x2 2 30x 1 200

90x 2 x2 5 200 1 60x

Cost5Revenue

97. (a)

(c) Answers will vary.

26 5 x 212 5 x

23 5 x 1 3 52 5 x 1 3

u 5 x 1 3 u 5 x 1 3

u 552

u 5 23 2u 5 5

u 1 3 5 0 2u 2 5 5 0

s2u 2 5dsu 1 3d 5 0

2u2 1 u 2 15 5 0

2sx 1 3d2 1 sx 1 3d 2 15 5 0 (b)

x 5 212

x 5 26 2x 5 21

x 1 6 5 0 2x 1 1 5 0

s2x 1 1dsx 1 6d 5 0

2x2 1 13x 1 6 5 0

2x2 1 12x 1 18 1 x 1 3 2 15 5 0

2sx2 1 6x 1 9d 1 sx 1 3d 2 15 5 0

2sx 1 3d2 1 sx 1 3d 2 15 5 0

99. (d) Verbal model:

Labels: Area

Length

Width

Equation:

reject

(e)

(f ) Domain of VBsxd

5 1026 cubic feet

5 504 1 636 2 120 1 6

5 73s216d 1

533 s36d 2 120 1 6

VSs6d 573s6d3 1

533 s6d2 2 20s6d 1 6

Volume 5 VSsxd 573x3 1

533 x2 2 20x 1 6

5 3 feet 5 5 feet 5 6 feet

Height 5 6 2 3 Width 5 6 2 1 Length 5 2s6d 2 6

x 5 22 6 5 x

x 1 2 5 0 0 5 x 2 6

0 5 sx 2 6dsx 1 2d

0 5 x2 2 4x 2 12

0 5 2x2 2 8x 2 24

30 5 2x2 2 8x 1 6

30 5 s2x 2 6dsx 2 1d

5 x 2 1

5 2x 2 6

5 30

Width?Length5Area


101. False. This is not an application of the Zero Factor Property because there are unlimited number of factors whose product is 1.

103. The maximum number of solutions of an degree polynomial equation is n. The third-degree equation hasonly one solution, x 5 21.

sx 1 1d3 5 0nth


1. is not a polynomial because the exponentof a variable must be a natural number.x2 1 2 1 3x1/2 3. Standard form:

Leading coefficient:

Degree:4

21

2x4 1 6x3 1 5x2 2 4x

5. Standard form:

Leading coefficient:

Degree:3

27

27x3 1 3x2 2 6x 1 14 7. Binomial of degree 4:3x 4 2 2

9. Monomial of degree 3 and leading coefficient 5:5x3

11. s5x 1 3x2d 1 s6 2 x 2 4x2d 5 s3x2 2 4x2d 1 s5x 2 xd 1 6 5 2x2 1 4x 1 6

13. s5x3 2 6x 1 11d 1 s5 1 6x 2 x2 2 8x3d 5 s5x3 2 8x3d 2 x2 1 s26x 1 6xd 1 s11 1 5d 5 23x3 2 x2 1 16

15. s3t 2 5d 2 st2 2 t 2 5d 5 s3t 2 5d 1 s2t2 1 t 1 5d 5 2t 2 1 s3t 1 td 1 s25 1 5d 5 2t 2 1 4t

17.

5 x5 2 4x3 1 7x2 2 9x 1 3

s3x5 1 4x2 2 8x 1 12d 2 s2x5 1 xd 1 s3x2 2 4x3 2 9d 5 s3x5 2 2x5d 2 4x3 1 s4x2 1 3x2d 1 s28x 2 xd 1 s12 2 9d

19.

5 29x3 1 9x 2 4

5 s2x3 2 8x3d 1 s23x 1 12xd 1 s24d

s2x3 2 3xd 2 4s2x3 2 3x 1 1d 5 2x3 2 3x 2 8x3 1 12x 2 4

21.

5 22y 2 15

5 s3y2 2 3y2d 2 2y 2 15

5 3y2 2 2y 2 3y2 2 15

3y2 2 f2y 1 3sy2 1 5dg 5 3y2 2 f2y 1 3y2 1 15g 23. x2 ? x3 5 x213 5 x5

25. su2d3 5 u2?3 5 u6 27.

5 28z3

s22zd3 5 s22d3z3 29.

5 4u7v3

5 4u413v211

2su2vd2s24u3vd 5 2su4v2ds24u3vd

31.12z5

6z2 5 1126 2 ? z522 5 2z3 33.

5 8u2v2

120u5v3

15u3v5

12015

?u5

u3 ?v3

v35.

5 144x4

5 s12x2d2

172x4

6x2 22

5 s12x422d2


37.

5 28x4 2 32x3

s22xd3sx 1 4d 5 28x3sx 1 4d 39. 3xs2x2 2 5x 1 3d 5 6x3 2 15x2 1 9x

41.

5 x2 1 5x 2 14

sx 2 2dsx 1 7d 5 x2 1 7x 2 2x 2 14

43. s5x 1 3ds3x 2 4d 5 15x2 2 20x 1 9x 2 12 5 15x2 2 11x 2 12

45. 5 24x4 1 22x2 1 3 s4x2 1 3ds6x2 1 1d 5 24x4 1 4x2 1 18x2 1 3

47.

5 4x3 2 5x 1 6

5 4x3 1 s6x2 2 6x2d 1 s29x 1 4xd 1 6

5 4x3 1 6x2 2 6x2 2 9x 1 4x 1 6

s2x2 2 3x 1 2ds2x 1 3d 5 2x2s2x 1 3d 2 3xs2x 1 3d 1 2s2x 1 3d

49.

5 u2 2 8u 1 7

5 s2u2 2 u2d 1 s214u 1 7u 2 ud 1 7

5 2u2 2 14u 2 u2 1 7u 2 u 1 7

2usu 2 7d 2 su 1 1dsu 2 7d 5 2usu 2 7d 2 usu 2 7d 2 1su 2 7d

51.

5 16x2 2 56x 1 49

s4x 2 7d2 5 s4xd2 2 2s4xds7d 1 s27d2 53.

5 4x2 1 12xy 1 9y2

s2x 1 3yd2 5 s2xd2 1 2s2xds3yd 1 s3yd2

55.

5 25u2 2 64

s5u 2 8ds5u 1 8d 5 s5ud2 2 82 57. s2u 1 vds2u 2 vd 5 s2ud2 2 v2 5 4u2 2 v2

59. fsu 2 3d 1 vgfsu 2 3d 2 vg 5 su 2 3d2 2 v2 5 u2 2 2suds3d 1 s23d2 2 v2 5 u2 2 6u 1 9 2 v2

61. 6x2 1 15x3 5 3x2s2 1 5xd 63.

5 214sx 1 5ds5x 1 23d

5 14sx 1 5ds25x 2 23d

5 14sx 1 5ds2 2 5x 2 25d

28sx 1 5d 2 70sx 1 5d2 5 14sx 1 5df2 2 5sx 1 5dg

65.

5 sv 2 2dsv 2 1dsv 1 1d

5 sv 2 2dsv2 2 1d

v3 2 2v2 2 v 1 2 5 v2sv 2 2d 2 1sv 2 2d 67.

5 st2 1 3dst 1 3d

t3 1 3t2 1 3t 1 9 5 t2st 1 3d 1 3st 1 3d

69. x2 2 36 5 x2 2 62 5 sx 2 6dsx 1 6d 71. 9a2 2 100 5 s3a 2 10ds3a 1 10d

73.

5 su 2 3dsu 1 15d

su 1 6d2 2 81 5 su 1 6 2 9dsu 1 6 1 9d 75. u3 2 1 5 su 2 1dsu2 1 u 1 1d


77.

5 s2x 1 3ds4x2 2 6x 1 9d

8x3 1 27 5 s2xd3 1 s3d3 79. x2 2 18x 1 81 5 x2 2 2s9dx 1 92 5 sx 2 9d2

81.

5 s2s 1 10td2

4s2 1 40st 1 100t2 5 s2sd2 1 2s2sds10d 1 s10td2 83. x2 1 2x 2 35 5 sx 1 7dsx 2 5d

85. 2x2 2 7x 1 6 5 s2x 2 3dsx 2 2d 87. 18x2 1 27x 1 10 5 s3x 1 2ds6x 1 5d

89.

5 4as1 2 4ads1 1 4ad

4a 2 64a3 5 4as1 2 16a2d 91.

5 4s2x 2 3ds2x 2 1d

5 s2x 2 3ds2x 2 1d

8xs2x 2 3d 2 4s2x 2 3d 5 s2x 2 3ds8x 2 4d

93.

5 s12x 1 yd2

14x2 1 xy 1 y2 5 s12xd2

1 2s12dxy 1 y2 95.

5 s2u 2 7d2

4u2 2 28u 1 49 5 s2ud2 2 2s2uds7d 1 72

97.

5 sx 2 y 2 5dsx 1 y 2 5d

5 sx 2 5 2 ydsx 2 5 1 yd

5 fsx 2 5d 2 ygfsx 2 5d 1 yg

x2 2 10x 1 25 2 y2 5 sx 2 5d2 2 y2 99.

s 5 2 s 5 243

s 2 2 5 0 3s 1 4 5 0

s3s 1 4dss 2 2d 5 0

3s2 2 2s 2 8 5 0

101.

v 5 210 v 5 10

v 1 10 5 0 v 2 10 5 0

sv 2 10dsv 1 10d 5 0

v2 2 100 5 0 103.

x 5 3 x 5 0

x 2 3 5 0 10x 5 0

10xsx 2 3d 5 0

105.

z 5 24 z 5 9

z 1 4 5 0 z 2 9 5 0

sz 2 9dsz 1 4d 5 0

z2 2 5z 2 36 5 0

5z 2 z2 1 36 5 0

zs5 2 zd 1 36 5 0 107.

y 5 3 y 5 24 y 5 0

y 2 3 5 0 y 1 4 5 0 2y2 5 0

2y2sy 1 4dsy 2 3d 5 0

2y2sy2 1 y 2 12d 5 0

2y4 1 2y3 2 24y2 5 0

109. Keystrokes:

10 21



10

0 10

−5


115. (a) Keystrokes:

.022 1.33 270.71

2.386 274.857

.028 3.4 278.18

(b)

This graph is most like the graph of , the averageof the high and low projections.

(c)

Keystrokes: .003 2.365 274.495

Keystrokes: .05 2.07 7.47

The difference between the high and lowprojections is increasing.

200

0 500

5 0.05t2 1 2.07t 1 7.47

5 s0.028t2 1 0.022t2d 1 s3.40t 2 1.33td 1 s278.18 2 270.71d

PH 2 PL 5 s0.028t2 1 3.40t 1 278.18d 2 s20.022t2 1 1.33t 1 270.71d

PM550

0 50200

5 0.003t2 1 2.365t 1 274.445

50.006t2 1 4.73t 1 548.89

2

PL 1 PH

25

20.022t2 1 1.33t 1 270.71 1 0.028t2 1 3.40t 1 278.182

y3

y2

y1

550

0 50200

PH

PM

PL

Y5 x2c X,T,u X,T,ux2 1 1 ENTER

X,T,u 1 ENTER

X,T,u X,T,ux2 1 1 GRAPH



117. Verbal model:

Equation:

5 $2000

5 3000 2 1000

Ps5000d 5 0.6s5000d 2 1000

5 0.6x 2 1000

5 1.1x 2 0.5x 2 1000

Psxd 5 1.1x 2 s0.5x 1 1000d

Cost2Revenue5Profit 119.

Area 5 1.3x 1 3.4x 5 3x 1 12x 5 15x

Perimeter 5 x 1 3x 1 4 1 4x 1 3 1 1 5 8x 1 8

111.

x2 1 4x 2 45 5 0

x2 1 9x 2 5x 2 45 5 0

sx 2 5dsx 1 9d 5 0

x 5 5 x 5 29 113.

2x3 2 x2 2 6x 5 0

xs2x2 2 4x 1 3x 2 6d 5 0

xs2x 1 3dsx 2 2d 5 0

xsx 132dsx 2 2d 5 0

x 5 0 x 5 232 x 5 2


121. Verbal model:

Labels: Width of larger rectangle

Length of larger rectangle

Width of smaller rectangle

Length of smaller rectangle

Equation:

5 14x 1 3

5 4x2 1 10x 2 4x2 1 4x 1 3

5 4x2 1 10x 2 s4x2 2 6x 1 2x 2 3d

Area 5 2xs2x 1 5d 2 s2x 1 1ds2x 2 3d

5 2x 1 1

5 2x 2 3

5 2x 1 5

5 2x

Area ofsmaller

rectangle2

Area oflarger

rectangle5

Area ofshadedregion

123. (a)

5 4l 2 10

5 2l 1 2l 2 10

5 2l 1 2sl 2 5d

Perimeter 5 2l 1 2w (b)

5 l 2 2 5l

5 lsl 2 5d

Area 5 l ? w

125. Verbal model:

Labels: Width

Front

Back

Equation:

400 2 x 5 300 feet

x 5 100 feet

25x 5 2500

20x 1 6000 2 15x 1 4000 2 10x 5 9500

10s2xd 1 15s400 2 xd 1 10s400 2 xd 5 9500

5 400 2 x

5 400 2 x

5 x

95005

Cost offencingback

1

Cost offencingfront

1

Cost offencing2 widths

127.

reject

The object reaches the ground after 2 seconds.

t 5 21 t 5 2

t 1 1 5 0 t 2 2 5 0

216st 2 2dst 1 1d 5 0

216st2 2 t 2 2d 5 0

216t2 1 16t 1 32 5 0


129. Verbal model:

Labels: First even integer

Second even integer

Equation:

reject

2n 1 2 5 16

2n 5 14

n 5 7 n 5 28

n 2 7 5 0 n 1 8 5 0

4sn 1 8dsn 2 7d 5 0

4sn2 1 n 2 56d 5 0

4n2 1 4n 2 224 5 0

2n ? s2n 1 2d 5 224

5 2n 1 2

5 2n

2245Secondeven?

First eveninteger


1.

Degree Leading coefficient5 25.25 3

25.2x3 1 3x2 2 8 2. is not a polynomial because the variable appears in

the denominator.

4x2 1 2

3. (a) s5a2 2 3a 1 4d 1 sa2 2 4d 5 6a2 2 3a (b)

5 22y2 2 2y

5 16 2 y2 2 16 2 2y 2 y2

s16 2 y2d 2 s16 1 2y 1 y2d

4. (a)

5 8x2 2 4x 1 10

5 24x4 2 10 1 4x4 1 8x2 2 4x

22s2x4 2 5d 1 4xsx3 1 2x 2 1d (b)

5 11t 1 7

5 4t 2 3t 1 10t 1 7

4t 2 f3t 2 s10t 1 7dg 5 4t 2 f3t 2 10t 2 7g

5. (a)

5 224u6v5

s22u2vd3s3v2d 5 s28u6v3ds3v2d (b)

5 60x3y2

3s5xds2xyd2 5 3s5xds4x2y2d

6. (a)

5y3

8

2y1y42

2

5 2y1 y2

162 (b)

527x6y4

2

s23x2yd4

6x2 581x8y4

6x2

7. (a) 23xsx 2 4d 5 23x2 1 12x (b) s2x 2 3ydsx 1 5yd 5 2x2 1 7xy 2 15y2

8. (a)

(b)

5 6s3 2 17s2 1 26s 2 21

s2s 2 3ds3s2 2 4s 1 7d 5 6s3 2 8s2 1 14s 2 9s2 1 12s 2 21

sx 2 1df2x 1 sx 2 3dg 5 sx 2 1ds3x 2 3d 5 3x2 2 6x 1 3


9. (a)

(b)

5 16 2 a2 2 2ab 2 b2

5 16 2 sa2 1 2ab 1 b2d

f4 2 sa 1 bdgf4 1 sa 1 bdg 5 16 2 sa 1 bd2

s4x 2 3d2 5 16x2 2 24x 1 9 10. 18y2 2 12y 5 6ys3y 2 2d

11. v2 2169 5 sv 2

43dsv 1

43d 12.

5 sx 2 3dsx 2 2dsx 1 2d

5 sx 2 3dsx2 2 4d

x3 2 3x2 2 4x 1 12 5 x2sx 2 3d 2 4sx 2 3d

13. 9u2 2 6u 1 1 5 s3u 2 1ds3u 2 1d or s3u 2 1d2 14.

5 2s3x 1 2dsx 2 5d

6x2 2 26x 2 20 5 2s3x2 2 13x 2 10d

15. x3 1 27 5 sx 1 3dsx2 2 3x 1 9d 16.

y 5 25 y 5 1

y 1 5 5 0 y 2 1 5 0

fsy 1 2d 2 3gfsy 1 2d 1 3g 5 0

sy 1 2d2 2 9 5 0

17.

243 5 y 3 5 y

4 1 3y 5 0 3 2 y 5 0

s3 2 yds4 1 3yd 5 0

12 1 5y 2 3y2 5 0 18. Area

5 x2 1 26x

Shaded region 5 2x2 1 30x 2 x2 2 4x

5 2xsx 1 15d 2 xsx 1 4d

19. Verbal model:

Labels: Length

Width

Equation:

9 centimeters 5 length

6 centimeters 5 width

36 5 w2

108 5 3w2

54 532w ? w

5 w

532w

Width?Length5Area rectangle

20.

reject

t 5 2 sec t 5 292

t 2 2 5 0 2t 1 9 5 0

0 5 s2t 1 9dst 2 2d

0 5 2t2 1 5t 2 18

0 5 216t2 2 40t 1 144


21. Verbal model:

Labels: Base

Height

Equation:

reject 2x 1 4 5 14 feet; height

x 5 5 feet; base x 5 27

x 2 5 5 0 x 1 7 5 0

0 5 sx 1 7dsx 2 5d

0 5 x2 1 2x 2 35

0 5 2x2 1 4x 2 70

70 5 2x2 1 4x

35 512 ? x ? s2x 1 4d

5 2x 1 4

5 x

Height?Base?12

5Area

Cumulative Test for Chapters P–3

1. (a)

(b)

(c) |2.3| > 2|24.5|

13 < 1

2

22 < 5 2. “The number n is tripled and the product is decreased by8,” is expressed by 3n 2 8.

3. (a)

(b)

5 2x3 2 11x

3xsx2 2 2d 2 xsx2 1 5d 5 3x3 2 6x 2 x3 2 5x

5 t2 2 9t

ts3t 2 1d 2 2tst 1 4d 5 3t2 2 t 2 2t2 2 8t 4. (a)

(b) 12x4y2

4x3y 22

5 1xy2 2

2

5x2y2

4

5 8a8b7

s2a2bd3s2ab2d2 5 s8a6b3dsa2b4d

5. (a)

(b)

5 4 1 4x 2 4y 1 x2 2 2xy 1 y2

f2 1 sx 2 ydg2 5 4 1 4sx 2 yd 1 sx 2 yd2

5 2x2 2 9x 2 5

s2x 1 1dsx 2 5d 5 2x2 2 10x 1 x 2 5

6. (a)

x 532

4x4

564

4x 5 6

3 2 3 1 4x 5 3 1 3

23 1 5x 2 x 5 x 1 3 2 x

23 1 5x 5 x 1 3

12 2 15 1 5x 5 x 1 3

12 2 5s3 2 xd 5 x 1 3 (b)

x 5 232

22x22

5 232

22x 5 3

4 2 4 2 2x 5 7 2 4

4 2 2x 5 7

8 2 2x 2 4 5 7

8 2 2sx 1 2d 5 7

831 2x 1 2

4 4 5 37848

1 2x 1 2

45

78

Cumulative Test for Chapters P–3 221

7. (a)

(b)

t 5 3 t 5 212

t 2 3 5 0 2t 1 1 5 0

s2t 1 1dst 2 3d 5 0

2t2 2 5t 2 3 5 0

x 5 223 x 5 4

3x 5 22 3x 5 12

3x 2 5 5 27 3x 2 5 5 7

|3x 2 5| 5 7 8. (a)

(b)

232 ≤ x < 4

264 ≤ x < 4

26 ≤ 4x < 16

212 ≤ 4x 2 6 < 10

x < 21

23x > 3

3 2 3x > 6

3s1 2 xd > 6

9. Verbal model:

Labels: Total annual premium

Equation:

x 5 $1408.75

x 5 1225 1 183.75

x 5 1225 1 0.15s1225d

5 x

Surcharge1Annual

premium5Total annual

premium

10.

x 5 6.5

x 513s4.5d

9

9x 5 13s4.5d

9

4.55

13x

11. or

−4 −2 0 2 4 6 8

−1 5x

x ≥ 5 x ≤ 21

x 2 2 ≥ 3 x 2 2 ≤ 23

12. Verbal model:

Equation:

x ≥ 103

x ≥ 102.27273

4.4x ≥ 450

12.90x ≥ 8.50x 1 450

Cost>Revenue 13. does represent y as a function of x.x 2 y3 5 0

14.

2 ≤ x < `

x 2 2 ≥ 0D 5 x ≥ 2f sxd 5 !x 2 2 15.

(a)

(b)

5 c2 1 3c

5 c2 1 6c 1 9 2 3c 2 9

f sc 1 3d 5 sc 1 3d2 2 3sc 1 3d

f s4d 5 42 2 3s4d 5 16 2 12 5 4

f sxd 5 x2 2 3x


16.

5 10

5 !100

5 !64 1 36

d 5 !s24 2 4d2 1 s0 2 6d2

m 56 2 04 1 4

568

534

18. (a)

(b)

5 9sx 2 4dsx 1 4d

5 3sx 2 4d 3sx 1 4d

5 s3x 2 12ds3x 1 12d

9x2 2 144 5 s3xd2 2 s12d2

3x2 2 8x 2 35 5 s3x 1 7dsx 2 5d 19. (a)

(b)

5 2ts2t 2 5d2

5 2tfs2td2 2 2s2tds5d 1 52g

8t3 2 40t2 1 50t 5 2ts4t2 2 20t 1 25d

5 sy 2 3dsy 2 3dsy 1 3d

5 sy 2 3dsy2 2 9d

y3 2 3y2 2 9y 1 27 5 y2sy 2 3d 2 9sy 2 3d

20.

s3, 0d x 5 3s0, 4d y 5 4

4x 5 12 3y 5 12

4x 1 3s0d 2 12 5 0 4s0d 1 3y 2 12 5 0

4x 1 3y 2 12 5 0

–6 –4 –2 2 6

–4

–2

2

4

8

x

y

21.

–1 2 4 5

–3

–2

–1

1

x

y

y 5 1 2 sx 2 2d2

17. (a)

2x 2 y 1 5 5 0

y 2 1 5 2x 1 4

y 2 1 5 2sx 1 2d

m 5 2

y 5 2x 2 1

2y 5 22x 1 1

2x 2 y 5 1 (b)

2x 2 3y 1 7 5 0

3y 5 2x 1 7

y 523x 1

73

y 2 1 523x 1

43

y 2 1 523sx 1 2d

m 523

y 5 232x 1 5

2y 5 23x 1 5

3x 1 2y 5 5


1. If the product of two real numbers is and one of the factors is 12, the other factor is negative.296

2. The sum of the digits of 576 is divisible by 9 and 3.576 5 5 1 7 1 6 5 18.

3. is positive is a false statement. 262 5 21 ? 62 5 21 ? 36 5 236262

4. is positive is a true statement. s26d2 5 s26ds26d 5 36s26d2

5.

x ≥ 6

2x2

≥122

2x ≥ 12

2x 2 12 1 12 ≥ 12

2x 2 12 ≥ 0 6.

x >32

22x

2>

2322

22x < 23

7 2 7 2 2x < 4 2 7

7 2 2x < 4

7 2 3x 1 x < 4 2 x 1 x

7 2 3x < 4 2 x

7.

23 < x < 3

3 > x > 23

2622

>22x22

>6

22

26 < 22x < 6

22 2 4 < 4 2 4 2 2x < 10 2 4

22 < 4 2 2x < 10 8.

21 ≤ x < 3

4 2 5 ≤ x 1 5 2 5 < 8 2 5

4 ≤ x 1 5 < 8

9.

1 < x < 5

22 1 3 < x 2 3 1 3 < 2 1 3

22 < x 2 3 < 2

|x 2 3| < 2 10.

or

or x < 2 x > 8

x 2 5 1 5 < 23 1 5 x 2 5 1 5 > 3 1 5

x 2 5 < 23 x 2 5 > 3

|x 2 5| > 3

11. Verbal model:

Proportion:

x 5 $1489.66

x 5s2400ds90,000d

145,000

2400

145,0005

x90,000

5Tax

Assessed ValueTax

Assessed Value

CHAPTER 3 Polynomials and Factoring

SECTION 3.1 Adding and Subtracting Polynomials


12. Verbal model:

Proportion:

or 1138

gallons x 5 11.375 gallons

x 57.325200

7

2005

x325

5GallonsMiles

GallonsMiles

1. The point is 2 units to the left of the y-axis and 3 units above the x-axis.s22, 3d

2. Point 3 units from x-axis and 4 units from y-axis s4, 3d, s24, 3d, s24, 23d, s4, 23d

3.

y 5 13

y 5 9 1 4

y 535

s15d 1 4

s15, d

y 535

x 1 4 4.

y 5 2113

y 593

2203

y 5 3 2203

y 5 3 259

s12d

s12, d

y 5 3 259

x

5.

6.84 < x

26.5

20.955 x

26.5 5 20.95x

21 5 5.5 2 0.95x

s , 21d

y 5 5.5 2 0.95x 6.

7 5 x

1.40.2

5 x

1.4 5 0.2x

4.4 5 3 1 0.2x

s , 4.4d

y 5 3 1 0.2x

7.

(a)

(b)

53

16

f 1342 5

131

342

2

513

?916

f s6d 513

s6d2 513

? 36 5 12

f sxd 513

x2 8.

(a)

(b)

5 22x

5 3 2 2x 2 6 2 3 1 6

f sx 1 3d 2 f s3d 5 3 2 2sx 1 3d 2 f3 2 2s3dg

5 27

5 3 2 10

f s5d 5 3 2 2s5d

f sxd 5 3 2 2x

SECTION 3.2 Multiplying Polynomials


9.

(a)

(b) g sc 2 6d 5c 2 6

sc 2 6d 1 105

c 2 6c 1 4

g s5d 55

5 1 105

515

513

g sxd 5x

x 1 1010.

(a)

(b)

5 !t 2 1

h st 1 3d 5 !t 1 3 2 4

5 2!3

5 !4 ? 3

5 !12

h s16d 5 !16 2 4

h sxd 5 !x 2 4

11.

y

x2−2−4 4 6

2

−2

4

6

g sxd 5 7 232

x 12.

y

x1 2 3 5−1 4 6

2

4

6

1

3

5

h sxd 5 |3 2 x|

1. A function f from a set A to a set B is a rule of correspondence that assigns to each element x in the set A exactly one elementy in the set B.

2. The set A (see Exercise 1) is called the domain (or set of inputs) of the function f, and the set B (see Exercise 1) contains therange (or set of outputs) of the function f.

3. y is not a function of x.

Answers will vary.

x 5 y2 2 1

y

x3−2 421

3

2

−3

−2

4. y is a function of x.

Answers will vary.

y

x3−2 −1 421

3

2

1

−1

SECTION 3.3 Factoring Polynomials


5.

Function

y

x6−2 842

6

8

10

4

2

−2

y 5 6 223

x 6.

Function

y

x3−2 −1 421

1

−1

−3

−4

−2

y 552

x 2 4

7.

Function

y

x3−2 −1 421

1

2

−1

−4

2y 2 4x 1 3 5 0 8.

Function

y

x−6 −5 −4 −3 −2 −1 1

1

−1

−2

−3

−4

−6

3x 1 2y 1 12 5 0

9.

Not a function

y

x3−2 −1 421

3

2

1

−3

−2

−1

|y| 2 x 5 0 10.

Not a function

y

x3−2 −1

−1

−2

−3

421

3

2

1

|y| 5 2 2 x

11. y

t42 531

100,000

50,000

150,000

200,000


12. (a)Verbal model:

Function:

(b) y

x200 300100

8,000

4,000

12,000

16,000

A sxd 5 x ? s250 2 xd

Width?Length5Area

1. A function can have only one value of y correspondingto x 5 0.

2. Leading coefficient of is 6.6t3 1 3t2 1 5t 2 4

3. The set of all real numbers x whose distance from 0 is less than 5 can be represented by |x| < 5.

4. The set of all real numbers x whose distance from 6 is more than 3 is represented by |x 2 6| > 3.

5.

m 524 2 2

5 2 s23d 5268

5234

y

x2−2

2

4

−4 4 6

−2

−4

−6

(−3, 2)

(5, −4)

s23, 2d, s5, 24d 6.

m 523 2 87 2 2

5211

5

y

x2−2

−2−4

−4

2

6

8

4

4 8

(2, 8)

(7, −3)

s2, 8d, s7, 23d

7.

5212 2 2114 2 15

523321

5 33 m 5

22 272

73

252

?66

y

x1−1

3

4

1

2

−2 3 42−1

−2 ), −2(73

),(52 72

152

, 722, 17

3, 222 8.

m 5

92

2 12142

23 2 1294 2

?44

518 1 1

212 1 95

1923

5 2193

y

x1−1

3

4

5

1

2

2−1

)−3,( 92

),( 94

14

−−

1294

, 2142, 123,

922

SECTION 3.4 Factoring Trinomials


9.

m is undefined

m 523 2 46 2 6

5270

y

x2 4−2

−2

−4

−6

−4

2

6

4

8

(6, 4)

(6, −3)

s6, 4d, s6, 23d 10.

m 55 2 5

7 2 s24d 50

115 0

y

x2−2

−2−4

−4

2

6

8

4

4 6 8

(7, 5)(−4, 5)

s24, 5d, s7, 5d

11. Verbal model:

Equation:

Verbal model:

Equation:

x 5 $12,720

x 5 12,000 1 720

Interest1Principal5Paymen

i 5 $720

i 5 12,000 ? 0.12 ? 12

Time?Rate?Principal5Interest

12. Verbal model:

Equation:

x < 49.1 mph

x 5540001100

1100x 5 54000

500x 1 600x 5 54000

10054

110045

5200

x

5Time

Time?Rate5Distance

DistanceRate

1. illustrates the Additive Inverse Property.3uv 2 3uv 5 0 2. illustrates the Multiplicative Identity Property.5z ? 1 5 5z

3. illustrates the Distributive Property.2ss1 2 sd 5 2s 2 2s2 4. illustrates the Associative Property ofMultiplication.s3xdy 5 3sxyd

SECTION 3.5 Solving Polynomial Equations


11. (a)

Keystrokes:

1 4 8 12

(c)

5 52

5 264 1 128 2 12

5 214s256d 1 128 2 12

Ps16d 5 214s16d2 1 8s16d 2 12

5 214x2 1 8x 2 12

5 16x 214x2 2 12 2 8x

P 5 116x 214

x22 2 s12 1 8xd

P 5 R 2 C

(b) 60

0 20

−15

Y5 x2c 4 X,T,u X,T,ux2 1 2 GRAPH

12.

t 5 6 seconds

t2 5 36

216t2

2165

2576216

216t2 5 2576

216t2 1 576 2 576 5 0 2 576

216t2 1 576 5 0

5.

x 5 24

s22d1212

x2 5 s2ds22d

212

x 5 2

4 2 4 212

x 5 6 2 4

4 212

x 5 6 6.

x < 353.33

20.75x20.75

5226520.75

20.75x 5 2265

500 2 500 2 0.75x 5 235 2 500

500 2 0.75x 5 235 7.

No solution

217 Þ 0

4x 2 12 2 4x 2 5 5 0

4sx 2 3d 2 s4x 1 5d 5 0

8.

x 5 219

2x2

5 2382

2x 5 238

36 2 36 1 2x 5 22 2 36

36 1 2x 5 22

36 2 12x 1 14x 5 22 2 14x 1 14x

36 2 12x 5 22 2 14x

36 2 12x 5 5 2 14x 2 7

12s3 2 xd 5 5 2 7s2x 1 1d 9.

x 5 40

12 2 12 1 x 5 52 2 12

12 1 x 5 52

4112 1 x4 2 5 s13d4

12 1 x

45 13 10.

t 5 24

t 2 24 5 0

8st 2 24d 5 0

C H A P T E R 4Rational Expressions, Equations, and Functions

Section 4.1 Integer Exponents and Scientific Notation . . . . . . . . .224

Section 4.2 Rational Expressions and Functions . . . . . . . . . . . .227

Section 4.3 Multiplying and Dividing Rational Expressions . . . . . .231

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . .235

Section 4.4 Adding and Subtracting Rational Expressions . . . . . . .236

Section 4.5 Dividing Polynomials . . . . . . . . . . . . . . . . . . .243

Section 4.6 Solving Rational Equations . . . . . . . . . . . . . . . .248


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .268

C H A P T E R 4Rational Expressions, Equations, and Functions

Section 4.1 Integer Exponents and Scientific NotationSolutions to Odd-Numbered Exercises

224

1. 522 5152 5

125

3. 21023 5 21

103 5 21

10005. s23d0 5 1

7.1

423 51

s1y4d3 5 43 5 64 9.

51

21

32

5 232

1

s22d25 51

121225

11. 1232

21

532

13. 1 3162

0

5 1 15.

5 1

5 30

5 331(23)

27 ? 323 5 33 ? 323 17.

5 729

5 36

5 3412

34

322 5 342(22)

19.

5 100,000

5 105

5 10312

103

1022 5 1032(22) 21.

51

16

5142

5 422

5 s41d22

s42 ? 421d225s421(21)d22 23.

51

64

5126

s223d2 5 226

25.

53

16

5216

11

16

518

11

16

223 1 224 5123 1

124 27.

564

121

5 1 8112

2

5 1118 2

22

134

1582

22

5 168

1582

22

29.

51615

5 115162

21

5 11616

21

16221

s50 2 422d21511 21422

21

31. y 4 ? y22 5 y 41(22) 5 y2 33. z5 ? z23 5 z51(23) 5 z2 35. 7x24 57x4

Section 4.1 Integer Exponents and Scientific Notation225

45. s2x2d22 51

s2x2d2 51

4x4 47.

5 212xy3

5 212x21y23

s23x23y2ds4x2y25d 5 23 ? 4 ? x2312 ? y21(25)

49. s3x2y22d 5 322x24y4 51y4

9x451. 1 x

10221

510x

53.

5x5

2y4

5x5y24

2

6x3y23

12x22y5

6x32(22)y2321

6 ? 2

55.

581v8

u6

534

u6v28

5 1 32

u3v2422

5 1u3v24

32 222

1 3u2v21

33u21v3222

5 13u22(21)v2123

33 222

57.

5b5

a5

1a22

b2221ba2

3

5 1b2

a221b3

a32

59.

51

2x8y3

54x28y23

8

54x2911y31(26)

23

s2x3y21d23s4xy26d 5 s223x29y3ds4xy26d 61.

5 6u

5 6u41(23)

u4s6u23v0ds7vd0 5 u4s6u23ds1d

37. s4xd23 51

s4xd3 51

64x3 39.1

x26 5 x6 41.

543

a

543

a2617

8a26

6a27 543

a(26)2(27) 43.s4td0

t22 51

t22 5 t 2

63. fsx24y26d21g2 5 sx4y6d2 5 x8y12 65.

52b11

25a12

52a212b11

25

58a2626b121122

100

s2a22b4d3bs10a3bd2 5

23a26b12 ? b102a6b2

67.

5v2

uv2 1 1

51

u 1 1 1v22

?v2

v2

su 1 v22d21 51

u 1 v22

226 Chapter 4 Rational Expressions, Equations, and Functions226 Chapter 4 Rational Expressions, Equations, and Functions

69.

5ab

b 2 a

5absa 1 bd

sb 2 adsb 1 ad

5a2b 1 ab2

b2 2 a2

a 1 b

ba21 2 ab21 5a 1 bba

2ab

?abab

71. 3,600,000 5 3.6 3 106 73. 47,620,000 5 4.762 3 107

75. 0.00031 5 3.1 3 1024 77. 0.0000000381 5 3.81 3 1028 79. 57,500,000 5 5.75 3 107

81. 9,461,000,000,000,000 5 9.461 3 1015 83. 0.0000899 5 8.99 3 1025 85. 6 3 107 5 60,000,000

87. 1.359 3 1027 5 0.0000001359 89. $3.17 3 1010 5 31,700,000,000 91. 1.3 3 107 5 13,000,000

93. 4.8 3 10210 5 0.00000000048 95.

5 6.8 3 105

s2 3 109ds3.4 3 1024d 5 s2ds3.4ds105d 97.

5 2.5 3 109

5 25 3 108

s5 3 104d2 5 52 3 108

99.

5 6.0 3 106

5 0.6 3 107

3.6 3 1012

6 3 105 53.66

3 101225 101.

5 9 3 1015

5 s4.5ds2d 3 1015

s4,500,000ds2,000,000,000d 5 s4.5 3 106ds2 3 109d

103.

5 1.6 3 1012

5 1.6 3 1072(25)

64,000,000

0.000045

6.4 3 107

4.0 3 1025

105.

< 3.46 3 1010

< 3.4629032 3 1010

5s5.65ds2.85d

4.653 1010

s0.0000565ds2,850,000,000,000d

0.004655

s5.65 3 1025ds2.85 3 1012d4.65 3 1023

107.

< 4.70 3 1011

5 4.70299 3 1011

5 0.0470299 3 1013

1.357 3 1012

s4.2 3 102ds6.87 3 1023d 51.357

s4.2ds6.87d 3 1013 109.

< 1.67 3 1014

5 1.6652 3 1014

5 16.652 3 1013

5 16.652 3 10419

72,400 3 2,300,000,000 5 s7.24 3 104ds2.3 3 109d

Section 4.2 Rational Expressions and Functions227

111.

< 2.74 3 1020

5 2.738 3 1020

5 2.738 3 1018210112

5s125ds13.69d

6253 10181(210)2(212)

5s53 3 1018ds3.72 3 10210d

54 3 10212

s5,000,000d3s0.000037d2

s0.005d4 5s5.0 3 106d3s3.7 3 1025d2

s5.0 3 1023d4

113. 93,000,000 5 9.3 3 107 115.

< 8.3 minutes

< 1.58 3 1025

< 0.157672 3 1024

1.49 3 1011

9.45 3 1015 51.499.45

3 1024

117.

< 3.33 3 105

< 0.3330544 3 106

1.99 3 1030

5.975 3 1024 51.99

5.9753 106 119.

< $20,393

< 2.03925 3 104

55.506 3 1012

2.7 3 108

$5506 billion270 million

5$5,506,000,000,000

270,000,000

121. In is called the base and 4 is called theexponent.

s3xd4, 3x 123. You can “move” a factor from the numerator to thedenominator by changing the sign of the exponent of the factor.

125. Scientific notation is an efficient way of writing andcomputing real numbers when the numbers are very largeor very small.

Section 4.2 Rational Expressions and Functions

1.

D 5 s2`, 8d < s8, `d

x Þ 8

x 2 8 Þ 0 3.

D 5 s2`, 24d < s24, `d

x Þ 24

x 1 4 Þ 0

9.

D 5 s2`, `d

x2 1 4 Þ 0 11.

D 5 s2`, 23d < s23, 0d < s0, `d

y Þ 0 y Þ 23

ysy 1 3d Þ 0

5.

D 5 s2`, `d

4 Þ 0 7. D 5 s2`, `d

228 Chapter 4 Rational Expressions, Equations, and Functions

13.

D 5 s2`, 24d < s24, 4d < s4, `d

t Þ 4 t Þ 24

st 2 4dst 1 4d Þ 0

t2 2 16 Þ 0 15.

D 5 s2`, 0d < s0, 3d < s3, `d

y Þ 0 y Þ 3

ysy 2 3d Þ 0

y2 2 3y Þ 0

21. (a)

(c)

5 not possible; undefined

f s23d 54s23d

23 1 35

2120

f s1d 54s1d

1 1 35

44

5 1 (b)

(d) f s0d 54s0d

0 1 35

03

5 0

f s22d 54s22d

22 1 35

281

5 28

23. (a)

(c)


gs3d 532 2 4s3d

32 2 95

9 2 129 2 9

5230

gs0d 502 2 4s0d

02 2 95 0 (b)

(d)


gs23d 5s23d2 2 4s23d

s23d2 2 95

9 1 129 2 9

5210

gs4d 542 2 4s4d

42 2 95

16 2 1616 2 9

507

5 0

17.

D 5 s2`, 2d < s2, 3d < s3, `d

x Þ 3 x Þ 2

sx 2 3dsx 2 2d Þ 0

x2 2 5x 1 6 Þ 0 19.

D 5 s2`, 21d < s21, 53d < s53, `d

u Þ53 u Þ 21

s3u 2 5dsu 1 1d Þ 0

3u2 2 2u 2 5 Þ 0

25. (a)

(c)


hs21d 5s21d2

s21d2 2 s21d 2 25

11 1 1 2 2

510

hs10d 5102

102 2 10 2 25

10088

52522

(b)

(d)


hs2d 522

22 2 2 2 25

44 2 2 2 2

540

hs0d 502

02 2 0 2 25

022

5 0

27. Since length must be positive,

Since must be defined,

Therefore, the domain isor s0, `d.x > 0

x Þ 0.

500x

x ≥ 0.

29.

D 5 H1, 2, 3, 4, . . .J

x 5 units of a product 31. Since is the percent of air pollu-tants in the stack emission of autility, Since

must be defined, Therefore, the domain is f0, 100d.

p Þ 100.

80,000p100 2 p

0 ≤ p ≤ 100.

p

33.56

55sx 1 3d6sx 1 3d, x Þ 23 35.

x2

53xsx 1 16d2

2s3sx 1 16d2d, x Þ 216 37.x 1 5

3x5

sx 1 5dsxsx 2 2dd3x2sx 2 2d , x Þ 2

39.8x

x 2 55

8xsx 1 2dx2 2 3x 2 10

, x Þ 22 41.5x25

55x

5 ? 55

x5

43.

5 6y, y Þ 0

12y2

2y5

2 ? 6 ? y ? y2 ? y

Section 4.2 Rational Expressions and Functions229

45.

56x5y3, x Þ 0

18x2y15xy4 5

3 ? 6 ? x ? x ? y3 ? 5 ? x ? y ? y3 47.

3x2 2 9x12x2 5

3xsx 2 3d12x2 5

sx 2 3d4x

49.

5 x, x Þ 0, x Þ 8

x2sx 2 8dxsx 2 8d 5

x ? xsx 2 8dxsx 2 8d

51.2x 2 34x 2 6

52x 2 3

2s2x 2 3d 512

, x Þ32 53.

5 213

, x Þ 5

5 2 x

3x 2 155

21sx 2 5d3sx 2 5d

55.

51

a 1 3

a 1 3

a2 1 6a 1 95

a 1 3sa 1 3dsa 1 3d

57.

5x

x 2 7

x2 2 7x

x2 2 14x 1 495

xsx 2 7dsx 2 7dsx 2 7d 59.

5ysy 1 2d

y 1 6, y Þ 2

5ysy 2 2dsy 1 2dsy 1 6dsy 2 2d

y3 2 4y

y2 1 4y 2 125

ysy2 2 4dsy 1 6dsy 2 2d 61.

5xsx 1 2d

x 2 3, x Þ 2

5xsx 2 2dsx 1 2dsx 2 3dsx 2 2d

x3 2 4x

x2 2 5x 1 65

xsx2 2 4dsx 2 3dsx 2 2d

63.

5 23x 1 5x 1 3

, x Þ 4

5s3x 1 5dsx 2 4d

21sx 2 4dsx 1 3d

3x2 2 7x 2 2012 1 x 2 x2 5

s3x 1 5dsx 2 4d21sx2 2 x 2 12d 65.

5x 1 8x 2 3

, x Þ 232

2x2 1 19x 1 24

2x2 2 3x 2 95

s2x 1 3dsx 1 8ds2x 1 3dsx 2 3d

67.

53x 2 15x 2 4

, x Þ 245

15x2 1 7x 2 4

25x2 2 165

s5x 1 4ds3x 2 1ds5x 1 4ds5x 2 4d 69.

53y2

y2 1 1, x Þ 0

3xy2

xy2 1 x5

3xy2

xsy2 1 1d

71.

5y 2 8x

15, y Þ 28x

y2 2 64x2

5s3y 1 24xd 5sy 2 8xdsy 1 8xd

15sy 1 8xd 73.

55 1 3xy

y2 , x Þ 0

5xy 1 3x2y2

xy3 5xys5 1 3xyd

xy ? y2

75.

5u 2 2vu 2 v

, u Þ 22v

u2 2 4v2

u2 1 uv 2 2v2 5su 2 2vdsu 1 2vdsu 2 vdsu 1 2vd 77.

53sm 2 2nd

m 1 2n

53sm 2 2ndsm 1 2ndsm 1 2ndsm 1 2nd

3m2 2 12n2

m2 1 4mn 1 4n2 53sm2 2 4n2d

sm 1 2ndsm 1 2nd

79.

64

Þ 9

10 2 4

4Þ 10 2 1

x 2 4

4Þ x 2 1 81.

1 Þ34

3s0d 1 24s0d 1 2

Þ34

3x 1 24x 1 2

Þ34

Choose a value such as 10for and evaluate both sides.x

Choose a value such as 0 forand evaluate both sides.x


83.

Domain of is

Domain of is

The two expressions are equal for all replacements of the variable except 2.x

s2`, `d.x 1 1

s2`, 2d < s2, `d.x2 2 x 2 2x 2 2

0 1 2 3 4

0 1 2 Undefined 4 5

0 1 2 3 4 521x 1 1

21x2 2 x 2 2

x 2 2

2122x

85.Area of shaded portion

Area of total figure5

xsx 1 1dsx 1 1dsx 1 3d 5

xx 1 3

, x > 0

87. (a)

Labels:

Equation:

(b)

Label:

Equation:

(c)

(d)2500 1 9.25s100d

1005 $34.25

Domain 5 H1, 2, 3, 4, . . .J

C 52500 1 9.25x

x

Average cost 5 C

2500 1 9.25x 5 C

Number of units 5 x

Total cost 5 C

Totalcost

VerbalModel:

15Numberof units

?Cost perunit

Initialcost

Totalcost

VerbalModel:

45Numberof units

Averagecost

89. (a)

Van:

Car:

(b)

(c)Distance of carDistance of van

560t

45st 1 3d 54t

3st 1 3d

5 15s9 2 td 5 135 2 15t

5 45t 1 135 2 60t

5 45st 1 3d 2 60t

Distance between van and car 5 d

60t

45st 1 3d


5 Rate ? Time 91.

5 p

5p s3dd2sd 1 2ds3dd2sd 1 2d

5p s3dd2sd 1 2d3d2 ? 3sd 1 2d

Circular pool volume

Rectangular pool volume5

p s3dd2sd 1 2dds3dds3d 1 6d

93. Average cost of Medicare per person 5107.30 1 15.09t billion34.26 1 0.65t million

5s10,730 1 1509td1000

3426 1 65t

95. Let and be polynomials. Thealgebraic expression is a rational expression.

uyvvu 97. The rational expression is in

simplified form if the numeratorand denominator have no factorsin common (other than ).±1

99. You can cancel only common factors.

Section 4.3 Multiplying and Dividing Rational Expressions231

Section 4.3 Multiplying and Dividing Rational Expressions

1. (a)

(c) x 5 22: 22 2 104s22d 5

21228

532

x 5 10: 10 2 104s10d 5

040

5 0 (b)

(d) x 5 12: 12 2 104s12d 5

248

51

24

x 5 0: 0 2 104s0d 5

2100

5 undefined

3.73y

57x2

3ysx2d , x Þ 0 5.3x

x 2 45

3xsx 1 2d2

sx 2 4dsx 1 2d2, x Þ 22

7.3u7v

53usu 1 1d7vsu 1 1d, u Þ 21 9.

13xx 2 2

513xss21ds2 1 xdd

4 2 x2 , x Þ 22

11.4528

?7760

59 ? 5 ? 7 ? 117 ? 4 ? 6 ? 10

53316

13. 7x ?9

14x5

7x ? 3 ? 37 ? 2 ? x

592

15. 8s3

9s?

6s2

32s5

8s3 ? 3 ? 2s ? s3 ? 3 ? s ? 8 ? 2 ? 2 ? s

5s3

6, s Þ 0 17. 16u 4 ?

128u2 5

8 ? 2 ? u2 ? u2 ? 128 ? u2 5 24u2, u Þ 0

19.

5 24, x Þ 234

8

3 1 4x? s9 1 12xd 5

8 ? 3s3 1 4xd3 1 4x

21.

52uvsu 1 vd3s3u 1 vd , u Þ 0

8u2v

3u 1 v?

u 1 v12u

54 ? 2 ? u ? u ? vsu 1 vd

s3u 1 vd ? 4 ? 3 ? u

23.12 2 r

3?

3r 2 12

521sr 2 12d ? 3

3sr 2 12d 5 21, r Þ 12 25.

5x 1 82x2 , x Þ

32

s2x 2 3dsx 1 8d

x3 ?x

3 2 2x5

s2x 2 3dsx 1 8dxx ? x2 ? 21s2x 2 3d

27. 4r 2 12r 2 2

?r2 2 4r 2 3

54sr 2 3dsr 2 2dsr 1 2d

sr 2 2d ? sr 2 3d 5 4sr 1 2d, r Þ 3, r Þ 2

29. 2t 2 2 t 2 15

t 1 2?

t 2 2 t 2 6t 2 2 6t 1 9

5s2t 1 5dst 2 3dst 2 3dst 1 2d

st 1 2dst 2 3dst 2 3d 5 2t 1 5, t Þ 3, t Þ 22

31. sx2 2 4y2d ?xy

sx 2 2yd2 5 sx 2 2ydsx 1 2yd ?xy

sx 2 2yd2 5sx 1 2ydxy

x 2 2y

33.x2 1 2xy 2 3y2

sx 1 yd2 ?x2 2 y2

x 1 3y5

sx 1 3ydsx 2 ydsx 1 yd2 ?

sx 2 ydsx 1 ydx 1 3y

5sx 2 yd2

x 1 y, x Þ 23y

35.

5s2x 1 1dsx 2 1ds3x 2 2dsx 1 2d, x Þ ±5, 21

5sx 1 5ds2x 1 1dsx 2 5dsx 2 1dsx 1 1dsx 2 5ds3x 2 2dsx 1 1dsx 1 5d(x 1 2d

x 1 5x 2 5

?2x2 2 9x 2 53x2 1 x 2 2

?x2 2 1

x2 1 7x 1 105

x 1 5x 2 5

?s2x 1 1dsx 2 5ds3x 2 2dsx 1 1d ?

sx 2 1dsx 1 1dsx 1 5dsx 1 2d


37.

5sx2 2 9ds2x 1 5dx2s3x 2 1ds2x 1 3ds3 2 2xd2s2x 1 1d , x Þ 0,

12

521sx 2 3dsx 1 3ds2x 1 5dx2s3x 2 1d

s2x 1 3ds2x 2 3d2s2x 1 1d

9 2 x2

2x 1 3?

4x2 1 8x 2 54x2 2 8x 1 3

?6x 4 2 2x3

8x2 1 4x5

s3 2 xds3 1 xd2x 1 3

?s2x 1 5ds2x 2 1ds2x 2 3ds2x 2 1d ?

2x3s3x 2 1d4xs2x 1 1d

39.

5sx 1 3d2

x, x Þ 22, 2, 3

5sx 1 3dsx2 2 4dsx 1 3dsx 2 3d

sx 2 3dsx2 2 4d ? x

x3 1 3x2 2 4x 2 12x3 2 3x2 2 4x 1 12

?x2 2 9

x5

x2sx 1 3d 2 4sx 1 3dx2sx 2 3d 2 4sx 2 3d ?

sx 1 3dsx 2 3dx

41. 2512

44532

52512

?3245

525 ? 8 ? 4

4 ? 3 ? 5 ? 95

2827

43. x2 43x4

5 x2 ?43x

54x3

, x Þ 0

45.

53y2

2ux2, v Þ 0

57xy2 ? 3 ? 3 ? 5 ? u ? v

5 ? 2 ? u ? u ? v ? 7 ? 3x ? x2

7xy2

10u2v4

21x3

45uv5

7xy2

10u2v?

45uv21x3 47.

53

2sa 1 bd

53sa 1 bd ? 2

2 ? 2 ? sa 1 bdsa 1 bd

3sa 1 bd

44

sa 1 bd2

25

3sa 1 bd4

?2

sa 1 bd2

49.

5 x 4ysx 1 2yd, x Þ 0, y Þ 0, x Þ 22y

5sx3ydsx2 ? xydsx 1 2yd

x2y

5sx3ydsx3ydsx 1 2yd2sx 1 2yd

sx 1 2yd2x2y

sx3yd2

sx 1 2yd2 4x2y

sx 1 2yd3 5sx3yd2

sx 1 2yd2 ?sx 1 2yd3

x2y

51.

53x10

, x Þ 0

5x2 ? 3 ? 3 ? 2

2 ? 2 ? 3 ? 5 ? x

5x2

12?

185x

1 x2

12215x

1825

x2

124

5x18

53.

525xsx 1 1d

2, x Þ 0, 5, 21

55 ? x ? s21dsx 2 5dsx 1 1d

sx 2 5d2

55 ? 5 ? x ? x ? s21dsx2 2 4x 2 5d

sx 2 5d ? 5 ? 2 ? x

525x2

x 2 5?

5 1 4x 2 x2

10x

1 25x2

x 2 521 10x

5 1 4x 2 x225

25x2

x 2 54

10x5 1 4x 2 x2

Section 4.3 Multiplying and Dividing Rational Expressions233

55.

5s4x 1 1dsx 1 3ds3x 2 1dsx 2 1d, x Þ 23, 2

14

5s4x 1 1ds4x 1 1dsx 1 3dsx 1 3ds3x 2 1dsx 1 3ds4x 1 1dsx 2 1d

5s4x 1 1ds4x 1 1ds3x 2 1dsx 1 3d ?

sx 1 3dsx 1 3ds4x 1 1dsx 2 1d

16x2 1 8x 1 13x2 1 8x 2 3

44x2 2 3x 2 1x2 1 6x 1 9

516x2 1 8x 1 13x2 1 8x 2 3

?x2 1 6x 1 94x2 2 3x 2 1

57.

5 sx 1 2d, x Þ ±2, 23

5sx 1 3dsx 2 2dsx 1 2dsx 1 2d

sx 2 2dsx 1 2dsx 1 3d

5sx 1 3dsx 2 2dsx 2 2dsx 1 2d ?

sx 1 2dsx 1 2dx 1 3

x2 1 3x 2 2x 2 6

x2 2 44

x 1 3x2 1 4x 1 4

5xsx 1 3d 2 2sx 1 3d

x2 2 4?

x2 1 4x 1 4x 1 3

59.

5 2sx2 2 3x 2 10dsx 1 2dsx2 2 4x 1 4dsx 1 3d , x Þ ±2, 7

5sx 2 5dsx 1 2dsx 2 2dsx 2 2d ?

sx 2 7dsx 1 2d21sx 2 7dsx 1 3d

5x2 2 3x 2 10x2 2 4x 1 4

?x2 2 5x 2 14

1sx2 2 4x 2 21d

1x2 2 3x 2 10

x2 2 4x 1 4 2121 1 4x 2 x2

x2 2 5x 2 1425

x2 2 3x 2 10x2 2 4x 1 4

421 1 4x 2 x2

x2 2 5x 2 1461.

5x 1 4

3, x Þ 22, 0

3x2

9?

3sx 1 4dx2 1 2x 4 4

xx 1 2

5x2

9?

3sx 1 4dxsx 1 2d ?

x 1 2x

63. 3xy 1 y4x

4 s3x 1 3d4 4y

3x5

ysx 1 1d4x

?1

3sx 1 1d ?3xy

514

, x Þ 21, 0, y Þ 0

65.

5s2x 2 5dsx 1 1d

x, x Þ 21, 25, 2

23

5s2x 2 5dsx 1 5dxs3x 1 2dsx 1 1dsx 1 1d

s3x 1 2dsx 1 1dsx 1 5dx ? x

2x2 1 5x 2 253x2 1 5x 1 2

?3x2 1 2x

x 1 54 1 x

x 1 122

5s2x 2 5dsx 1 5ds3x 1 2dsx 1 1d ?

xs3x 1 2dx 1 5

? 1x 1 1x 2

2

67.

5x 4

sx n 1 1d2, x n Þ 23, 3, 0

x3 ?x2n 2 9

x2n 1 4x n 1 34

x2n 2 2x n 2 3x

5 x3 ?sx n 2 3dsx n 1 3dsx n 1 3dsx n 1 1d ?

xsx n 2 3dsx n 1 1d


69. Keystrokes:

3 2

9 4

3 2y2

y1

−4

−2

5

4

71. Keystrokes:

3 15 4

5

3y2

y1

−9

−2

9

10

73. Area 5 12w 1 33 21w

22 5s2w 1 3dw

675.

5x

4s2x 1 1d

5x2

?x2

?1

xs2x 1 1d

Unshaded Area

Total Area5

x2

?x2

xs2x 1 1d 5 3x2

?x24 4 fxs2x 1 1dg

77.Unshaded Area

Total Area5

x ?x2

xs2x 1 1d 5x

2s2x 1 1d 79. (a)

(b)

(c)

5 105 seconds or 74

minutes

3 seconds

1 page? 35 pages 5 3 ? 35 seconds

3 seconds1 page

? x pages 5 3x seconds or x

20 minutes

t 5 3 seconds or 120

minutes

20 pages1 minute

520 pages

60 seconds5

1 page3 seconds

,

Y5 x x

x

x

x

Y5 x x

x x

X,T,u X,T,u

X,T,u X,T,u

X,T,u X,T,u

X,T,u X,T,

u

X,T,

u

X,T,

u X,T,u

1 d d

d

d d

3

x2 x2

4

4 2 ENTER

4 2 GRAPH

1

1

c

c

c

c

4

44

4

>

x2

x2

ENTER

GRAPH

81. (a) Keystrokes:

6357 1070

6115.2 590.7

(b)

Equation:

(c)

(d) The number of subscribers was increasing at a faster rate than the revenue.

83. Divide a rational expression by a polynomial by multiplying the rational expression by the reciprocal of the polynomial.

85. Invert the divisor, not the dividend.

AMB 5s6115.2 1 590.7t 2d1000

s6357 1 1070t 2d125

6115200 1 590700t 2

s6357 1 1070t 2d12

y2

y1

50,000

70

0

N

R

Y5 1 x2 ENTER

1 x2 GRAPH

Averagemonthly bill

VerbalModel:

5Revenue

Number of subscribers

Year, t 0 2 4 6

Monthly bill $80.16 $66.40 $55.21 $50.70

X,T,u

X,T,u


16.

55x

x 2 2

sx2 1 2xd ?5

x2 2 45

xsx 1 2d5sx 2 2)sx 1 2d 17.

58x

3sx 2 1dsx2 1 2x 2 3d

58x

3sx 2 1d2sx 1 3d

4

3sx 2 1d ?12x

6sx2 1 2x 2 3d 54s12xd

3sx 2 1d6sx 1 3dsx 2 1d

18.

54su 2 vd2

5uv

5u

3su 1 vd ?2su2 2 v2d

3v4

25u2

18su 2 vd 55u ? 2su 2 vdsu 1 vd ? 18su 2 vd

3su 1 vds3vds25u2d

19.

9t 2

3 2 t

6tt 2 3

?t 2 3t 2 3

529t 2

6t5 2

3t2

20.

52sx 1 1d

3x

510sx 1 1d

15x

10x2 1 2x

15

x2 1 3x 1 2

5

10xsx 1 2d

15

sx 1 2dsx 1 1d ?

xsx 1 2dsx 1 1dxsx 1 2dsx 1 1d

21. (a)

Equation:

(b) Average cost when x 5 500 units are produced 56000 1 10.50s500d

5005 $22.50

Average cost 56000 1 10.50x

x

Averagecost

VerbalModel:

45Numberof units

Totalcost

Section 4.4 Adding and Subtracting Rational Expressions

1.58

178

55 1 7

85

128

532

3.5x8

27x8

522x

85

2x4

5.23a

2113a

52 2 11

3a5

293a

523a 7.

x9

2x 1 2

95

x 2 sx 1 2d9

5x 2 x 2 2

95 2

29

9.z2

31

z2 2 23

5z2 1 z2 2 2

35

2z2 2 23

11.2x 1 5

31

1 2 x3

52x 1 5 1 1 2 x

35

x 1 63

13.

5 243

53y 2 3y 1 3 2 7

3

3y3

23y 2 3

32

73

53y 2 s3y 2 3d 2 7

315.

5 1, y Þ 6

5y 2 6y 2 6

53y 2 22 2 2y 1 16

y 2 6

3y 2 22y 2 6

22y 2 16y 2 6

53y 2 22 2 s2y 2 16d

y 2 6



1.

53t 9

5 3t29

5 3t21213

st 3d24s3t 3d 5 t212s3t 3d 2.

53x6

16y3

53

16x6y23

5 3 ?142 ? x214y211(22)

s3x2y21ds4x22yd22 5 3x2y21 ? 422x4y22

6. (a)

(b)3.2 3 104

16 3 107 53.216

3 1023 5 0.2 3 1023 5 2 3 1024

s3 3 103d4 5 34 3 1012 5 81 3 1012 5 8.1 3 1013 7.

D 5 s2`, 0d < s0, 4d < s4, `d

y Þ 4

y 2 4 Þ 0

y Þ 0

ysy 2 4d Þ 0

8.

(a)

(c)

51 2 9

1 1 1 2 25

280

5 undefined

hs21d 5s21d2 2 9

s21d2 2 s21d 2 2

59 2 9

9 1 3 2 25

010

5 0

hs23d 5s23d2 2 9

s23d2 2 s23d 2 2

h sxd 5x2 2 9

x2 2 x 2 2(b)

(d)

525 2 9

25 2 5 2 25

1618

589

hs5d 552 2 9

52 2 5 2 2

52922

592

hs0d 502 2 9

02 2 0 2 2

12.

5z 1 3

2z 2 1

sz 1 3d2

2z2 1 5z 2 35

sz 1 3dsz 1 3ds2z 2 1dsz 1 3d

13.

57 1 3ab

a

7ab 1 3a2b2

a2b5

abs7 1 3abda2b

9.9y2

6y5

3y2

10.8u3v2

36uv3 52u2

9v11.

52x 1 1

2x

5s2x 2 1ds2x 1 1d

2xs2x 2 1d

4x2 2 1x 2 2x2 5

s2x 2 1ds2x 1 1dxs1 2 2xd

3.10u22

15u5

1015

u2221 523

u23 52

3u3 4.s10xd0x22

sx2d21 5x22

x22 5 1 5. (a)

(b) 0.00075 5 7.5 3 1024

13,400,000 5 1.34 3 107

14.

5n2

m 1 n

2mn2 2 n3

2m2 1 mn 2 n2 5n2s2m 2 nd

s2m 2 ndsm 1 nd 15.11t 2

6?

933t

511t 2s9d6s33td 5

t2


16.

55x

x 2 2

sx2 1 2xd ?5

x2 2 45

xsx 1 2d5sx 2 2)sx 1 2d 17.

58x

3sx 2 1dsx2 1 2x 2 3d

58x

3sx 2 1d2sx 1 3d

4

3sx 2 1d ?12x

6sx2 1 2x 2 3d 54s12xd

3sx 2 1d6sx 1 3dsx 2 1d

18.

54su 2 vd2

5uv

5u

3su 1 vd ?2su2 2 v2d

3v4

25u2

18su 2 vd 55u ? 2su 2 vdsu 1 vd ? 18su 2 vd

3su 1 vds3vds25u2d

19.

9t 2

3 2 t

6tt 2 3

?t 2 3t 2 3

529t 2

6t5 2

3t2

20.

52sx 1 1d

3x

510sx 1 1d

15x

10x2 1 2x

15

x2 1 3x 1 2

5

10xsx 1 2d

15

sx 1 2dsx 1 1d ?

xsx 1 2dsx 1 1dxsx 1 2dsx 1 1d

21. (a)

Equation:

(b) Average cost when x 5 500 units are produced 56000 1 10.50s500d

5005 $22.50

Average cost 56000 1 10.50x

x

Averagecost

VerbalModel:

45Numberof units

Totalcost

Section 4.4 Adding and Subtracting Rational Expressions

1.58

178

55 1 7

85

128

532

3.5x8

27x8

522x

85

2x4

5.23a

2113a

52 2 11

3a5

293a

523a 7.

x9

2x 1 2

95

x 2 sx 1 2d9

5x 2 x 2 2

95 2

29

9.z2

31

z2 2 23

5z2 1 z2 2 2

35

2z2 2 23

11.2x 1 5

31

1 2 x3

52x 1 5 1 1 2 x

35

x 1 63

13.

5 243

53y 2 3y 1 3 2 7

3

3y3

23y 2 3

32

73

53y 2 s3y 2 3d 2 7

315.

5 1, y Þ 6

5y 2 6y 2 6

53y 2 22 2 2y 1 16

y 2 6

3y 2 22y 2 6

22y 2 16y 2 6

53y 2 22 2 s2y 2 16d

y 2 6

Section 4.4 Adding and Subtracting Rational Expressions237

17.

51

x 2 3, x Þ 0

5x

xsx 2 3d

2x 2 1

xsx 2 3d 11 2 x

xsx 2 3d 52x 2 1 1 1 2 x

xsx 2 3d 19.

LCM 5 20x3

20x3 5 5 ? 2 ? 2 ? x ? x ? x

5x2 5 5 ? x ? x

21.

LCM 5 3 ? 3 ? 2 ? 2 ? y ? y ? y 5 36y3

12y 5 2 ? 2 ? 3 ? y

9y3 5 3 ? 3 ? y ? y ? y 23.

LCM 5 15x2sx 1 5d

3sx 1 5d 5 3 ? sx 1 5d

15x2 5 5 ? 3 ? x ? x

25.

LCM 5 126z2sz 1 1d4

14sz 1 1d4 5 7 ? 2 ? sz 1 1d4

63z2sz 1 1d 5 7 ? 9 ? z ? zsz 1 1d 27.

LCM 5 2 ? 2 ? 2 ? 7 ? t ? st 1 2dst 2 2d 5 56tst 2 2 4d

14st 2 2 4d 5 2 ? 7 ? st 1 2dst 2 2d

8tst 1 2d 5 2 ? 2 ? 2 ? t ? st 1 2d

29.

LCM 5 6xsx 2 2dsx 1 2d

2xsx 1 2d 5 2 ? x ? sx 1 2d

6sx2 2 4d 5 6sx 2 2dsx 1 2d 31.7x2

4asx2d 57

4a, x Þ 0

33.5rsu 1 1d3vsu 1 1d 5

5r3v

, u Þ 21 35.

5 21sx 2 2ds2 1 xd

4 2 x2 5 s2 2 xds2 1 xd

7ys21sx 1 2dd

4 2 x2 57y

x 2 2, x Þ 22

37.

LCD 5 6n2sn 2 4d

106n2 5

103 ? 2n2 5

10sn 2 4d3 ? 2n2sn 2 4d 5

10sn 2 4d6n2sn 2 4d

n 1 8

3n 2 125

n 1 83sn 2 4d 5

n 1 8s2n2d3sn 2 4ds2n2d 5

2n2sn 1 8d6n2sn 2 4d 39.

LCD 5 x2sx 2 3dsx 1 3d 5 x2sx2 2 9d

5xsx 1 3d 5

5xsx 2 3dx2sx 1 3dsx 2 3d

2x2sx 2 3d 5

2sx 1 3dx2sx 2 3dsx 1 3d

41.

LCD 5 6v2sv 1 1d

43v2 5

4s2sv 1 1dd3v2s2sv 1 1dd 5

8v 1 86v2sv 1 1d

v2v2 1 2v

5v

2vsv 1 1d 5vs3vd

2vsv 1 1ds3vd 53v2

6v2sv 1 1d 43.

LCD 5 sx 2 5d2sx 1 5d

59xsx 1 5d

sx 2 5d2sx 1 5d 59xsx 1 5d

sx 2 5d2sx 1 5d

9x

x2 2 10x 1 255

9xsx 2 5d2

5sx 2 8dsx 2 5d

sx 2 5dsx 1 5dsx 2 5d 5sx 2 8dsx 2 5dsx 2 5d2sx 1 5d

x 2 8

x2 2 255

x 2 8sx 2 5dsx 1 5d

45.54x

235

55s5d4xs5d 2

3s4xd5s4xd 5

2520x

212x20x

525 2 12x

20x47.

7a

114a2 5

7sadasad 1

14s1da2s1d 5

7aa2 1

14a2 5

7a 1 14a2


49.

520 2 20

x 2 45 0, x Þ 4

520

x 2 42

20x 2 4

20

x 2 41

204 2 x

520s1d

sx 2 4ds1d 120s21d

s4 2 xds21d 51.

53x 1 6x 2 8

53x

x 2 81

6x 2 8

3x

x 2 82

68 2 x

53xs1d

sx 2 8ds1d 26s21d

s8 2 xds21d

53.

525x 1 100 1 10

x 1 45

25x 1 110x 1 4

525sx 1 4d

x 1 41

10x 1 4

25 110

x 1 45

25sx 1 4d1sx 1 4d 1

10s1dsx 1 4ds1d 55.

53x 2 23x 2 2

5 1, x Þ23

53x

3x 2 21

223x 2 2

3x

3x 2 21

22 2 3x

53xs1d

3x 2 2s1d 12s21d

s2 2 3xds21d

57.

51

2xsx 2 3d

53

6xsx 2 3d

52x 1 3 1 x

6xsx 2 3d

52sx 2 3d6xsx 2 3d 1

x6xsx 2 3d

216x

11

6sx 2 3d 521sx 2 3d6xsx 2 3d 1

1sxd6sx 2 3dx 59.

5x2 2 7x 2 15sx 1 3dsx 2 2d

5x2 2 2x 2 5x 2 15

sx 1 3dsx 2 2d

5xsx 2 2d

sx 1 3dsx 2 2d 25sx 1 3d

sx 2 2dsx 1 3d

x

x 1 32

5x 2 2

5xsx 2 2d

sx 1 3dsx 2 2d 25sx 1 3d

sx 2 2dsx 1 3d

61.

5x 2 2

xsx 1 1d

53x 2 2x 2 2

xsx 1 1d

53x

xsx 1 1d 22sx 1 1dxsx 1 1d

3

x 1 12

2x

53x

sx 1 1dx 22sx 1 1dxsx 1 1d 63.

55x 1 5

sx 2 5dsx 1 5d

53x 1 15 1 2x 2 10

sx 2 5dsx 1 5d

53sx 1 5d

sx 2 5dsx 1 5d 12sx 2 5d

sx 1 5dsx 2 5d

3

x 2 51

2x 1 5

53sx 1 5d

sx 2 5dsx 1 5d 12sx 2 5d

sx 1 5dsx 2 5d

65.

54

x2sx2 1 1d

54x2 1 4 2 4x2

x2sx2 1 1d

54sx2 1 1dx2sx2 1 1d 2

4x2

x2sx2 1 1d

4x2 2

4x2 1 1

54sx2 1 1dx2sx2 1 1d 2

4x2

sx2 1 1dx2


67.

5x2 1 x 1 9


5x2 2 2x 1 3x 1 9


5xsx 2 2d

sx 2 2dsx 2 3dsx 1 3d 13sx 1 3d


x

x2 2 91

3x2 2 5x 1 6

5x

sx 2 3dsx 1 3d 13

sx 2 3dsx 2 2d

69.

54x

sx 2 4d2

54x 2 16 1 16

sx 2 4d2

54x 2 16sx 2 4d2 1

16sx 2 4d2

4

x 2 41

16sx 2 4d2 5

4sx 2 4dsx 2 4dsx 2 4d 1

16s1dsx 2 4d2s1d 71.

5sy 2 xdsy 1 xd

xysx 1 yd 5y 2 x

xy, x Þ 2y

5y2 2 x2

xysx 1 yd

5y2

xysx 1 yd 2x2

xysx 1 yd

5ysyd

xsx 1 ydsyd 2xsxd

ysx 1 ydsxd

y

x2 1 xy2

xxy 1 y2 5

yxsx 1 yd 2

xysx 1 yd

73.

52s4x2 1 5x 2 3d

x2sx 1 3d

58x2 1 10x 2 6

x2sx 1 3d

54x2 1 12x 2 2x 2 6 1 4x2

x2sx 1 3d

54x2 1 12xx2sx 1 3d 2

2x 1 6x2sx 1 3d 1

4x2

x2sx 1 3d

4x

22x2 1

4x 1 3

54xsx 1 3d

xsxdsx 1 3d 22sx 1 3dx2sx 1 3d 1

4sx2dsx 1 3dx2

75.

5 2u2 2 uv 2 5u 1 2v

su 2 vd2

55u 2 2v 2 u2 1 uv

su 2 vd2

53u 1 2u 2 2v 2 u2 1 uv

su 2 vd2

53u

su 2 vd2 12u 2 2v 2 u2 1 uv

su 2 vd2

53us1d

su 2 vd2s1d 1s2 2 udsu 2 vdsu 2 vdsu 2 vd

53u

su 2 vd2 12 2 uu 2 v

3u

u2 2 2uv 1 v2 12

u 2 v2

uu 2 v

77.

5x

x 2 1, x Þ 26

5xsx 1 6d

sx 2 1dsx 1 6d

5x2 1 6x

sx 2 1dsx 1 6d

5x2 1 8x 1 12 2 2x 1 2 2 14

sx 2 1dsx 1 6d

5x2 1 8x 1 12sx 2 1dsx 1 6d 2

2x 2 2sx 1 6dsx 2 1d 2

14sx 1 6dsx 2 1d

x 1 2x 2 1

22

x 1 62

14x2 1 5x 2 6

5sx 1 2dsx 1 6dsx 2 1dsx 1 6d 2

2sx 2 1dsx 1 6dsx 2 1d 2

14s1dsx 1 6dsx 2 1ds1d


79. Keystrokes:

2 4

2

6 4 2

y1 5 y2

2x

14

sx 2 2d 52sx 2 2dxsx 2 2d 1

4xxsx 2 2d 5

2x 2 4 1 4xxsx 2 2d 5

6x 2 4xsx 2 2d

y2

y1

−8 10

−6

6

Y5 x 4

x

41 x

x 2

x x

ENTER

x 2

x

4 x x 2

x x GRAPH

81.

5x

2s3x 1 1d, x Þ 0

5x

6x 1 2

5

12

? 2x

3s2xd 11xs2xd

12

13 11x2

5

12

13 11x2

?2x2x

83.

54 1 3x4 2 3x

, x Þ 0

14

x1 32

14x

2 325

14x

1 3214

x2 32

?xx

X,T,u

X,T,u

X,T,u X,T,u X,T,u

85.

5 24x 2 1, x Þ 0, 14

54x 1 1

21

5s4x 2 1ds4x 1 1d

21s4x 2 1d

516x2 2 11 2 4x

516xsxd 2

1xsxd

1xsxd 2 4sxd

116x 2

1x2

11x

2 425

116x 21x2

11x

2 42?

xx

87.

53x4x

534

, x Þ 0, 3

53x 2 9 1 9

4x 2 12 1 12

53sx 2 3d 1

9x 2 3

sx 2 3d

4sx 2 3d 112

x 2 3sx 2 3d

13 1

9x 2 32

14 112

x 2 325

13 19

x 2 3214 1

12x 2 32

?x 2 3x 2 3


93.

5 2sy 2 3dsy 2 1d

ys4y 2 1d , y Þ 3

5y2 2 4y 1 3

2ys21 1 4yd

5y2 2 3y 2 y 1 3

y 2 4y2

5ysy 2 3d 2 sy 2 3d

ys1 2 4yd

11 2

1y2

11 2 4yy 2 3 2

511 2

1y2

11 2 4yy 2 3 2

?ysy 2 3dysy 2 3d 95.

5x2 1 6x

3x3 1 10x 2 30, x Þ 0, x Þ 3

53x2 2 2x2 1 6x10x 2 30 1 3x3

53x2 2 2xsx 2 3d10sx 2 3d 1 3x2

1 x

x 2 32

232

1103x

1x2

x 2 325

1 xx 2 3

2232

1103x

1x2

x 2 32?

3xsx 2 3d3xsx 2 3d

89.

55s3 1 xd

2xs5x 2 2d

515 1 5x

10x2 2 4x

1 3

x2 11x2

12 245x2

51 3

x2 11x2

12 245x2

?5x2

5x2 91.

5 y 2 x, x Þ 0, y Þ 0, x Þ 2y

5sy 2 xdsy 1 xd

x 1 y

5y2 2 x2

x 1 y

5

yx

sxyd 2xy

sxyd

1x 1 yxy 2xy

1y

x2

xy2

1x 1 yxy 2

51y

x2

xy2

1x 1 yxy 2

?xyxy

97.

521

2s2 1 hd

52h

2hs2 1 hd

52 2 2 2 h2hs2 1 hd

52 2 s2 1 hd2hs2 1 hd

5

12 1 h

212

h?

2s2 1 hd2s2 1 hd

f s2 1 hd 2 f s2d

h5

12 1 h

212

h


99.

Keystrokes:

1 1

1 1

1

Zero and one are not in the domain of

but are in the domain of The two expressions are equivalent except at x 5 0 and x 5 1.x

x 1 1.

1 21x

1 21x2

y2

y1

−7

−4

5

4

0 1 2 3

2 Undef. Undef. Undef.

2 Undef. 034

23

12

32

xx 1 1

34

23

32

11 21x2

11 21x22

212223x

Y5 x 2 4

x

4

x 2 4 x2

x

ENTER

4 x 1

x

GRAPH

101.

55t12

53t12

12t12

t4

1t6

5ts3d4s3d 1

ts2d6s2d 103.

55x24

53x 1 2x

24

5

x4

s12d 1x6

s12d

2s12d

x4

1x6

25

1x4

1x62

2?

1212

105.

Thus,

x2 511x45

12x45

513x45

.

x1 5x5

12x45

59x45

12x45

511x45

x3

2x5

3?

1515

5 5x 2 3x 52x45

X,T,u

X,T,u

X,T,uX,T,u

107.

5R1R2

R2 1 R1

5R1R2

1R1

sR1R2d 11R2

sR1R2d

1

1 1R1

11R22

51

1 1R1

11R22

?R1R2

R1R2

109. (a)

Equation: Upstream

Downstream

(b)

(c)

5100

s5 2 xds5 1 xd

550 1 10x 1 50 2 10x

s5 2 xds5 1 xd

t sxd 510s5 1 xd

s5 2 xds5 1 xd 110s5 2 xd

s5 2 xds5 1 xd

Total time 5 tsxd 510

5 2 x1

105 1 x

t 510

5 1 x

t 510

5 2 x

DistanceRate

5 Time


5 ?Rate Time

Section 4.5 Dividing Polynomials 243

111. Rewrite each fraction in terms of the lowest commondenominator, combine the numerators, and place theresult over the lowest common denominator.

113.

The subtraction must be distributed to both terms of the numerator of the second fraction.

523x 1 10

x 1 4

5x 2 1 2 4x 1 11

x 1 4

x 2 1x 1 4

24x 2 11x 1 4

5sx 2 1d 2 s4x 2 11d

x 1 4

Section 4.5 Dividing Polynomials

1.6z 1 10

25

6z2

1102

5 3z 1 5 3.

55z2

21 z 2 3

10z2 1 4z 2 12

45

10z2

41

4z4

2124

5.

5 7x2 2 2x, x Þ 0

57x3

x2

2x2

x

s7x3 2 2x2d 4 x 57x3 2 2x2

x7.

5 m3 1 2m 27m

, m Þ 0

m4 1 2m2 2 7

m5

m4

m1

2m2

m2

7m

9.

5 210z2 2 6, z Þ 0

50z3 1 30z

25z5

50z3

25z1

30z25z

11.

5 4z2 132

z 2 1, z Þ 0

8z3 1 3z2 2 2z

2z5

8z3

2z1

3z2

2z2

2z2z

13.

55x2

2 4 172

y, x Þ 0, y Þ 0

55x2y2xy

28xy2xy

17xy2

2xy

s5x2y 2 8xy 1 7xy2d 4 2xy 55x2y 2 8xy 1 7xy2

2xy15.

25x 1 15

25x 1 15

x2 2 3x

x2 2 8x 1 15

x 2 35 x 2 3 ) x2 2 8x 1 15

x 2 5, x Þ 3

17.

10x 1 50

10x 1 50

x2 1 5x

sx2 1 15x 1 50d 4 sx 1 5d 5 x 1 5 ) x2 1 15x 1 50

x 1 10, x Þ 25

19.

2

23x 1 6

23x 1 8

x2 2 2x

x 2 2 ) x2 2 5x 1 8

x 2 3 12

x 2 2

21.

27x 1 21

27x 1 21

2x2 1 3x

2x 1 3 ) 2x2 2 4x 1 21

x 1 7, x Þ 3


23.

19

28x 2 16

28x 1 3

5x2 1 10x

x 1 2 ) 5x2 1 2x 1 3

5x 2 8 119

x 1 225.

211

9x 1 6

9x 2 5

12x2 1 8x

3x 1 2 ) 12x2 1 17x 2 5

4x 1 3 1211

3x 1 2

27.

210t 1 25

210t 1 25

12t 2 2 30t

2t 2 5 ) 12t 2 2 40t 1 25

6t 2 5, t Þ52

29.

6y 1 3

6y 1 3

2y2 1 y

2y 1 1 ) 2y2 1 7y 1 3

y 1 3, y Þ 212

31.

4x 2 8

4x 2 8

x3 2 2x2

x 2 2 ) x3 2 2x2 1 4x 2 8

x2 1 4, x Þ 2

33.

6

4x 2 12

4x 2 6

x2 2 3x

x2 1 x

2x3 2 6x2

x 2 3 ) 2x3 2 5x2 1 x 2 6

2x2 1 x 1 4 16

x 2 3

35.

5

2x 1 4

x 1 2 ) 2x 1 9

2 15

x 1 237.

32

24x 2 16

24x 1 16

x2 1 4x

x 1 4 ) x2 1 0x 1 16

x 2 4 132

x 1 4

39.

4125

415

z 2 4125

415

z 1 0

6z2 2 65

z

5z 2 1 ) 6z2 1 7z 1 0

65

z 1 4125

141

25s5z 2 1d41.

24x 2 1

24x 2 1

16x2 1 4x

4x 1 1 ) 16x2 1 0x 2 1

4x 2 1, x Þ 214


43.

25x 1 125

25x 1 125

25x2 2 25x

25x2 1 0x

x3 1 5x2

x 1 5 ) x3 1 0x2 1 0x 1 125

x2 2 5x 1 25, x Þ 25

45.

2x2 1 4x 1 6

2x2 1 4x 1 6

x3 1 2x2 1 3x

x2 1 2x 1 3 ) x3 1 4x2 1 7x 1 6

x 1 2

47.

52x 2 55

25x2 2 75x 1 50

25x2 2 23x 2 5

12x3 2 36x2 1 24x

12x3 2 11x2 1 x

4x4 2 12x3 1 8x2

x2 2 3x 1 2 ) 4x4 1 0x3 2 3x2 1 x 2 5

4x2 1 12x 1 25 152x 2 55

x2 2 3x 1 2

49.

0

x 2 1

x 2 1

x2 2 x

x2

x3 2 x2

x3

x4 2 x3

x4

x5 2 x4

x5

x6 2 x5

x 2 1 ) x6 2 1

x5 1 x4 1 x3 1 x2 1 x 1 1, x Þ 1

51.

x

2x3 2 x

2x3

x5 1 x3

x2 1 1 ) x5

x3 2 x 1x

x2 1 1

53.4x4

x3 2 2x 5 4x 2 2x 5 2x, x Þ 0 55.

5 7uv, u Þ 0, v Þ 0

5 4uv 1 3uv

8u2v2u

13suvd2

uv5 4uv 1

3u2v2

uv

57.

x3 2 5x2 1 3x 2 4x 2 2

5 x2 2 3x 2 3 210

x 2 2

2 1

1

252

23

3 26

23

2426

210

x3 2 5x2 1 3x 2 4x 2 2

59.

x3 1 3x2 2 1x 1 4

5 x2 2 x 1 4 1217x 1 4

24 1

1

324

21

0 4

4

21216

217

x3 1 3x2 2 1x 1 4


61.

x4 2 4x3 1 x 1 10x 2 2

5 x3 2 2x2 2 4x 2 7 124

x 2 2

2 1

1

24 2

22

0 24

24

1 28

27

10214

24

x4 2 4x3 1 x 1 10x 2 2

63.

5x3 2 6x2 1 8x 2 4

5 5x2 1 14x 1 56 1232

x 2 4

4 5

5

2620

14

056

56

8224

232

5x3 2 6x2 1 8x 2 4

65.

10x4 2 50x3 2 800x 2 6

5 10x3 1 10x2 1 60x 1 360 11360x 2 6

6 10

10

250 60

10

0 60

60

0 360

360

28002160

1360

10x4 2 50x3 2 800x 2 6

67.

0.1x2 1 0.8x 1 1x 2 0.2

5 0.1x 1 0.82 11.164

x 2 0.2

0.2 0.1

0.1

0.80.02

0.82

10.164

1.164

0.1x2 1 0.8x 1 1x 2 0.2

69.

x3 2 13x 1 12 5 sx 2 3dsx 1 4dsx 2 1d

x2 1 3x 2 4 5 sx 1 4dsx 2 1d

3 1

1

0 3

3

213 9

24

12212

0

71.

6x3 2 13x2 1 9x 2 2 5 sx 2 1ds2x 2 1ds3x 2 2d

6x2 2 7x 1 2 5 s2x 2 1ds3x 2 2d

1 6

6

2136

27

927

2

222

0

73.

9x3 1 45x2 2 4x 2 20 5 sx 1 5ds3x 2 2ds3x 1 2d

9x2 2 4 5 s3x 2 2ds3x 1 2d

25 9

9

45245

0

240

24

22020

0

75.

x4 1 7x3 1 3x2 2 63x 2 108 5 sx 1 3d2sx 1 4dsx 2 3d

5 sx 1 4dsx 2 3dsx 1 3d

5 sx 1 4dsx2 2 9d

x3 1 4x2 2 9x 2 36 5 x2sx 1 4d 2 9sx 1 4d

23 1

1

7 23

4

3 212

29

263 27

236

2108108

0

77.

5 5s3x 1 2d1x 2452

15x2 2 2x 2 8 5 s15x 1 10d1x 2452

45 15

15

2212

10

288

0

15x2 2 2x 2 8

x 245

79.

c 5 28

c 1 8 5 0

2 1

1

22

4

248

4

c8

0

x3 2 2x2 2 4x 1 cx 2 2

81. Keystrokes:

4 2

1 2 2

So, y1 5 y2.

x 1 42x

5x2x

142x

512

12x

y2

y1

−6

−6

12

6

Y5 x X,T, u 1

x

4 X,T, u ENTER

x 4

x

1 x 4 X,T, u

x

GRAPH


83. Keystrokes:

3 1 1

1

x3 1 1x 1 1

5sx 1 1dsx2 2 x 1 1d

x 1 15 x2 2 x 1 1, x Þ 1

y2

y1

−4

−1

5

5

Y5 x X,T, u > 1 x x4 X,T, u 1 x

ENTER

X,T, u x2 2 X,T, u 1 GRAPH

85.

4xn 1 8

4xn 1 8

x2n 1 2xn

x2n 1 6xn

x3n 1 2x2n

xn 1 2 ) x3n 1 3x2n 1 6xn 1 8

x2n 1 xn 1 4, xn Þ 22

87.

5 x3 2 5x2 2 5x 2 10

5 x3 1 x2 1 x 2 6x2 2 6x 2 6 2 4

5 sx 2 6d ? sx2 2 x 1 1d 2 4

Dividend 5 Divisor ? Quotient 1 Remainder

89.

The polynomial values equal the remainders.

5 0

5 8 2 4 2 4

f s2d 5 23 2 22 2 2s2d

5 22

5 1 2 1 2 2

f s1d 5 13 2 12 2 2s1d

5 298

5 18 2

28 2

88

5 18 2

14 2 1

f s12d 5 s1

2d32 s1

2d22 2s1

2d

5 0

f s0d 5 03 2 02 2 2s0d

5 0

5 21 2 1 1 2

f s21d 5 s21d3 2 s21d2 2 2s21d

Polynomial value Divisor Remainder

0 0

0 0 0

1

2 0 0x 2 2

22x 2 122

298x 2

122

98

12

x

x 1 121

28x 1 22822

x

2 1

1

212

1

222

0

00

0

1 1

1

211

0

220

22

022

22

12 1

1

2112

212

22

214

294

0

298

298

0 1

1

210

21

220

22

00

0

21 1

1

2121

22

222

0

00

0


Section 4.6 Solving Rational Equations

1. (a)

Not a solution

0 Þ43

03

205

5? 4

3

x 5 0 (b)

Not a solution

2215

Þ2015

2515

1315

5? 20

15

2515

22315

5? 20

15

213

2215

5? 4

3

x 5 21 (c)

Not a solution

2

120Þ

160120

5

1202

3120

5? 160

120

1

242

140

5? 4

3

1y83

21y85

5? 4

3

x 518

(d)

Solution

2015

52015

5015

23015

5? 20

15

103

2105

5? 4

3

x 5 10

3. (a)

Not a solution

21 Þ 1

214

1234

5?

1

214

13

4s21d 5?

1

x 5 21 (b)

Solution

1 5 1

14

134

5?

1

14

13

4s1d 5?

1

x 5 1 (c)

Solution

1 5 1

34

114

5?

1

34

13

125?

1

34

13

4s3d 5?

1

x 5 3 (d)

Not a solution

78

Þ 1

48

138

5?

1

24

13

4s2d 5?

1

x 5 2

91. so

26x 2 9

26x 2 9

2x3 1 3x2

2x 1 3 ) 2x3 1 3x2 2 6x 2 9

x2 2 3

Length 52x3 1 3x2 2 6x 2 9

2x 1 35 x2 2 3

Width 5Area

Length.Area 5 Length ? Width,

95. is not a factor of the numerator.x 97. A divisor divides evenly into a dividend when the remainder is 0 and the divisor is a factor of the dividend.

99. True. If then nsxd 5 dsxd ? qsxd.nsxddsxd 5 qsxd,

93.

5 2x 1 8 or 2sx 1 4d

52sx2 1 6x 1 8d

x 1 2

Height 52sArea of triangled

Base

Area of triangle 512

? Base ? Height

5 x2 1 6x 1 8

5x3 1 18x2 1 80x 1 96

x 1 12

Area of triangle 5Volume

Height (of prism)

Volume 5 Area of triangle ? Height (of prism)

Section 4.6 Solving Rational Equations 249

5. Check:

x 5 10 23

523

x 2 6 5 4

53

233

5? 2

3 61x

62 12 5 12

326

106

2 1 5? 2

3 x6

2 1 523

7. Check:

z 5 8

5z 5 40

4z 1 8 5 48 2 z 103

5103

4sz 1 2d 5 48 2 z

103

5? 12

32

23

121z 1 23 2 5 14 2

z12212

8 1 2

35?

4 28

12 z 1 2

35 4 2

z12

9.

29

325 y

29 5 32y

4y 2 18 5 36y 2 9

2s2y 2 9d 5 36y 2 9

s12d12y 2 96 2 5 13y 2

342s12d

2y 2 9

65 3y 2

34 Check:

25132

5 25132

1612

30632 2 5

?2

5132

1612

1832

228832 2 5

?2

2732

22432

2s2 9

32d 2 9

65?

3129

322 234

11.

t 5 10

9t 5 90

8t 5 90 2 t

614t3 2 5 115 2

t626

4t3

5 15 2t6

Check:

403

5403

403

5? 45

32

53

4s10d

35?

15 2106

13.

y 5 229

9y 5 22

5y 2 1 1 4y 5 23

1215y 2 112

1y32 5 12

14212

5y 2 1

121

y3

5 214

15.

h 574

4h 5 7

4h 1 23 5 30

9h 1 18 2 5h 1 5 5 30

9sh 1 2d 2 5sh 2 1d 5 30

451h 1 25

2h 2 1

9 2 5 123245

h 1 2

52

h 2 19

523

Check:

23

523

812

5? 2

3

9

122

112

5? 2

3

34

21

125? 2

3

151

154 2 2

191

342 5

? 23

151

74

1842 2

191

74

2442 5

? 23

74 1 2

52

74 2 1

95? 2

3


17.

x 5438

28x 5 243

29x 1 47 5 4 2 x

3x 1 15 2 12x 1 32 5 4 2 x

3sx 1 5d 2 4s3x 2 8d 5 4 2 x

121x 1 54

23x 2 8

3 2 5 14 2 x12 212

x 1 5

42

3x 2 83

54 2 4

12Check:

1812

11122 5

1812

11122

181

24912

226012 2 5

? 1812

11122

181

834

2653 2 5

? 1812

11122

141

838 2 2

131

658 2 5

? 11212

118 2

141

438

1408 2 2

131

1298

2648 2 5

? 1121

328

2438 2

438 1 5

42

3s438 d 2 8

35? 4 2 s43

8 d12

19.

61 5 y

36 5 225 1 y

36 5 2s25 2 yd

4s25 2 yd1 925 2 y2 5 12

1424s25 2 yd

9

25 2 y5 2

14

Check:

214

5 214

29

365 2

14

9

25 2 615?

214

21.

a 5185

a 53610

10a 5 36

15a 2 36 5 5a

3a15 212a 2 5 15

323a

5 212a

553

Check:

53

553

153

2103

553

5 26018

553

5 212185

5? 5

323.

2625

5 x

26 5 25x

40 2 14 5 25x

10x14x

275x2 5 12

12210x

4x

275x

5 212

Check:

212

5 212

21326

5?

212

22026

17

265?

212

4

2625

27

51 26252

5?

212

25.

3 5 y

9 5 3y

y 1 29 5 4y 1 20

24 1 y 1 5 5 4sy 1 5d

2sy 1 5d1 12y 1 5

1122 5 s2d2sy 1 5d

12

y 1 51

12

5 2 Check:

2 5 2

42

5 2

32

112

5 2

12

3 1 51

12

5?

2


27.

3 5 x

30 5 10x

15x 1 30 5 25x

15sx 1 2d 5 25x

3xsx 1 2d15x2 5 1 25

3sx 1 2d23xsx 1 2d

5x

525

3sx 1 2d Check:

53

553

53

52515

53

5? 25

3s3 1 2d

29.

x 5 2115

5x 5 211

8x 1 16 5 3x 1 5

8sx 1 2d 5 3x 1 5

s3x 1 5dsx 1 2d1 83x 1 52 5 1 1

x 1 22s3x 1 5dsx 1 2d

8

3x 1 55

1x 1 2

Check:

25 5 25

8

285

51

215

8

2335 1

255

51

2115 1

105

8

3s2115 d 1 5

5? 1

2115 1 2

31.

x 543

x 5129

9x 5 12

10x 2 10 5 x 1 2

15x 2 5x 2 10 5 x 1 2

15x 2 5sx 1 2d 5 x 1 2

5xsx 1 2d1 3x 1 2

21x2 5 1 1

5x25xsx 1 2d

3

x 1 22

1x

515x

Check:

3

205

320

1820

21520

53

20

9

102

34

53

20

3103

2143

51203

1

43 1 2

2143

5? 1

5s43d

33.

x 5 6 x 5 26

sx 2 6dsx 1 6d 5 0

x2 2 36 5 0

x2 5 36

2x21122 5 118

x2 22x2

12

518x2 Check:

12

512

12

512

12

51836

12

51836

12

5? 18

s26d2 12

5? 18

62


35.

t 5 4 t 5 24

0 5 st 2 4dst 1 4d

0 5 t 2 2 16

16 5 t 2

32 5 2t 2

t132t 2 5 s2tdt

32t

5 2t Check:

28 5 28 8 5 8

3224

5?

2s24d 324

5?

2s4d

37.

x 5 29 x 5 8

sx 1 9dsx 2 8d 5 0

x2 1 x 2 72 5 0

x2 1 x 5 72

xsx 1 1d 5 172x 2x

x 1 1 572x

Check:

9 5 9 28 5 28

8 1 1 5? 72

8 29 1 1 5

? 7229

39.

y 5 13 y 5 3

sy 2 13dsy 2 3d 5 0

y2 2 16y 1 39 5 0

y2 5 16y 2 39

y2s1d 5 316y

239y2 4y2

1 516y

239y2 Check:

1 5 1 1 5 1

1 5163

2133

1 51613

23

13

1 5? 16

32

3932 1 5

? 1613

239132

41.

No real solution

2x2 2 15x 1 50 5 0

2x2 2 5s3x 2 10d 5 0

xs3x 2 10d1 2x3x 2 10

25x2 5 s0dxs3x 2 10d

2x

3x 2 102

5x

5 0

43.

x 5 25

x 5 0 x 1 5 5 0

xsx 1 5d 5 0

x2 1 5x 5 0

2x2 5 x2 2 5x

5x12x5 2 5 1x2 2 5x

5x 25xCheck:

so is extraneous.x 5 0

0 Þ undefined

2s0d

55? 02 2 5s0d

5s0d

x 5 0

22 5 22

210

55

25 1 25225

2s25d

55? s25d2 2 5s25d

5s25d

x 5 25


45.

5 5 q

30 5 6q

35 5 6q 1 5

56 2 21 5 6q 1 5

28s2d 2 7s3d 5 6q 1 5

28s6q 1 5d1 26q 1 5

23

4s6q 1 5d2 5 1 128228s6q 1 5d

2

6q 1 52

34s6q 1 5d 5

128

Check:

1

285

128

5

1405? 1

28

8

1402

3140

5? 1

28

235

23

4s35d 5? 1

28

2

30 1 52

34s30 1 5d 5

? 128

2

6s5d 1 52

34f6s5d 1 5g 5

? 128

47.

x 5 21110

x 5 2

0 5 s10x 1 11dsx 2 2d

0 5 10x2 2 9x 2 22

0 5 30x2 2 27x 2 66

20x 2 12 1 34x 1 51 5 30x2 1 27x 2 27

4s5x 2 3d 1 17s2x 1 3d 5 3s10x2 1 9x 2 9d

s5x 2 3ds2x 1 3d1 42x 1 3

117

5x 2 32 5 s3ds2x 1 3ds5x 2 3d

4

2x 1 31

175x 2 3

5 3 Check:

Check:

3 5 3

47

1177

5?

3

4

2s2d 1 31

175s2d 2 3

5?

3

3 5 3

5 1 22 5?

3

445

117

2172

5?

3

48

10

117

28510

5?

3

4

22210 1

3010

117

25510 2

3010

5?

3

4

2s21110d 1 3

117

5s21110d 2 3

5?

3

49.

x 5 20

2x 5 220

2x 1 26 5 6

2x 2 4 2 3x 1 30 5 6

2sx 2 2d 2 3sx 2 10d 5 6

sx 2 10dsx 2 2d1 2x 2 10

23

x 2 22 5 1 6sx 2 10dsx 2 2d2sx 2 10dsx 2 2d

2

x 2 102

3x 2 2

56

sx 2 10dsx 2 2d

2

x 2 102

3x 2 2

56

x2 2 12x 1 20Check:

1

305

130

630

25

305

130

15

216

5? 6

180

2

102

318

5? 6

400 2 240 1 20

2

20 2 102

320 2 2

5? 6

s20d2 2 12s20d 1 20


51.

x 532

22x 5 23

23 2 2x 1 6 5 0

1 2 4 2 2sx 2 3d 5 0

sx 2 3d1 1x 2 3

24

x 2 32 22 5 0sx 2 3d

x 1 3

sx 2 3dsx 1 3d 24

x 2 32 2 5 0

x 1 3x2 2 9

14

3 2 x2 2 5 0 Check:

0 5 0

223

183

263

5?

0

92

2274

1432

2 2 5?

0

32 1

62

94 2

364

14

62 2

32

2 2 5?

0

32 1 3

s32d2

2 91

4

3 232

2 2 5?

0

53.

x 5 3 x 5 21

sx 2 3dsx 1 1d 5 0

x2 2 2x 2 3 5 0

4x2 2 8x 2 12 5 0

x2 2 4x 1 3x2 2 6x 5 22x 1 12

xsx 2 4d 1 3xsx 2 2d 5 22sx 2 6d

sx 2 2dsx 2 4d1 xx 2 2

13x

x 2 42 5 1 22sx 2 6dsx 2 4dsx 2 2d2sx 2 2dsx 2 4d

x

x 2 21

3xx 2 4

522sx 2 6d

x2 2 6x 1 8Check:

Check:

1415

51415

515

19

155? 14

15

13

135

5? 14

15

21

21 2 21

3s21d21 2 4

5? 22s21 2 6d

s21d2 2 6s21d 1 8

26 5 26

3 19

215? 6

21

3

3 2 21

3s3d3 2 4

5? 22s3 2 6d

32 2 6s3d 1 8

55. Check:

Division by zero is undefined. Solution is extraneous, so equation has no solution.

24 5 x

28 5 2x

12 5 2x 1 20

4x 1 28 2 4x 2 16 5 2x 1 20

4sx 1 7d 2 2 ? 2sx 1 4d 5 2x 1 20

60

2 2 Þ120

2sx 1 4d12sx 1 7dx 1 4

2 22 5 12x 1 202sx 1 4d22sx 1 4d

2f24 1 7g

24 1 42 2 5

? 2s24d 1 202s24d 1 8

2sx 1 7d

x 1 42 2 5

2x 1 202x 1 8


57.

x 5 3 x 5 2

sx 2 3dsx 2 2d 5 0

x2 2 5x 1 6 5 0

x2 2 x 5 4x 2 6

xsx 2 1d 5 s2x 2 3d2

2sx 2 1d1x22 5 12x 2 3

x 2 1 22sx 2 1d

x2

5

2 23x

1 21x

?xx

x2

5

2 23x

1 21x

Check:

32

532

32

51

23

32

5?

2 233

1 213

1 5 1

1 5

12

12

22

5?

2 232

1 212

59. intercept:

22 5 x

0 5 x 1 2

sx 2 2ds0d 5 1x 1 2x 2 22sx 2 2d

0 5x 1 2x 2 2

s22, 0dx- 61. intercepts:

x 5 21 x 5 1

x 1 1 5 0 x 2 1 5 0

0 5 sx 2 1dsx 1 1d

0 5 x2 2 1

xs0d 5 1x 21x2x

0 5 x 21x

s21, 0d and s1, 0dx-

63. (a) Keystrokes:

4 5

intercept:

(b)

4 5 x

0 5 x 2 4

0 5x 2 4x 1 5

s4, 0dx- −40

−8

30

8

Y5 x 2

x

4 x 1

x

GRAPH

65. Keystrokes:

1 4 5

intercept:

1 5 x

5 5 5x

0 5 x 2 5 1 4x

xsx 2 5ds0d 5 11x

14

x 2 52xsx 2 5d

0 51x

14

x 2 5

s1, 0dx- −6

−6

12

6

Y5 4 x x

GRAPH14 2

X,T,u

X,T,u X,T,u

X,T,u


67. Keystrokes:

1 6

intercepts:

−15

−10

15

10

x 5 2 x 5 23

x 2 2 5 0 x 1 3 5 0

0 5 sx 1 3dsx 2 2d

0 5 x2 1 x 2 6

xs0d 5 3sx 1 1d 26x4x

0 5 sx 1 1d 26x

s23, 0d and s2, 0dx-

Y5 x x

GRAPH1 2 4

69. (a) Keystrokes:

1 12

intercepts: and

(b)

−12

−8

12

8

x 5 23 x 5 4

x 1 3 5 0 x 2 4 5 0

0 5 sx 2 4dsx 1 3d

0 5 x2 2 x 2 12

0 5 sx 2 1d 212x

s4, 0ds23, 0dx-

Y5 x x

GRAPH2 42

71.

Labels:

Equation:

x 518 x 5 8

s8x 2 1dsx 2 8d 5 0

8x2 2 65x 1 8 5 0

8x2 1 8 5 65x

8x1x 11x2 5 165

8 28x

x 11x

5658

Reciprocal 51x

Number 5 x

Verbal Model: 1 5Number658

Reciprocal

73.

Labels:

Equation:

x 5 40 miles per hour

21200x 5 248,000

204,000 2 680x 5 156,000 1 520x

680s300 2 xd 5 520s300 1 xd

s300 1 xds300 2 xd1 680300 1 x2 5 1 520

300 2 x2s300 1 xds300 2 xd

680

300 1 x5

520300 2 x

Speed of the wind 5 x

Verbal Model: 4 5Distance Rate Time

5

Distance

RateTime

Time traveledwith wind

5Time traveledwithout wind

X,T,u X,T,u X,T,u X,T,u


75.

Labels:

Equation:

x 1 2 5 10 mph person 1

x 5 8 mph person 2

5x 5 4x 1 8

5x 5 4sx 1 2d

xsx 1 2d1 5x 1 22 5 14

x2xsx 1 2d

5

x 1 25

4x

Rate person 2 5 x

Rate person 1 5 x 1 2

Verbal Model: ?Distance Rate Time5

5Distance person 1

Rate person 1Distance person 2

Rate person 2

77.

Labels:

Equation:

x 5 4 mph

x2 5 16

5x2 5 80

1920 5 2000 2 5x2

960 1 48x 1 960 2 48x 5 2000 2 5x2

48s20 1 xd 1 48s20 2 xd 5 5s400 2 x2d

s20 2 xds20 1 xd1 4820 2 x

148

20 1 x2 5 s5ds20 2 xds20 1 xd

48

20 2 x1

4820 1 x

5 5

Speed of the current 5 x

DistanceRate

5 Time

Verbal Model: ?Distance Rate Time5

Time traveledupstream 1 5

Time traveleddownstream

Totaltime

79.

Labels:

Equation:

x 5 10 people x 5 212

x 2 10 5 0 x 1 12 5 0

0 5 sx 1 12dsx 2 10d

0 5 x2 1 2x 2 120

0 5 4000x2 1 8000x 2 480,000

240,000x 1 480,000 2 240,000x 5 4000x2 1 8000x

240,000sx 1 2d 2 240,000x 5 4000sx2 1 2xd

xsx 1 2d1240,000x

2240,000x 1 2 2 5 s4000dxsx 1 2d

240,000

x2

240,000x 1 2

5 4000

Persons in new group 5 x 1 2

Persons in current group 5 x

Cost per personcurrent group

VerbalModel:

52Cost per personnew group 4000


81.

Labels:

Equation:

persons x 5 215 x 5 12

0 5 sx 1 15dsx 2 12d

0 5 x2 1 3x 2 180

0 5 1300x2 1 3900x 2 234,000

78,000x 1 234,000 2 78,000x 5 1300x2 1 3900x

78,000sx 1 3d 2 78,000x 5 1300xsx 1 3d

xsx 1 3d178,000x

278,000x 1 3 2 5 s1300dxsx 1 3d

78,000

x2

78,000x 1 3

5 1300

Persons in new group 5 x 1 3

Persons in current group 5 x

Cost per personoriginal group

VerbalModel:

52Cost per personnew group 1300

83. (a) Keystrokes:

120,000 100

(b)

Equation:

85% 5 p

68,000,000 5 800,000p

68,000,000 2 680,000p 5 120,000p

s100 2 pds680,000d 5 1120,000p100 2 p 2s100 2 pd

680,000 5120,000p100 2 p

00

100

1,000,000

Y5 x x

4 x 2

x

GRAPH

CostVerbalModel:

5120,000p100 2 p

85.

t 5 3 hours

2t 5 6

t 1 t 5 6

16

116

51t

t 5158

minutes

8t 5 15

5t 1 3t 5 15

13

115

51t

t 553

hours

3t 5 5

t 1 2t 5 5

15

125

51t

15

11 52

51t

15

11

212

51t

Person #1 Person #2 Together

6 hours 6 hours 3 hours

3 minutes 5 minutes minutes

5 hours hours hours5321

2

158

X,T,u X,T,u


87.

Labels:

Equation:

2212

hours 5452

532

x

15 hours 5 x

9 1 6 5 x

9x11x

123x2 5 11

929x

1x

11

32 x

519

First landscaper's time 532

x

Second landscaper's time 5 x

Verbal Model: 1 5Rate Person 1 Rate TogetherRate Person 2

89.

Labels:

Equation:

1114

hours 5454

554

x

9 hours 5 x

5 1 4 5 x

5x11x

145x2 5 11

525x

1x

11

54 x

515

First pipe's time 554

x

Second pipe's time 5 x

Verbal Model: 1 5Rate Pipe 1 Rate TogetherRate Pipe 2

91.

y 587,709 2 1236s5d

1000 2 93s5d < 152.4y 587,709 2 1236s2d

1000 2 93s2d < 104.7

y 587,709 2 1236s4d

1000 2 93s4d < 131.8y 587,709 2 1236s1d

1000 2 93s1d < 95.3

y 587,709 2 1236s3d

1000 2 93s3d < 116.5y 587,709 2 1236s0d

1000 2 93s0d < 87.7

93. (a)

(b) 700

200

4

Domain 5 H4, 6, 8, 10, . . .J (c)

x < 10d

x2 5935391.6

91.6 59353

x2

135 5 43.4 19353

x2


95. (d)

x 5 3 miles per hour

0 5 x2 2 9

0 5 25x2 2 225

625 2 25x2 5 200 1 40x 1 200 2 40x

25s25 2 x2d 5 40s5 1 xd 1 40s5 2 xd

4s5 2 xds5 1 xd1254 2 5 1 10

5 2 x1

105 1 x24s5 2 xds5 1 xd

6 14 5

105 2 x

110

5 1 x

t sxd 510

5 2 x1

105 1 x

(e)

Yes

t sxd 5 1119 or 11.1 hours

t sxd 5 10 1109

5909

1109

51009

t sxd 510

5 2 41

105 1 4

97. Solve a rational equation by multiplying both sides ofthe equation by the lowest common denominator. Thensolve the resulting equation, checking for any extraneous solutions.

99. (a) Simplify each side by removing symbols of grouping,combining like terms, and reducing fractions on oneor both sides.

(b) Add (or subtract) the same quantity to (from) bothsides of the equation.

(c) Multiply (or divide) both sides of the equation by thesame nonzero real number.

(d) Interchange the two sides of the equation.

101. When the equation involves only two fractions, one on each side of the equation, the equation can be solved by cross-multiplication.


1.

5 172

5 7221

s23 ? 32d21 5 s8 ? 9d21 3.

5 1258

s25d23

5 s52d3

5.

5 36,000,000

5 36 3 106

s6 3 103d2 5 62 3 106

7.

5 500

5 5 3 102

5 0.5 3 103

3.5 3 107

7 3 104 53.57

3 10724 9. 0.0000538 5 5.38 3 1025 11. 4.833 3 108 5 483,300,000

13.

5 12y

5 12y1

s6y4ds2y23d 5 12y41(23) 15.

52x3

5 2x23

4x22

2x5 2x2221 17.

5x6

y8

sx3y24d2 5 x6y28


25.

D 5 s2`, 1d < s1, 6d < s6, `d

u Þ 6, u Þ 1

su 2 6dsu 2 1d Þ 0

u2 2 7u 1 6 Þ 0 27.

52x3

5, x Þ 0, y Þ 0

6x4y2

15xy2 52 ? 3x ? x3 ? y2

5 ? 3x ? y2 29.

5b 2 3

6sb 2 4d

55sb 2 3d

5 ? 6sb 2 4d

5b 2 15

30b 2 1205

5sb 2 3d30sb 2 4d

31.

5 29, x Þ y

9x 2 9yy 2 x

59sx 2 yd

21sx 2 yd 33.

5x

2sx 1 5d, x Þ 5

5xsx 2 5d

2sx 2 5dsx 1 5d

x2 2 5x

2x2 2 505

xsx 2 5d2sx2 2 25d 35.

5 3x5y2

3xsx2yd2 5 3xsx4y2)

37.24x4

15x5

8x3

539.

5y8x

, y Þ 0

78

?2xy

?y2

14x2 57 ? 2 ? x ? y ? y

2 ? 2 ? 2 ? y ? 7 ? 2 ? x ? x

19.

51t 3

5 t23

5 t2512

t25

t22 5 t (25)2(22) 21.

527y3

1y32

23

5 13y2

323.

D 5 s2`, 8d < s8, `d

y Þ 8

y 2 8 Þ 0

41.

5 12zsz 2 6d, z Þ 26

60z

z 1 6?

z2 2 365

55 ? 12zsz 2 6dsz 1 6d

sz 1 6d5 43.

5 214

, u Þ 0, u Þ 3

u

u 2 3?

3u 2 u2

4u2 5u

u 2 3?

2usu 2 3d4u2

45.

5 3x2, x Þ 0

53 ? 2

x?

x ? x2

2

6yx2yx3 5

6x

42x3

47.

5125y

x, y Þ 0

25y2 4xy5

5 25y ? y ?5xy

49.

5xsx 2 1d

x 2 7, x Þ 21, x Þ 1

x2 2 7xx 1 1

4x2 2 14x 1 49

x2 2 15

xsx 2 7dx 1 1

?sx 2 1dsx 1 1dsx 2 7dsx 2 7d


51.

56sx 1 5dxsx 1 7d, x Þ 5, x Þ 25

56x2sx 1 5dx3sx 1 7d

1 6x2

x2 1 2x 2 3521 x3

x2 2 2525

6x2

sx 1 7dsx 2 5d

x3

sx 2 5dsx 1 5d ?

sx 1 7dsx 2 5dsx 1 5dsx 1 7dsx 2 5dsx 1 5d

53.49

2119

54 2 11

95 2

79

55.

5 21348

545 2 10 2 48

48

1516

25

242 1 5

15s3d16s3d 2

5s2d24s2d 2

1s48d1s48d

57.

54x 1 3

sx 1 5dsx 2 12d

5x 2 12 1 3x 1 15

sx 1 5dsx 2 12d

5x 2 12

sx 1 5dsx 2 12d 13sx 1 5d

sx 2 12dsx 1 5d

1

x 1 51

3x 2 12

51

x 1 51x 2 12x 2 122 1

3x 2 121

x 1 5x 1 52

59.

55x3 2 5x2 2 31x 1 13

sx 2 3dsx 1 2d

55x3 2 5x2 2 30x 1 2x 1 4 2 3x 1 9

sx 2 3dsx 1 2d

5x 12

x 2 32

3x 1 2

55xsx 2 3dsx 1 2dsx 2 3dsx 1 2d 1

2sx 2 3d1

x 1 2x 1 22 2

3sx 1 2d1

x 2 3x 2 32

61.

524 1 x

xsx2 1 4d

56x2 1 24 2 6x2 1 x

xsx2 1 4d

6x

26x 1 1x2 1 4

56sx2 1 4dxsx2 1 4d 2

6x 2 1sxdx2 1 4sxd

63.

56x 2 54

sx 1 3d2sx 2 3d

55x2 2 45 2 4x2 1 12x 2 x2 2 6x 2 9

sx 1 3d2sx 2 3d

5

x 1 32

4xsx 1 3d2 2

1x 2 3

55

x 1 31sx 1 3dsx 2 3dsx 1 3dsx 2 3d2 2

4xsx 1 3d21x 2 3

x 2 32 21

x 2 31sx 1 3d2

sx 1 3d22


65.3t

15 22t 2

?tt

53t 2

5t 2 2, t Þ 0

67.

52a2 1 a 1 164a3 2 63a 2 4

, a Þ 0, a Þ 24

5a 2 a2 1 16

a 2 4 1 4a3 2 64a

5a 2 sa2 2 16d

a 2 4 1 4asa2 2 16d

1 1

a2 2 162

1a2

1 1a2 1 4a

1 42?

asa 2 4dsa 1 4dasa 2 4dsa 1 4d 5

a 2 sa 2 4dsa 1 4da 2 4 1 4asa 2 4dsa 1 4d

69. Keystrokes:

6 9

3 3

9

x2 1 6x 1 9x2 ?

x2 2 3xx 1 3

5sx 1 3dsx 1 3dxsx 2 3d

x2sx 1 3d 5x2 2 9

x

y2

y1

−15

−10

15

10

Y5 x x X,T, u x2 1 X,T, u 1 x

4 X,T, u x2 x

3

x x X,T, u x2 2 X,T, u

x

4 x X,T, u 1

x x

ENTER

x X,T, u x2 2

x

4 X,T, u GRAPH

71. Keystrokes:

1 1 2 2

2 4

11x

2122

2x?

2x2x

52 2 x

4x2

y2

y1

−6

−2

6

6

Y5 x x 4 X,T, ux

2 x 4x x

4 X,T, u ENTER

x 2 X,T, u

x

4 X,T, u x2 GRAPH

73.

5 2x2 212

, x Þ 0

54x3

2x2

x2x

s4x3 2 xd 4 2x 54x3 2 x

2x75.

109

283

x 1 89

283

x 1 2

4x2 2 43

x

4x2 2 4x

6x3 2 2x2

3x 2 1 ) 6x3 1 2x2 2 4x 1 2

2x2 1 43

x 2 89

110

9s3x 2 1d

77.

22x2 1 2

22x2 1 2

x4 2 x2

x2 2 1 ) x4 1 0x3 2 3x2 1 2, x Þ ±1

x2 2 2


79.

23x2 1 2x 1 3

23x3 1 6x2 2 3x 1 3

23x3 1 3x2 2 x 1 6

2x4 1 2x3 2 x2 1 x

2x4 2 x3 1 2x2 1 0x

x5 2 2x4 1 x3 2 x2

x3 2 2x2 1 x 2 1 ) x5 2 3x4 1 0x3 1 x2 1 0x 1 6

x2 2 x 2 3 123x2 1 2x 1 3

x3 2 2x2 1 x 2 1

81.

22 1

1

7 22

5

3210

27

21414

0

x3 1 7x2 1 3x 2 14x 1 2

5 x2 1 5x 2 7, x Þ 22 83.

3 1

1

0 3

3

23 9

6

0 18

18

22554

29

5 x3 1 3x2 1 6x 1 18 129

x 2 3

sx4 2 3x2 2 25d 4 sx 2 3d

85.

x 5 2120

3x 5 2120 1 2x

8138

x2 5 s215d8 1 1x428

3x8

5 215 1x4

Check:

245 5 245

2360

85?

215 1 230

3s2120d

85?

215 12120

4

87.

t 53623

23t 5 36

24t 2 36 5 t

s3td18 212t 2 5

13

s3td Check:

13

513

243

2233

5? 1

3

8 2233

5? 1

3

8 212

s3623d 5

? 13

89.

5 5 y

6 2 1 5 y

3y12y

213y2 5 11

323y

2y

213y

513

Check:

13

513

515

5? 1

3

6

152

115

5? 1

3

25

21

155? 1

3

25

21

3s5d 5? 1

3

91.

r 5 6, r 5 24

sr 2 6dsr 1 4d 5 0

r2 2 2r 2 24 5 0

r2 5 2r 1 24

rsrd 5 12 124r 2r

r 5 2 124r

Check:

Check:

24 5 24

24 5?

2 2 6

24 5?

2 12424

6 5 6

6 5?

2 1 4

6 5?

2 1246


93.

x 5 2163

3x 5 216 x 5 3

3x 1 16 5 0 x 2 3 5 0

0 5 5s3x 1 16dsx 2 3d

0 5 5s3x2 1 7x 2 48d

0 5 15x2 1 35x 2 240

240 1 40x 5 15x2 1 75x

48x 1 240 2 8x 5 15x2 1 75x

48sx 1 5d 2 8x 5 15xsx 1 5d

8xsx 1 5d16x

21

x 1 52 5 1158 28xsx 1 5d

16x

21

x 1 52 5158

816x

21

x 1 52 5 15 Check:

15 5 15

81158 2 5

?15

81298

1248 2 5

?15

81298

1 32 5?

15

81298

21

2132 5

?15

8121816

21

2163 1

1532 5

?15

81 6

2163

21

2163 1 52 5

?15

Check:

15 5 15

81158 2 5

?15

81168

2182 5

?15

812 2182 5

?15

8163

21

3 1 52 5?

15

95.

x 5 252

, x 5 1

s2x 1 5dsx 2 1d 5 0

2x2 1 3x 2 5 5 0

4x2 1 6x 2 10 5 0

4x2 1 2x 2 10 1 4x 5 0

4x2 1 2sx 2 5d 5 24x

sx 2 5d1 4xx 2 5

12x2 5 12

4x 2 52xsx 2 5d

4x

x 2 51

2x

5 24

x 2 5Check:

8

155

815

2015

21215

5? 8

15

210

2152

245

5?

24

2152

4s25

2ds25

2d 2 51

2

s252d 5

?2

4

s252d 2 5

Check:

1 5 1

4

241 2 5

?2

424

4s1d

1 2 51

21

5?

24

1 2 5

97.

x 5 2 x 5 22

x 2 2 5 0 x 1 2 5 0

sx 2 2dsx 1 2d 5 0

sx2 2 4d 5 0

12 2 x 2 4 5 2x2 2 x 1 12

12 2 sx 1 4d 5 21sx2 1 x 2 12d

sx 2 3dsx 1 4d1 12x2 1 x 2 12

21

x 2 32 5 s21dsx 2 3dsx 1 4d

12

x2 1 x 2 122

1x 2 3

5 21


99.

x 5 295 x 5 3

5x 1 9 5 0 x 2 3 5 0

s5x 1 9dsx 2 3d 5 0

5x2 2 6x 2 27 5 0

5 2 6x 2 12 5 25x2 1 20

5 2 6sx 1 2d 5 25sx2 2 4d

sx 2 2dsx 1 2d1 5x2 2 4

26

x 2 22 5 s25dsx 2 2dsx 1 2d

5

x2 2 42

6x 2 2

5 25

101. Keystrokes:

1 1 2 3

intercepts:

23 5 x

0 5 x 1 3

0 5 2x 1 3 2 x−7

−4

5

4 0 51x

21

2x 2 3

x 5 23x-

Y5 4 X,T, u 2 4 x X,T, u 1 x

GRAPH

103. Domain:

P 5 21w 136w 2, w Þ 0

s0, 6g

105.

Labels:

Equation:

x 1 8 5 56 miles per hour

x 5 48, x 5 256

0 5 sx 2 48dsx 1 56d

0 5 x2 1 8x 2 2688

336x 1 2688 5 336x 1 x2 1 8x

336sx 1 8d 5 336x 1 xsx 1 8d

6xsx 1 8d156x 2 5 1 56

x 1 81

1626xsx 1 8d

56x

556

x 1 81

16

Rate of return trip 5 x 1 8

Rate of original trip 5 x

DistanceVerbalModel: 5 ?Rate Time

Original trip distanceRate

5Return trip distance

Rate1

16


107.

Label:

Equation:

Thus, the player must hit safely 25 consecutive times to obtain a batting average of .400.

x 5 25

x 5150.6

0.6x 5 15

45 1 0.6x 5 60

45 1 x 5 60 1 0.4x

45 1 x 5 0.4s150 1 xd

s150 1 xd1 45 1 x150 1 x2 5 0.4s150 1 xd

45 1 x

150 1 x5 0.4

Consecutive times 5 x

HitsAt bats

VerbalModel:

5 0.400

109.

Labels:

Equation:

x 5 26, x 5 4 people

0 5 sx 1 6dsx 2 4d

0 5 x2 1 2x 2 24

0 5 5000x2 1 10,000x 2 120,000

60,000x 1 120,00 5 60,000x 1 5000x2 1 10,000x

60,000sx 1 2d 5 60,000x 1 5000xsx 1 2d

xsx 1 2d160,000x 2 5 160,000

x 1 21 50002xsx 1 2d

60,000

x5

60,000x 1 2

1 5000

People in new group 5 x 1 2

People presently in group 5 x

Share per person now


Share per person later


111.

Labels:

Equation:

x 5203

5 623

min or 6 min 40 sec

x 5609

9x 5 60

5x 1 4x 5 60

60x1 112

11

152 5 11x260x

112

11

155

1x

Time together 5 x

Your time 5 15

Supervisor’s time 5 12

Rate ofperson 1

VerbalModel: 1 5

Ratetogether

Rate ofperson 2

113. (a)

N 520f4 1 3(25dg1 1 0.05s25d < 702,222

N 520f4 1 3s10dg1 1 0.05s10d < 453,333

N 520f4 1 3s5dg1 1 0.05s5d 5 304,000 (b)

29.8 years < t

672 5 22.4t

752 1 37.6t 5 80 1 60t

752s1 1 0.05td 5 20s4 1 3td

752 520s4 1 3td1 1 0.05t


1.

538

528

118

514

118

222 1 223 5122 1

123 2.

5 3 3 1025

6.3 3 1023

2.1 3 102 5 3 3 102322

3.

510a

5 10a21

s5a23ds2a2d 5 10a2312 4.

51

r3s5

5 r23s25

r2s23

r5s2 5 r225s2322

5.

5x8

y12

5 x8y212

sx2y23d4 5 x2?4y23?4 6.

5 108x2y8

5 27 ? 4x61(24)y612

s3x2y2d3s2x22yd2 5 s33x6y6ds22x24y2d


11. (a)

(b)

52a 1 3

5, a Þ 4

2a2 2 5a 2 12

5a 2 205

s2a 1 3dsa 2 4d5sa 2 4d

2 2 x3x 2 6

52 2 x

23s2x 1 2d 5 213

, x Þ 2 12.4z3

5?

2512z2 5

4 ? z2 ? z ? 5 ? 55 ? 4 ? 3 ? z2 5

5z3

, z Þ 0

13.

54

y 1 4, y Þ 2

y2 1 8y 1 16

2sy 2 2d ?8y 2 16sy 1 4d3 5

sy 1 4d2 ? 8sy 2 2d2sy 2 2dsy 1 4d2sy 1 4d 14.

5s2x 1 3d2

x 1 1, x Þ

32

s4x2 2 9d ?2x 1 3

2x2 2 x 2 35

s2x 2 3ds2x 1 3ds2x 1 3ds2x 2 3dsx 1 1d

9.

D 5 s2`, 25d < s25, 5d < s5, `d

y Þ 5, 25

sy 2 5dsy 1 5d Þ 0

y2 2 25 Þ 0 10. Least common denominator:x3sx 2 3dsx 1 3d

7. 0.000032 5 3.2 3 1025 8. 3.04 3 107 5 30,400,000

15.

514y6

15, x Þ 0

58x3y6 ? 7 ? 3

5 ? 3 ? 4 ? 3x3

s2xy2d3

154

12x3

215

s2xy2d3

15?

2112x3 16.

5x3

4, x Þ 0, 22

53x

x 1 2?

x2sx 1 2d12

1 3x

x 1 221 12

x3 1 2x225

3xx 1 2

412

x3 1 2x2

17.

5 2s3x 1 1d, x Þ 0, 13

5s3x 2 1ds3x 1 1d

21s21 1 3xd

59x2 2 11 2 3x

59xsxd 2

1xsxd

1xsxd 2 3sxd

19x 2

1x2

11x

2 325

19x 21x2

11x

2 32?

xx

18.

522x2 1 2x 1 1

x 1 1

52x2 1 2x

x 1 11

1 2 4x2

x 1 1

2x 11 2 4x2

x 1 15 2x1x 1 1

x 1 12 11 2 4x2

x 1 1


19.

55x2 2 15x 2 2sx 1 2dsx 2 3d

55x

x 1 21x 2 3x 2 32 2

2sx 2 3dsx 1 2d

5x

x 1 22

2x2 2 x 2 6

55x

x 1 22

2sx 2 3dsx 1 2d

20.

55x3 1 x2 2 7x 2 5

x2sx 1 1d2

53x3 1 6x2 1 3x 2 5x2 2 10x 2 5 1 2x3

x2sx 1 1d2

53xsx2 1 2x 1 1d 2 5sx2 1 2x 1 1d 1 2x3

x2sx 1 1d2

53x3

xsx 1 1d2

xsx 1 1d24 25x23xsx 1 1d2

xsx 1 1d24 12x

sx 1 1d21x2

x22

3x

25x2 1

2xx2 1 2x 1 1

53x

25x2 1

2xsx 1 1d2

21.

5 4, x Þ 21

54s1 1 xd

x 1 1

4

x 1 11

4xx 1 1

54 1 4xx 1 1

22.

26t 1 6

3t2 2 6

3t 2 2 6t

t 4 2 2t 2

t 4 1 t 2 2 6t

t 2 2 25 t 2 2 2 ) t 4 1 0t3 1 t 2 2 6t 1 0

t 2 1 3 26t 2 6t 2 2 2

23.

2x4 2 15x2 2 7x 2 3

5 2x3 1 6x2 1 3x 1 9 120

x 2 3

3 2

2

06

6

21518

3

09

9

2727

20

2x 4 2 15x2 2 7x 2 3

24.

22 5 h

24 5 h 1 2

3s8d 5 h 1 2

3

h 1 25

18

Check:

18

518

324

518

3

22 1 25

18


25.

x 5 2152

22x 5 15 x 5 21

22x 2 15 5 0 x 1 1 5 0

s22x 2 15dsx 1 1d 5 0

22x2 2 17x 2 15 5 0

2x2 1 6x 2 3x2 2 15x 5 x2 1 3x 1 5x 1 15

2xsx 1 3d 2 3xsx 1 5d 5 sx 1 5dsx 1 3d

2

x 1 52

3x 1 3

51x

Check:

Check:

21 5 21

12

232

5?

21

24

232

5?

1

2

21 1 52

321 1 3

5?

211

2215

5 2215

21215

11015

5?

22

15

2

2152 1 5

23

2152 1 3

5? 1

2152

26.

Division by zero is undefined. Solution is extraneous,so equation has no solution.

x 5 1

2x 5 2

x 2 1 1 x 1 1 5 2

1

x 1 11

1x 2 1

52

x2 2 1Check:

11 1 1

11

1 2 1Þ

21 2 1

27.

Labels: Time of painter 1

Time of painter 2

Equation:

10 hours 532

x

203

5 623

hours 5 x

20 5 3x

12 1 8 5 3x

12x11x

123x2 5

14

s12xd

1x

11

32 x

514

532

x

5 x

Rate ofpainter 1

VerbalModel: 1 5

Ratetogether

Rate ofpainter 2


CHAPTER 4 Rational Expressions, Equations, and Functions

SECTION 4.1 Integer Exponents and Scientific Notation

1. The graph of an equation is the set of solution points of the equation on a rectangular coordinate system.

2. The point-plotting method for graphing an equation begins by creating a table of solution points of the equation. Plot thesepoints on a rectangular coordinate system, and connect the points with a smooth curve or line.

3.

s6, 2d g s6d 5 !6 2 2 5 !4 5 2

s2, 0d g s2d 5 !2 2 2 5 !0 5 0

g sxd 5 !x 2 2

4. To find the x-intercept, let and solve the equation for x. To find the y-intercept, let and solve the equation for y.x 5 0y 5 0

5. s7x2ds2x3d 5 14x213 5 14x5 6. sy2z3dsz2d4 5 sy2z3dsz8d 5 y2z318 5 y2z11

7.a4b2

ab2 5 a421b222 5 a3b0 5 a3 8. sx 1 2d4 4 sx 1 2d3 5 sx 1 2d423 5 sx 1 2d1 5 x 1 2

9.

Keystrokes:

5 2

−1

−3

4

6

(2.5, 0)

(0, 5)

f sxd 5 5 2 2x

Y5 2 X,T,u GRAPH

10.

Keystrokes:

1 2

−4

−1

4

6

(0, 0)

h sxd 512x 1 |x|

Y5 x 4 d X,T,u 1 ABS X,T,u GRAPH

11.

Keystrokes:

4

−1

−6

5

6

(4, 0)(0, 0)

g sxd 5 x2 2 4x 12.

Keystrokes:

2 1

−2

−1

8

8

(0, 2)

( 1, 0)−

f sxd 5 2!x 1 1

Y5 X,T,u x2 2 X,T,u GRAPH Y5 ! x X,T,u 1 d GRAPH


SECTION 4.2 Rational Expressions and Functions

1. Slope 5 m 5y2 2 y1

x2 2 x1

2. (a) (b)

(c) (d) is undefined.mm 5 0

m < 0m > 0

3.

5 10

2sx 1 5d 2 3 2 s2x 2 3d 5 2x 1 10 2 2x 4.

5 12

3sy 1 4d 1 5 2 s3y 1 5d 5 3y 1 12 1 5 2 3y 2 5

5.

5 210 2 8x

5 4 2 14 2 8x

5 4 2 2f7 1 4xg

4 2 2f3 1 4sx 1 1dg 5 4 2 2f3 1 4x 1 4g 6.

5 22x2 1 14x

5 5x 1 9x 2 2x2

5 5x 1 xf9 2 2xg

5x 1 xf3 2 2sx 2 3dg 5 5x 1 xf3 2 2x 1 6g

7. 1 5x22

2

525x 4 8. 2

s2u2vd2

23uv2 5 24u 4 v2

23uv2 54u3

3

9.

Labels:

Equation:

20 2 x 5 623 gallons at 60%


20.30x 5 24

0.30x 1 12 2 0.60x 5 8

0.30x 1 0.60s20 2 xd 5 20s0.40d

Total gallons 5 20

Gallons solution 2 5 20 2 x

Gallons solution 1 5 x

Gallonssolution 1

VerbalModel: ? 30% 1

Gallonssolution 2 ? 60% 5

Total gallons ? 40%

10.

Labels:

Equation:

x 5 $500

x ? 0.75 5 375

Sale price 5 $375

Original price 5 x

Originalprice

VerbalModel: ? 75% 5

Saleprice

SECTION 4.3 Multiplying and Dividing Rational Expressions

1. 9t 2 2 4 5 s3t 2 2ds3t 1 2d 2. 4x2 2 12x 1 9 5 s2x 2 3d2

3. 8x3 1 64 5 s2x 1 4ds4x2 2 8x 1 16d 4. 3x2 1 13x 2 10 5 s3x 2 2dsx 1 5d

5. 5x 2 20x2 5 5xs1 2 4xd 6.

5 s14 2 xds2 1 xd

5 s8 2 x 1 6ds8 1 x 2 6d

64 2 sx 2 6d2 5 f8 2 sx 2 6dgf8 1 sx 2 6dg


7. 15x2 2 16x 2 15 5 s5x 1 3ds3x 2 5d 8. 16t 2 1 8t 1 1 5 s4t 1 1d2

9. y3 2 64 5 sy 2 4dsy2 1 4y 1 16d 10. 8x3 1 1 5 s2x 1 1ds4x2 2 2x 1 1d

11.

x

y

−1 1 3 4 5−1

−4

−5

−6

m = 2

m = 0

m = −

m is undefined.

13

12.

x

y

−2−4 1 2

2

6

1

3

4

m = 1− m = 2

m =

m is undefined.

12

SECTION 4.4 Adding and Subtracting Rational Expressions

1. (a)

(b)

Let

(Many answers)

y1 575 y 2

75 5

35sx 2 1d

y 2 y1 535sx 2 1d 1

45

x1 5 1. y 2 y1 535sx 2 x1d 1

45

y 535 x 1

45

5y 5 3x 1 4

5y 2 3x 2 4 5 0

y 535 x 1

45

5y 5 3x 1 4

5y 2 3x 2 4 5 0 2. If the line rises from left to right.

If the line falls from left to right.m < 0,

m > 0,

3.

5 42x2 2 60x

26xs10 2 7xd 5 260x 1 42x2 4.

5 6 1 y 2 2y2

s2 2 yds3 1 2yd 5 6 1 4y 2 3y 2 2y2

5. s11 2 xds11 1 xd 5 121 2 x2 6.

5 16 2 25z2

s4 2 5zds4 1 5zd 5 16 1 20z 2 20z 2 25z2

7.

5 x2 1 2x 1 1

sx 1 1d2 5 sx 1 1dsx 1 1d 8.

5 2t

tst 2 1 1d 2 tst 2 2 1d 5 t 3 1 t 2 t 3 1 t

9.

5 x3 2 8

sx 2 2dsx2 1 2x 1 4d 5 x3 1 2x2 1 4x 2 2x2 2 4x 2 8 10.

5 2t3 2 5t 2 2 12t

5 ts2t 2 2 5t 2 12d

tst 2 4ds2t 1 3d 5 ts2t 2 1 3t 2 8t 2 12d


11.

5 5x2 1 9x

5 x2 1 3x2 1 9x 1 x2

5 sx ? xd 1 sx 1 3ds3xd 1 sx ? xd

Area 5 Area rectangle 1 1 Area rectangle 2 1 Area rectangle 3

5 12x 1 6

5 7sxd 1 sx 1 3d 1 s2xd 1 s2x 1 3d

Perimeter 5 Sum of all sides

12.

5 6x2

5 12 ? 3x ? 4x

Area 512 ? Base ? Height

5 12x

5 3x 1 4x 1 5x

Perimeter 5 Sum of all sides

SECTION 4.5 Dividing Polynomials

1. Divide the numerator and the denominator by 30.120y90

530 ? 4y30 ? 3

54y3

2. s2n 1 1ds2n 1 3d 5 4n2 1 6n 1 2n 1 3 5 4n2 1 8n 1 3

3. s2n 1 1d 1 s2n 1 3d 5 4n 1 4 4. 2ns2n 1 2d 5 4n2 1 4n

5.

34

5 x

68

58x8

6 5 8x

6 2 3x 1 3x 5 5x 1 3x

6 2 3x 5 5x

3s2 2 xd 5 5x 6.

x 552

250x250

52125250

250x 5 2125

125 2 125 2 50x 5 0 2 125

125 2 50x 5 0

7.

y 5 252 y 5

52

2y 1 5 5 0 2y 2 5 5 0

2s2y 2 5ds2y 1 5d 5 0

2s4y2 2 25d 5 0

8y2 2 50 5 0 8.

t 5 8

t 2 8 5 0t 5 0

tst 2 8d 5 0

t2 2 8t 5 0


9.

x 5 6 x 5 27

x 2 6 5 0 x 1 7 5 0

sx 1 7dsx 2 6d 5 0

x2 1 x 2 42 5 0 10.

x 5 5

x 2 5 5 0

sx 2 5d2 5 0

x2 2 10x 1 25 5 0

2x2 1 10x 2 25 5 0

10x 2 x2 5 25

xs10 2 xd 5 25

11. Verbal model:

Labels: Monthly wage

Salary

Commission

Function:

y

x20,00010,000

2000

4000

1000

3000

y 5 1500 1 0.12x

5 0.12x

5 1500

5 y

Commission1Salary5Salary 12. Verbal model: 3500

Labels: Total enrollment

Enrollment per year

Function:

3400

4000

N

4200

6t

2 8

3800

3600

4 10

n 5 60t 1 3500

5 60t

5 n

1Enrollment

per year5Total

enrollment

SECTION 4.6 Solving Rational Equations

1. can be located in quadrants II or III.s22, yd 2. can be located in quadrants I or II.sx, 3d

3. Points whose coordinates are 0 are located on the axis.x-

y- 4. s9, 26d

5.

x < 32

22x > 23

7 2 3x > 4 2 x 6.

x < 5

2x < 10

2x 2 8 < 2

2x 1 12 2 20 < 2

2sx 1 6d 2 20 < 2

7.

1 < x < 5

22 < x 2 3 < 2

|x 2 3| < 2 8.

x > 8 or x < 2

x 2 5 > 3 or x 2 5 < 23

|x 2 5| > 3


9.

x ≥ 16 or x ≤ 28

14 x ≥ 4 14 x ≤ 22

14 x 2 1 ≥ 3 or 14 x 2 1 ≤ 23

|14 x 2 1| ≥ 3 10.

224 ≤ x ≤ 36

36 ≥ x ≥ 224

212 ≤ 213x ≤ 8

210 ≤ 2 213 x ≤ 10

|2 213 x| ≤ 10

11.

Labels:

2nd jogger’s rate 5 8; 2nd jogger’s time 5 x

1st jogger’s rate 5 6; 1st jogger’s time 5 x 1560

Distance 5 d

12.

Labels:

Amount at 9% 5 y

Amount at 7.5% 5 x

Amountat 7.5%

VerbalModel:

1 5 24,000Amountat 9%

Amountat 7.5%

1 9% ? 5 1935Amountat 9%

7.5% ?

Equation:

d 5 8s14 hourd 5 2 miles

14 5 x hours, or 15 minutes

12

5 2x

6x 112

5 8x

61x 11

122 5 8x

d 5 8x

d 5 61x 11

122

System:

y 5 $9000 at 9%

x 5 $15,000 at 7.5%

20.015x 5 2225

0.075x 1 2160 2 0.09x 5 1935

0.075x 1 0.09s24,000 2 xd 5 1935

y 5 24,000 2 x

0.075x 1 0.09y 5 1935

x 1 y 5 24,000

CHAPTER 5 Radicals and Complex Numbers

SECTION 5.1 Radicals and Rational Exponents

1. am ? an 5 am1n 2. sabdm 5 ambm 3. samdn 5 amn4. if m > n

am

an 5 am2n,

C H A P T E R 5Radicals and Complex Numbers

Section 5.1 Radicals and Rational Exponents . . . . . . . . . . . . .273

Section 5.2 Simplifying Radical Expressions . . . . . . . . . . . . . .277

Section 5.3 Multiplying and Dividing Radical Expressions . . . . . .280

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284

Section 5.4 Solving Radical Equations . . . . . . . . . . . . . . . . .285

Section 5.5 Complex Numbers . . . . . . . . . . . . . . . . . . . . .292


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303

273

C H A P T E R 5Radicals and Complex Numbers

Section 5.1 Radicals and Rational ExponentsSolutions to Odd-Numbered Exercises

1. because 8 ? 8 5 64!64 5 8 3. because 7 ? 7 5 492!49 5 27

5. because 22 ? 22 ? 22 5 283!28 5 22 7. is not a real number because no real number multi-plied by itself yields 21.!21

9. Because 7 is a square root of 49.72 5 49, 11. Because 4.2 is a cube root of 74.088.4.23 5 74.088,

13. Because 45 is called the square root of 2025.452 5 2025,

15.

(index is even)

!82 5 |8| 5 8 17.

(index is even)

!s210d2 5 |210| 5 10 19. not a real number

(even root of a negative number)

!292 5

21.

(index is even)

2!s23d2

5 223 23. not a real number


!2s 310d2

5 25.

(inverse property of powers androots)

s!5d25 5

27.


2s!23d25 223 29.

(index is odd)

3!s5d3 5 5 31.

(index is odd)

3!103 5 10

33.

(index is odd)

2 3!s26d3 5 6 35.

(index is odd)

3!s214d3

5 214 37.


s 3!11 d35 11

39.


s2 3!24 d35 224 41.


4!34 5 3 43. not a real number


2 4!254 5

45. is not rational because 6 is not a perfect square.!6 47. is rational because a perfect square.30 ? 30 5 900,!900

49. Radical Form Rational Exponent Form

161y2 5 4!16 5 4


272y3 5 93!272 5 9


2563y4 5 644!2563 5 64

55.

Root is 2. Power is 1.

251y2 5 !25 5 5

57.


2361y2 5 2!36 5 26 59.


2s16d3y4 5 2s 4!16d35 28

274 Chapter 5 Radicals and Complex Numbers

61.


3222y5 51

s 5!32d2 5122 5

14

63.


519

s227d22y3 51

s227d2y3 51

s 3!227d2

65.


s 827d2y3

5 s 3! 827 d2

5 s23d2

549 67.


s1219 d21y2

5 s 9121d1y2

5 ! 9121 5

311

69.


s33d2y3 5 33?2y3 5 32 5 9 71.


2s44d3y4 5 244?3y4 5 243 5 264

73.


1 1532

22y3

5 s53d2y3 5 53?2y3 5 52 5 25 75.


!t 5 t1y2

77.


x 4!x3 5 x ? x3y4 5 x113y4 5 x7y4 79.


u2 3!u 5 u2 ? u1y3 5 u211y3 5 u7y3

81.


s 4!s5 5 s 4 ? s5y2 5 s 415y2 5 s13y2 83.!x

!x35

x1y2

x3y2 5 x1y223y2 5 x21 51x

85.

51

t 9y4

5 t 29y4

5 t1y4210y4

4!t

!t 55

t1y4

t 5y2 5 t1y425y2 87.

5 x3

3!x2 ? 3!x7 5 x2y3 ? x7y3 5 x2y317y3 5 x9y3

89.

5 y 9y1214y12 5 y13y12

4!y3 ? 3!y 5 y3y4 ? y1y3 5 y3y411y3 91.

5 x3y4 y1y4

4!x3y 5 sx3yd1y4

93.

5 z212y5y2 5 z 4y 5y2

z2!y5z 4 5 z2 ? sy5z 4d1y2 5 z2y 5y2 z2 95.

5 31

5 34y4

31y4 ? 33y4 5 31y413y4

97.

5 3!2

5 21y3

s21y2d2y3 5 21y2?2y3 99.

5 221 512

5 225y5

21y5

26y5 5 21y526y5 101.

5 !c

5 c1y2

sc3y2d1y3 5 c3y2?1y3

Section 5.1 Radicals and Rational Exponents275

103.

53y2

4z 4y3

53y 6y3z24y3

4

18y 4y3z21y3

24y22y3z5

6 ? 3y 4y32s22y3dz21y321

6 ? 4105.

59y3y2

x2y3

s3x21y3y 3y4d2 5 32x22y3y 3y2

107.

5 x1y4

5 x3y12

5 sx1y12d3

5 sx3y1222y12d3

1x1y4

x1y623

5 sx1y421y6d3 109.

5 8!y

5 y1y8

5 y1y4?1y2

! 4!y 5 sy1y4d1y2

111.

5 x3y8

5 x3y2?1y4

5 sx3y2d1y4

4!!x3 5 4!x3y2 113.

5 !x 1 y

5 sx 1 yd1y2

5 sx 1 yd2y4

5 sx 1 yd3y421y4

sx 1 yd3y4

4!x 1 y5

sx 1 yd3y4

sx 1 yd1y4

115.

51

s3u 2 2vd5y6

5 s3u 2 2vd25y6

5 s3u 2 2vd4y629y6

s3u 2 2vd2y3

!s3u 2 2vd35

s3u 2 2vd2y3

s3u 2 2vd3y2 5 s3u 2 2vd2y323y2 117.

Scientific: 73

Graphing: 73

!73 < 8.5440

121.

Scientific: 1698 3 4

Graphing: 1698 3 4

169823y4 < 0.0038

123.

Scientific: 342 1 4

Graphing: 342 1 4

4!342 5 3421y4

4!342 < 4.3004 125.

Scientific: 545 2 3

Graphing: 545 2 3

3!5452 5 5452y3

3!5452 < 66.7213

!

! ENTER

119.

Scientific: 315 2 5

Graphing: 315 2 5

3152y5 5 s 5!315 d2 < 9.9845

x c

x c

4

4 ENTER>

5

x c

x c

4

4 ENTER>

5 x c

x c

4

4 ENTER>

5

x c

x c

4

4 ENTER>

5

x2c

127.

Scientific: 8 35 2

Graphing: 8 35 2

8 2 !352

< 1.0420 129.

Scientific: 3 17 9

Graphing: 3 17 9

3 1 !179

< 0.7915

2

2

x

x

!

!

c 4

4

5

ENTERc

x

x

c

c

51

1

!

! 4

4

ENTER

1/2y x

y x

y x

y x


131.

Domain5 f0, `d

x ≥ 0,f sxd 5 3!x, 133. The domain of is the set of all

nonnegative real numbers or s0, `d.

gsxd 52

4!x135.

Domain5 s2`, 0g

x ≤ 0

2x ≥ 0f sxd 5 !2x,

137. Keystrokes:

5 4

Domain is so graphing utility did complete the graph.s0, `d

Y5

139. Keystrokes:

2 3 5

Domain is so graphing utility did complete the graph.s2`, `d

Y5

MATH 5 MATH 34

4> GRAPHx c

GRAPH

−3

−2

15

10

−15

−10

15

10

141. x1y2s2x 2 3d 5 2x3y2 2 3x1y2 143.

5 1 1 5y

y21y3sy1y3 1 5y4y3d 5 y0 1 5y3y3

145.

< 0.128 < 12.8%

5 1 2 1132

1y8

r 5 1 2 125,00075,0002

1y8

147. Verbal Model:

Labels: Area

side

Equation:

23 feet 23 feet3

23 5 x

!529 5 x

529 5 x2

529 5 x ? x

5 x

5 529

side?side5Area

149.

< 10.49 cm

5 !110

5 !81 1 25 1 4

5 !92 1 52 1 22

d 5 !l2 1 w2 1 h2

151. (a)

r < 15 feet

r 5!2s35,350dp s100d

r 5!2vpl

(b)

h 5!152 2 1a22

2

h 5!r2 2 1a22

2

(c)

h < 8.29 feet

h 5 !68.75

h 5 !225 2 156.25

h 5!152 2 1252 2

2

(d)

< 25.38 feet

5 2!161

5 2!225 2 64

a 5 2!152 2 82

a 5 2! r2 2 h2

X,T,u

X,T,u

Section 5.2 Simplifying Radical Expressions277

Section 5.2 Simplifying Radical Expressions

153. Given a is the radicand and n is the index.n!a, 155. No. is an irrational number. Its decimal representa-tion is a nonterminating, nonrepeating decimal.

!2

157. (a) “Last digits:” 1 (Perfect square 81) (b) Yes, 4,322,788,986 ends in a 6, but it is not a perfect

4 (Perfect square 64)square.

5 (Perfect square 25)




1. !20 5 !4 ? 5 5 !22 ? 5 5 2!5 3. !50 5 !25 ? 2 5 !52 ? 2 5 5!2

5. !96 5 !16 ? 6 5 !42 ? 6 5 4!6 7. !216 5 !36 ? 6 5 !62 ? 6 5 66!6

9. !1183 5 !169 ? 7 5 !132 ? 7 5 13!7 11. !0.04 5 !4 ? 0.01 5 !4!0.01 5 2 ? 0.1 5 0.2

13.

5 0.06!2

5 6 ? 0.01!2

5 !36 ? !2 ? !0.0001

!0.0072 5 !36 ? 2 ? 0.0001 15.

5 1.1!2

5 11 ? 0.1!2

5 !121 ? !2 ? !0.01

!2.42 5 !121 ? 2 ? 0.01

17. !154

5!15

219. !13

255

!135

21.

5 3x2!x

5 3 ? x2 ? !x

!9x5 5 !32x 4 ? x

23. !48y 4 5 !16 ? 3 ? y 4 5 4y2!3 25. !117y6 5 !9 ? 13 ? y6 5 3|y3|!13

27. !120x2y3 5 !4 ? 30 ? x2 ? y2 ? y 5 2|x|y!30y 29.

5 8a2b3!3ab

!192a5b7 5 !64 ? 3 ? a4 ? a ? b6 ? b

31. 3!48 5 3!16 ? 3 5 3!24 ? 3 5 2 3!3 ? 2 5 2 3!6 33.

5 2 3!14

5 3!8 ? 3!14

3!112 5 3!8 ? 14

35.

5 2x 3!5x2

3!40x5 5 3!8 ? 5 ? x3 ? x2 37.

5 3|y|!2y

5 3|y| 4!22y 2

5 3|y| 4!4y2

4!324y6 5 4!81 ? 4 ? y 4 ? y2 39. 3!x 4y3 5 3!x3 ? x ? y3 5 xy 3!x

41.

5 |x| 4!3y2

4!3x 4y2 5 4!x 4 ? 4!3y2 43. 5!32x5y6 5 5!25 ? x5 ? y5 ? y 5 2xy 5!y


45. 3!3564

53!35

447. 5! 15

2435

5!153

49.

52y

5!x2

5!32x2

y5 5 5!25x2

y5

51. 3!54a 4

b9 5 3!33 ? 2 ? a3 ? ab9 5

3ab3

3!2a 53. !32a 4

b2 5!16 ? 2 ? a 4

!b25

4a2!2

|b|

55. 4!s3x2d4 5 3x2 57. !13

51

!3?!3

!35

!33

59.1

!75

1

!7?!7!7

5!77

61.12

!35

12

!3?!3

!35

12!33

5 4!3

63. 4!54

54!5

4!22?

4!22

4!225

4!5 ? 22

4!245

4!202

65.6

3!325

63!23 ? 22

56

2 3!22?

3!23!2

56 3!2

2 3!235

6 3!24

53 3!2

2

67.1

!y5

1

!y?!y

!y5

!y

!y25

!yy

69. !4x

5!4

!x5

2

!x?!x

!x5

2!xx

71.1

!2x5

1

!2x?!2x

!2x5

!2x2x

73.6

!3b35

6

b!3b?!3b

!3b5

6!3b3b2 5

2!3bb2

75. 3!2x3y

53!2x3!3y

?3!32y2

3!32y25

3!2x ? 32y2

3!33y35

3!18xy2

3y77.

a3

3!ab25

a3

3!ab2?

3!a2b3!a2b

5a3 3!a2b

3!a3b35

a3 3!a2bab

5a2 3!a2b

b

79. 3!2 2 !2 5 2!2 81. 12!8 2 3 3!8 5 12!4 ? 2 2 3 ? 2 5 24!2 2 6

83. 4!3 2 5 4!7 2 12 4!3 5 211 4!3 2 5 4!7 85.

5 6 3!2 1 24 3!2 5 30 3!2

2 3!54 1 12 3!16 5 2 3!27 ? 2 1 12 3!8 ? 2

87. 5!9x 2 3!x 5 15!x 2 3!x 5 12!x 89. !25y 1 !64y 5 5!y 1 8!y 5 13!y

91.

5 s10 2 zd 3!z

10 3!z 2 3!z 4 5 10 3!z 2 3!z3 ? z 5 10 3!z 2 z 3!z 93.

525!5

5 11 2352!5

!5 23

!55 !5 2 1 3

!5?!5

!52 5 !5 2

3!55

95.

595!5

5 12 2152!5

!20 2!15

5 !4 ? 5 2!15

?!5

!55 2!5 2

!55

97.

5!2xs2x 2 3d

2x

52x!2x 2 3!2x

2x

52x!2x

2x2

3!2x2x

!2x 23

!2x5 !2x 2 1 3

!2x?!2x

!2x2 5 !2x 23!2x

2x

Section 5.2 Simplifying Radical Expressions279

99. !7 1 !18 > !7 1 18 101. 5 > !32 1 22

103.

5 !36 1 9 5 !45 5 !9 ? 5 5 3!5

5 !62 1 32

c 5 !a2 1 b2

6

9

105.

5 3!13

5 !117

5 !81 1 36

5 !92 1 62

c 5 !a2 1 b2

3

6

107. (a)

c 5 5!10

c 5 !25 ? 10

c 5 !250

c 5 !225 1 25

c 5 !s15d2 1 s5d2

c 5 !a2 1 b2 (b) Area of roof

Thus, the total area of the roof is square feet.

400!10 < 1264.9

A 5 400!10

A 5 2 ? 40 ? 5!10

5 2 ? Length ? Width

15

5

109. cycles per secondf 51

100!400 3 106

5< 8.9443 3 101 < 89.443 < 89.44

111. (a) (b)

The average salary will reach $48,000 after 14 years so 1990 1 14 5 2004.

40

730

2

60

2030

2

113.

No. When you rationalize the denominator, the value of the number is not changed.

5

!35

5

!3?!3

!35

5!33

Þ253

1 5!32

2

55!3

?5!3

5253

115. Example: !6 ? !15 5 !6 ? 15 5 !3 ? 2 ? 3 ? 5 5 3!10

117. is not in simplest form because can be simplified to and then added to

!2 1 !18 5 !2 1 3!2 5 4!2

!2.3!2!18!2 1 !18

119. for all negative values of x.

Example: !s28d2 5 !64 5 8

!x2 Þ x

280 Chapter 5 Radicals and Complex Ñumbers

Section 5.3 Multiplying and Dividing Radical Expressions

1.

5 4

5 !16

!2 ? !8 5 !2 ? 8 3.

5 3!2

5 !9 ? 2

5 !18

!3 ? !6 5 !3 ? 6 5.

5 2 3!9

5 3!8 ? 9

3!12 ? 3!6 5 3!12 ? 6

7.

5 2 4!3

4!8 ? 4!6 5 4!8 ? 6 5 4!24 ? 3 9. !5s2 2 !3d 5 2!5 2 !5!3 5 2!5 2 !15

11.

5 2!10 1 8!2

5 !40 1 8!2

!2s!20 1 8d 5 !2!20 1 8!2 13.

5 3!2

5 6!2 2 3!2

5 !36 ? 2 2 !9 ? 2

5 !72 2 !18

!6s!12 2 !3d 5 !6!12 2 !6!3

15.

5 6 2 2!5

5 6 2 !4 ? 5

5 !36 2 !20

!2s!18 2 !10d 5 !2!18 2 !2!10 17.

5 y 1 4!y

!ys!y 1 4d 5 s!yd21 4!y

19.

5 4!a 2 a

!a s4 2 !ad 5 !a ? 4 2 !a!a 21.

5 2 2 7 3!4

5 3!8 2 7 3!4

3!4 s 3!2 2 7d 5 3!4 3!2 2 7 3!4

23.

5 21

5 3 2 4

s!3 1 2ds!3 2 2d 5 s!3 d22 22 25. s!5 1 3ds!3 2 5d 5 !15 2 5!5 1 3!3 2 15

27.

5 24 1 8!5

5 24 1 4!4 ? 5

5 20 1 4!20 1 4

s!20 1 2d25 s!20 d2

1 2 ? !20 ? 2 1 22 29.

5 2 3!3 1 3 3!6 2 3 3!4 2 9

5 3!8 ? 3 1 3 3!6 2 3 3!4 2 9

5 3!24 1 3 3!6 2 3 3!4 2 9

s 3!6 2 3ds 3!4 1 3d 5 3!6 3!4 1 3 3!6 2 3 3!4 2 9

31.

5 100 1 20!2x 1 2x

s10 1 !2x d25 102 1 2 ? 10 ? !2x 1 s!2x d2

33.

5 45x 2 17!x 2 6

s9!x 1 2ds5!x 2 3d 5 s9!x ds5!x d 2 27!x 1 10!x 2 6

35.

5 9x 2 25

s3!x 2 5ds3!x 1 5d 5 s3!xd22 52 37.

5 3!4x2 1 10 3!2x 1 25

5 3!s2xd2 1 10 3!2x 1 25

s 3!2x 1 5d25 s 3!2xd2

1 2 ? 5 3!2x 1 52

Section 5.3 Multiplying and Dividing Radical Expressions281

39.

5 y 2 5 3!y 1 2 3!y2 2 10

5 3!y3 2 5 3!y 1 2 3!y2 2 10

s 3!y 1 2ds 3!y2 2 5d 5 3!y ? 3!y2 2 5 3!y 1 2 3!y2 2 10

41.

5 t 1 5 3!t2 1 3!t 2 3

5 3!t3 1 4 3!t2 2 3 3!t 1 3!t2 1 4 3!t 2 3

s 3!t 1 1ds 3!t2 1 4 3!t 2 3d 5 3!t 3!t2 1 3!t ? 4 3!t 2 3 3!t 1 3!t2 1 4 3!t 2 3

43. 5x!3 1 15!3 5 5!3sx 1 3d 45.

5 2!3s4 2 3xd

5 8!3 2 6x!3

4!12 2 2x!27 5 4!4 ? 3 2 2x!9 ? 3

47.

5 3us2u 1 !2ud 5 6u2 1 3u!2u

6u2 1 !18u3 5 6u2 1 !9 ? 2u2 ? u 49. conjugate

product

5 4 2 5 5 21

5 22 2 s!5 d2

5 s2 1 !5ds2 2 !5 d5 2 2 !52 1 !5,

51. conjugate

product

5 11 2 3 5 8

5 s!11 d22 s!3 d2

5 s!11 2 !3ds!11 1 !3 d5 !11 1 !3!11 2 !3, 53.

conjugate

product

5 15 2 9 5 6

5 !15 ? !15 2 3!15 1 3!15 2 9

5 s!15 1 3ds!15 2 3d5 !15 2 3

!15 1 3,

55. conjugate

product

5 x 2 9

5 s!xd22 32

5 s!x 2 3ds!x 1 3d5 !x 1 3!x 2 3, 57. conjugate

product

5 2u 2 3

5 s!2ud22 s!3d2

5 s!2u 2 !3ds!2u 1 !3d5 !2u 1 !3!2u 2 !3,

59. conjugate

product

5 8 2 4 5 4

5 4 ? 2 2 4

5 s2!2d22 s!4d2

5 s2!2 1 !4ds2!2 2 !4d5 2!2 2 !42!2 1 !4, 61. conjugate

product

5 x 2 y

5 s!xd22 s!yd2

5 s!x 1 !y ds!x 2 !y d5 !x 2 !y!x 1 !y,

63.

51 2 2!x

3

4 2 8!x

125

4s1 2 2!xd12

65.

521 1 !3y

4

52ys21 1 !3yd

8y

22y 1 !12y3

8y5

22y 1 2y!3y8y


67. (a)

5 2!3 2 4

5 4 2 4!3 1 3 2 12 1 6!3 1 1

f s2 2 !3 d 5 s2 2 !3 d22 6 s2 2 !3d 1 1 (b)

5 0

5 9 2 12!2 1 8 2 18 1 12!2 1 1

f s3 2 2!2d 5 s3 2 2!2d22 6s3 2 2!2d 1 1

69. (a)

5 0

5 1 1 2!2 1 2 2 2 2 2!2 2 1

f s1 1 !2 d 5 s1 1 !2 d22 2s1 1 !2 d 2 1 (b)

5 21

5 4 2 4 2 1

f s!4 d 5 s!4 d22 2!4 2 1

71.6

!2 2 25

6

!2 2 2?!2 1 2

!2 1 25

6s!2 1 2ds!2d2 2 22

56s!2 1 2d

2 2 45

6 s!2 1 2d22

5 23s!2 1 2d

73.7

!3 1 55

7

!3 1 5?!3 2 5

!3 2 55

7s!3 2 5ds!3d2 2 52

57s!3 2 5d

3 2 255

7s!3 2 5d222

57s5 2 !3d

22

75.

52!10 1 5

5

53s2!10 1 5d

15

53s2!10 1 5d

40 2 25

53s2!10 1 5ds2!10d2 2 52

3

2!10 2 55

3

2!10 2 5?

2!10 1 5

2!10 1 577.

5!6 2 !2

2

52s!6 2 !2d

4

2

!6 1 !25

2

!6 1 !2?!6 2 !2

!6 2 !25

2s!6 2 !2d6 2 2

79.9

!3 2 !75

9

!3 2 !7?!3 1 !7

!3 1 !75

9s!3 1 !7ds!3d2 2 s!7d2 5

9s!3 1 !7d3 2 7

59s!3 1 !7d

245

29s!3 1 !7d4

81. s!7 1 2d 4 s!7 2 2d 5!7 1 2

!7 2 2?!7 1 2

!7 1 25

s!7d21 2!7 1 2!7 1 4

s!7d2 2 225

7 1 4!7 1 47 2 4

511 1 4!7

3

83. s!x 2 5d 4 s2!x 2 1d 5!x 2 5

2!x 2 1?

2!x 1 1

2!x 1 15

2x 1 !x 2 10!x 2 5

s2!xd22 12

52x 2 9!x 2 5

4x 2 1

87.2t2

!5 2 !t5

2t2

!5 2 !t?!5 1 !t

!5 1 !t5

2t2s!5 1 !tds!5d2 2 s!td2 5

2t2s!5 1 !td5 2 t

52t2s!5 1 !td

5 2 t

85.

53xs!15 1 !3d

125

x!15 1 x!34

3x

!15 2 !35

3x

!15 2 !3?!15 1 !3

!15 1 !35

3xs!15 1 !3ds!15d2 2 s!3d2 5

3xs!15 1 !3 d15 2 3

89.8a

!3a 1 !a5

8a

!3a 1 !a?!3a 2 !a

!3a 2 !a5

8as!3a 2 !ads!3ad2

2 s!ad2 58as!3a 2 !ad

3a 2 a5

8as!3a 2 !ad2a

5 4s!3a 2 ! ad

Section 5.3 Multiplying and Dividing Radical Expressions283

91.

53sx 2 4dsx2 1 !xd

xsx3 2 1d 53sx 2 4dsx2 1 !xd

xsx 2 1dsx2 1 x 1 1d

3sx 2 4dx2 2 !x

53sx 2 4dx2 2 !x

?x2 1 !x

x2 1 !x5

3sx 2 4dsx2 1 !xdsx2d2 2 s!xd2 5

3sx 2 4dsx2 1 !xdx 4 2 x

93.

5!u 1 v s!u 2 v 1 !ud

2v5

2!u 1 v s!u 2 v 1 !udv

!u 1 v

!u 2 v 2 !u5

!u 1 v

!u 2 v 2 !u?!u 2 v 1 !u

!u 2 v 1 !u5

!u 1 v s!u 2 v 1 !udu 2 v 2 u

95. Keystrokes:

10 1

10 1 1

, except at

x Þ 110

!x 1 15

10

!x 1 1?!x 2 1

!x 2 15

10s!x 2 1dx 2 1

,

x 5 1y1 5 y2

y2

y1 Y5 4

4

x

x x xcc

!

!

1 c ENTER

2 2 c GRAPH

−4

−2

14

10

97. Keystrokes:

2 2

2 2 4

52s2!x 1 xd

4 2 x

52!x s2 1 !xd

4 2 x

52!x s2 1 !xd

22 2 s!xd2

2!x

2 2 !x5

2!x

2 2 !x?

2 1 !x

2 1 !x

y1 5 y2

y2

y1

GRAPH

Y5 !

! x

4 c

c c

x 2 ! ENTER

4 x 2

99.!27

5!27

?!2

!25

2

7!2

101.

54

5s!7 2 !3d

57 2 3

5s!7 2 !3d

5s!7d2

2 s!3d2

5s!7 2 !3d

!7 1 !3

55

!7 1 !35

?!7 2 !3

!7 2 !3103.

5 192!2 square inches

5 8 ? 23 ? 3!2

5 8!27 ? 32

5 8!1152

5 !384 ? 8!3

5 !576 2 192 ? 8!3

5 !242 2 s8!3d2? 8!3

Area 5 h ? w

−2

−10

28

10

X,T,u

X,T,u X,T,u

X,T,u X,T,u

X,T,u X,T,u X,T,u1


105.

5500k!k2 1 1

1 1 k2

500k

1

!k2 1 11

k2

!k2 1 1

5500k

1 1 k2

!k2 1 1

107.


Multiplication Property of Radicals

Simplify radicals.

5 !3 2 3!2

5 !3 2 !9 ? 2

!3s1 2 !6d 5 !3 2 !3 ? !6

109.

Multilpying the number by its conjugate yields the difference of two squares. Squaring a square root eliminates the radical.

s3 2 !2ds3 1 !2d 5 9 2 2 5 7


1. because 15 ? 15 5 225!225 5 15 2. because 32

?32

?32

?32

58116

4!8116

532

5. !27x2 5 !9 ? 3 ? x2 5 3|x|!3 6.

5 3|x|!x

4!81x6 5 4!81 ? x4 ? x2 5 3|x| 4!x2

7. !4u3

95

!4 ? u2 ? u

!95

2|u|!u3

8. 3!16u6 5

3!163!u6

53!16u2 5

3!8 ? 2u2 5

2 3!2u2

9.

5 4!2y

5 10!2y 2 6!2y

!200y 2 3!8y 5 !100 ? 2y 2 3!4 ? 2y 10.

5 6x 3!5x2 1 4x 3!5x

6x 3!5x2 1 2 3!40x4 5 6x 3!5x2 1 2 3!8 ? 5 ? x3 ? x

11.

5 6!2 1 16

5 6!2 1 24

5 3!4 ? 2 1 !28

!8s3 1 !32d 5 3!8 1 !256 12.

5 10 2 4!2

5 !102 2 4!2

s!50 2 4d!2 5 !100 2 4!2

13.

5 3 1 5!6

5 24 1 5!6 2 21

s!6 1 3ds4!6 2 7d 5 !6 ? 4!6 2 7!6 1 12!6 2 21

14.

5 60 1 67!3

5 18 1 67!3 1 42

5 18 1 67!3 1 14s3d

s9 1 2!3ds2 1 7!3d 5 18 1 63!3 1 4!3 1 2!3 ? 7!3

3. because 8 ? 8 5 64641y2 5 !64 5 8 4. s227d2y3 5 3!s227d2 5 s 3!227d25 s23d2 5 9

Section 5.4 Solving Radical Equations 285

Section 5.4 Solving Radical Equations

1. (a) Not a solution

(b) Not a solution

(c) Not a solution

(d) A solution!100 2 10 5 0x 5 100

!!10 2 10 Þ 0x 5 !10

!2100 2 10 Þ 0x 5 2100

!24 2 10 Þ 0x 5 24 3. (a) Not a solution

(b) A solution

(c) Not a solution

(d) Not a solution3!0 2 4 Þ 4x 5 0

3!20 2 4 Þ 4x 5 20

3!68 2 4 5 4x 5 68

3!260 2 4 Þ 4x 5 260

15.

5!7 2 !21

225

!21 2 !72

!7

1 1 !3?

1 2 !3

1 2 !35

!7s1 2 !3d1 2 s!3d2 5

!7s1 2 !3d1 2 3

5!7s1 2 !3d

22

16.

5 23 2 3!2

524 1 24!2

285

24s1 1 !2d28

5 23s1 1 !2d

6!2

2!2 2 4?

2!2 1 4

2!2 1 45

6!2s2!2 1 4ds2!2d2 2 42

512s!2d2 1 24!2

8 2 16

17.

543

s3 2 !6d

54s!6 2 3d

23

4 4 s!6 1 3d 54

!6 1 3?!6 2 3

!6 2 35

4s!6 2 3ds!6d2 2 32

54s!6 2 3d

6 2 9

18.

52s4 2 4!3 2 !6 1 3!2d

245

4 2 4!3 2 !6 1 3!222

512

s4!3 1 !6 2 3!2 2 4d

54s2d 2 4!12 2 2!6 1 2!18

2 2 65

8 2 4!4 ? 3 2 2!6 1 2!9 ? 224

58 2 8!3 2 2!6 1 6!2

24

s4!2 2 2!3d 4 s!2 1 !6d 54!2 2 2!3

!2 1 !6?!2 2 !6

!2 2 !65

4!2 ? !2 2 4!2 ? !6 2 2!3 ? !2 1 2!3 ? !6

s!2d2 2 s!6d2

19. conjugate

product

5 23

5 1 2 4

5 12 2 s!4 d2

5 s1 1 !4ds1 2 !4d 5 1 2 !41 1 !4, 20. conjugate

product

5 215

5 10 2 25

5 s!10d22 52

5 s!10 2 5ds!10 1 5d5 !10 1 5!10 2 5,

21.

!52 1 122 Þ 17

13 Þ 17

5 13

5 !169

!52 1 122 5 !25 1 144 22.

Equation:

inches 5 23 1 8!2

5 14 1 9 1 8!2

P 5 2s7d 1 2s412d 1 4s!8 d

5 !4 1 4 5 !8

C 5 !22 1 22 11

8

22

22

2

2

2

2

12


5. Check:

x 5 400

20 5 20 s!x d25 202

!400 5?

20 !x 5 20 7. Check:

x 5 9

3 5 3 s!x d25 32

!9 5?

3 !x 5 3

9. Check:

z 5 27

3 5 3 s 3!zd35 33

3!27 5?

3 3!z 5 3 11. Check:

y 5 49

0 5 0 s!y d25 72

7 2 7 5?

0 !y 5 7

!49 2 7 5?

0 !y 2 7 5 0

13. Check:

No solution

u 5 169

s!ud25 s213d2

13 1 13 Þ 0 !u 5 213

!169 1 13 5?

0 !u 1 13 5 0 15. Check:

x 5 64

0 5 0 s!x d25 82

8 2 8 5?

0 !x 5 8

!64 2 8 5?

0 !x 2 8 5 0

17. Check:

x 5 90

30 5 30 10x 5 900

!900 5?

30 s!10xd25 302

!10 ? 90 5?

30 !10x 5 30 19. Check:

x 5 227

9 5 9 23x 5 81

!81 5?

9 s!23xd25 92

!23s227d 5?

9 !23x 5 9

21. Check:

t 545

0 5 0 5t 5 4

2 2 2 5?

0 s!5td25 22

!4 2 2 5?

0 !5t 5 2

!5s45d 2 2 5

?0 !5t 2 2 5 0 23. Check:

y 5 5

3y 5 15

4 5 4 3y 1 1 5 16

!16 5?

4 s!3y 1 1d25 42

!3s5d 1 1 5?

4 !3y 1 1 5 4

25. Check:

No solution

x 5 21

25x 5 5

3 Þ 23 4 2 5x 5 9

!9 5?

23 s!4 2 5xd25 s23d2

!4 2 5s21d 5?

23 !4 2 5x 5 23 27. Check:

y 5443

3y 5 44

4 5 4 3y 1 5 5 49

7 2 3 5?

4 s!3y 1 5d25 72

!49 2 3 5?

4 !3y 1 5 5 7

!3s443 d 1 5 2 3 5

?4 !3y 1 5 2 3 5 4

29. Check:

x 51425

25x 5 14

8 5 8 25x 1 50 5 64

5 ? 85 5

?8 25sx 1 2d 5 64

5!6425 5

?8 s5!x 1 2d2

5 82

5!1425 1 2 5

?8 5!x 1 2 5 8 31. Check:

No solution x 5233

10 Þ 0 3x 5 23

5 1 5 5?

0 3x 1 2 5 25

!25 1 5 5?

0 s!3x 1 2d25 s25d2

!23 1 2 1 5 5?

0 !3x 1 2 5 25

!3s233 d 1 2 1 5 5

?0 !3x 1 2 1 5 5 0


33.

4 5 x

x 1 3 5 2x 2 1

s!x 1 3d25 s!2x 2 1d2

!x 1 3 5 !2x 2 1 Check:

!7 5 !7

!4 1 3 5?!2s4d 2 1

35.

256 5 y

25 5 6y

3y 2 5 5 9y

s!3y 2 5d25 s3!yd2

!3y 2 5 5 3!y

!3y 2 5 2 3!y 5 0 Check:

No solution

!3s256d 2 5 2 3!2

56 5

?0

37.

x 5 7

2x 5 14

3x 2 4 5 x 1 10

s 3!3x 2 4d35 s 3!x 1 10d3

3!3x 2 4 5 3!x 1 10 Check:

3!17 5 3!17

3!3s7d 2 4 5? 3!7 1 10

39.

x 5 215

2x 1 15 5 x

s 3!2x 1 15d35 s 3!xd3

3!2x 1 15 5 3!x

3!2x 1 15 2 3!x 5 0 Check:

0 5 0

3!215 2 3!215 5?

0

3!2s215d 1 15 2 3!215 5?

0

41.

223 5 x

246 5 x

24 5 6x

x2 1 5 5 x2 1 6x 1 9

s!x2 1 5d25 sx 1 3d2

!x2 1 5 5 x 1 3 Check:

73 573

!499 5

? 73

!49 1

459 5

?2

23 1

93

!s223d2

1 5 5?

223 1 3

43.

,

Not a solution

x 5 2 8 5 x

0 5 sx 2 8dsx 2 2d

0 5 x2 2 10x 1 16

2x 5 x2 2 8x 1 16

s!2xd25 sx 2 4d2

!2x 5 x 2 4 Check:

2 Þ 22

!4 5?

22

!2s2d 5?

2 2 4

4 5 4

!16 5?

4

!2s8d 5?

8 2 4


45.

, x 5 1 3 5 x

0 5 sx 2 3dsx 2 1d

0 5 x2 2 4x 1 3

8x 1 1 5 x2 1 4x 1 4

s!8x 1 1d25 sx 1 2d2

!8x 1 1 5 x 1 2 Check:

3 5 3

!9 5?

3

!8s1d 1 1 5?

1 1 2

5 5 5

!25 5?

5

!8s3d 1 1 5?

3 1 2

47.

14 5 z

1 5 4z

12 5 s2!zd2

1 5 2!z

z 1 2 5 1 1 2!z 1 z

s!z 1 2d25 s1 1 !zd2

!z 1 2 5 1 1 !2 Check:

32 532

!94 5

?1 1

12

!14 1 2 5

?1 1 !1

4

49.

12 5 t

1 5 2t

12 5 s!2td2

1 5 !2t

26 5 26!2t

2t 1 3 5 9 2 6!2t 1 2t

s!2t 1 3d25 s3 2 !2td2

!2t 1 3 5 3 2 !2t Check:

2 5 2

!4 5?

3 2 1

!1 1 3 5?

3 2 !1

!2s12d 1 3 5

?3 2 !2s1

2d

51.

4 5 x

22 5 s!xd2

2 5 !x

4 5 2!x

x 1 5 5 1 1 2!x 1 x

s!x 1 5d25 s1 1 !xd2

!x 1 5 5 1 1 !x

!x 1 5 2 !x 5 1

53.

t 5 4

t3 5 64

s!t3d25 82

!t3 5 8

t3y2 5 8

Check:

1 5 1

3 2 2 5?

1

!9 2 !4 5?

1

!4 1 5 2 !4 5?

1

Check:

8 5 8

23 5?

8

s!4d35?

8

43y2 5?

8


55.

y 5 216

s 3!yd35 63

3!y 5 6

y1y3 5 6

3y1y3 5 18

57.

5 4, 212

x 5 24 ± 8

x 1 4 5 ±!64

sx 1 4d2 5 64

s 3!sx 1 4d2d35 s4d3

3!sx 1 4d2 5 4

sx 1 4d2y3 5 4

Check:

18 5 18

3 ? 6 5?

18

3 3!216 5?

18

3s216d1y3 5?

18

Check:

s22d2 5 4

s28d2y3 5?

4

s212 1 4d2y3 5?

4

22 5 4

82y3 5?

4

s4 1 4d2y3 5?

4

59.

x 5 216

2x 5 232

2x 1 5 5 227

s 3!2x 1 5d35 s23d3

3!s2x 1 5d 5 23

s2x 1 5d1y3 1 3 5 0 Check:

0 5 0

23 1 3 5?

0

s227d1y3 1 3 5?

0

s232 1 5d1y3 1 3 5?

0

s2s216d 1 5d1y3 1 3 5?

0

61. Keystrokes:

2 2

Approximate solution:

Check algebraically:

1.186 5 1.186

!1.407 5?

2s2 2 1.407d

x < 1.407

y2

y1 Y5 ! ENTER

GRAPHcx 2

−2

4

4−2

63. Keystrokes:

1

5 2



1.86 5 1.86

!1.5692 1 1 5?

5 2 2s1.569d

x < 1.569

y2

y1

65. Keystrokes:

3

5



2.8 5 2.8

!4.840 1 3 5?

5 2 !4.840

x < 4.840

y2

y1

Y5 ! ENTER

GRAPH2

x cx2 1

−2

4

4−2

−1

5

8−1

Y5 ! ENTER

GRAPH2

x c1

!

X,T,u

X,T,u

X,T,u

X,T,u

X,T,u

X,T,u


67. Keystrokes:

4

7



5.02 5 5.02

4 3!1.978 5?

7 2 1.978

x < 1.978

y2

y1 Y5 ENTER

GRAPH2

−1

10

5−1

MATH 4

69. Keystrokes:

15 4

2

Solution:


3 5 3

!9 5?

3

!15 2 4s1.5d 5?

2s1.5d

x 5 1.5

y2

y1 Y5 ! x 2 ENTERc

GRAPH

− 4

−1

5

5

71. (c) graph is shifted down 1 unit 73. (d) graph is shifted left 3 units and upward 1 unit

75. (f) graph is shifted down 1 unit

77.

9 5 x

!81 5 x2

81 5 x2

225 5 x2 1 144

152 5 x2 1 122 79.

12 5 x

!144 5 x

144 5 x2

169 5 x2 1 25

132 5 x2 1 52

c2 5 a2 1 b213

5

x

15

12

x

81.

x 5 11 inches

x 5 !121

x2 5 13.752 2 8.252

13.752 5 8.252 1 x2

13.75 in.8.25 in.

X,T,u

X,T,u

X,T,u

X,T,u


85.

x 5 15 feet

x 5 !225

x 5 !289 2 64

x2 5 172 2 82

172 5 x2 1 82

17 feet

8 feet

House

87.

46 2 w 5 l

46 5 l 1 w

92 5 2l 1 2w

P 5 2l 1 2w

30 inches 3 16 inches

l 5 30 l 5 16

w 5 16 w 5 30

0 5 sw 2 30dsw 2 16d

0 5 w2 2 46w 1 480

0 5 2w2 2 92w 1 960

1156 5 w2 1 2116 2 92w 1 w2

342 5 w2 1 s46 2 wd2

34 in.

l

w

89.

!S 2 2 p 2r 4

pr5 h

!S 2 2 p 2r 4

p2r2 5 h

S 2 2 p 2r 4

p 2r 2 5 h2

S2

p 2r2 2 r2 5 h2

S 2

p 2r 2 5 r 2 1 h2

1 Spr2

2

5 s!r2 1 h2d2

S

pr5 !r 2 1 h2

S 5 pr!r2 1 h2 91.

64 feet 5 d

4 5d16

22 5 1! d162

2

2 5! d16

93.

v < 56.57 feet per second

v 5 40!2

v 5 !3200

v 5 !2s32d50 95.

56.25 feet 5 h

3600

645 h

3600 5 64h

602 5 s!64hd2

60 5 !2s32dh

83.

< 41.23 feet

5 10!17

5 !1700

5 !1024 1 676

c 5 !322 1 262

26

32


Section 5.5 Complex Numbers

97.

1.82 feet < L

2.254p 2s32d 5 L

2.254p 2 5

L32

11.52p2

2

5 1! L32 2

2

1.5 5 2p! L32

99.

< 500 units

x 5 500.0005

0.8x 5 400.0004

0.8x 2 0.8 5 399.2004

0.8sx 2 1d 5 399.2004

s!0.8sx 2 1dd25 s19.98d2

!0.8sx 2 1d 5 19.98

30.02 5 50 2 !0.8sx 2 1d


133.5 9.3 18 y1

(b)

< 5 years from 1990 < 1995

0120

7

260

0120

7

260

103.

P < $12,708.73

P 525,0001.0710

1.0710 525,000

P

1.0710 5 1 10!25,000P 2

10

1.07 5 10!25,000P

1.07 5 125,000P 2

1y10

0.07 5 125,000P 2

1y10

2 1

R 5 1AP2

1yn

2 1

105. No. It is not an operation that necessarily yields anequivalent equation. There may be extraneous solutions.

107.

must be multiplied by FOIL.s!x 1 !6d2

s!x 1 !6d2Þ s!xd2

1 s!6 d2

Y5 1 1 ! GRAPH

1.

5 2i

5 !21 ? !4

!24 5 !21 ? 4 3.

5 212i

5 2!144 ? !21

2!2144 5 2!144 ? 21 5.

525

i

5! 425

? !21

!2425

5! 425

? 21

X,T,u X,T,u

Section 5.5 Complex Numbers 293

7.

5 0.3i

5 !0.09 ? !21

!20.09 5 !0.09 ? 21 9.

5 2i!2

5 !4 ? !2 ? !21

!28 5 !4 ? 2 ? 21 11.

5 3i!3

5 !21 ? !9 ? !3

!227 5 !21 ? 9 ? 3

13.

5 i!7

5 !7 ? !21

!27 5 !7 ? 21 15.

or

!212

!235!212

235 !4 5 2

5 !4 5 2

5!4 ? !3 ? !21

!3 ? !21

!212

!235

!4 ? 3 ? 21

!3 ? 2117.

5 i!5

5 !5 ? !21

5 !25

!220

!45!220

4

19.

53i8!2

!21864

5!21 ? 9 ? 264

21.

5 10i

5 s4 1 6di

!216 1 !236 5 4i 1 6i

23.

5 3!2i

5 s5!2 2 2!2di !250 2 !28 5 5i!2 2 2i!2

25.

5 3i!3

5 s4i 1 2i 2 3id!3

5 4i!3 1 2i!3 2 3i!3

!248 1 !212 2 !227 5 !16 ? 3 ? 21 1 !4 ? 3 ? 21 2 !9 ? 3 ? 21

27.

5 4s21d 5 24

5 2 ? 2 ? i2

!28!22 5 s2i!2dsi!2d 29.

5 23!6

5 3!6 ? i2

!218!23 5 s3i!2dsi!3d

31. !20.16!21.21 5 s0.4id(1.1id 5 0.44i2 5 20.44 33.

5 23 2 2!3

5 si!3d21 2!3i2

!23s!23 1 !24d 5 i!3si!3 1 2id

39.

5 216

5 16i2

s!216d25 s4id2 41.

5 28i

5 8i3

s!24d35 s2id3 43.

b 5 24 a 5 3

3 2 4i 5 a 1 bi

35.

5 5!2 2 4!5

5 24!5 1 5!2

5 i24!5 2 i2!50

!25s!216 2 !210d 5 i!5s4i 2 i!10d 37.

5 3!2 i 1 4

5 3!2i 2 2i2s2d

!22s3 2 !28d 5 i!2s3 2 2i!2d


45.

b 5 23 a 5 2

b 2 1 5 24 a 1 3 5 5

5 2 4i 5 sa 1 3d 1 sb 2 1di 47.

22!2 5 b

22i!2 5 bi 24 5 a

24 2 2i!2 5 a 1 bi

24 2 !28 5 a 1 bi

49.

b 5 22 a 5 2

b 2 1 5 23 a 1 5 5 7

sa 1 5d 1 sb 2 1di 5 7 2 3i 51.

5 10 1 4i

s4 2 3id 1 s6 1 7id 5 s4 1 6d 1 s23 1 7di

53.

5 214 2 40i

s24 2 7id 1 s210 2 33id 5 s24 2 10d 1 s27 2 33di 55.

5 214 1 20i

13i 2 s14 2 7id 5 s214d 1 s13 1 7di

57.

5 9 2 7i

s30 2 id 2 s18 1 6id 1 3i2 5 30 2 i 2 18 2 6i 2 3 59.

5 3 1 6i

6 2 s3 2 4id 1 2i 5 6 2 3 1 4i 1 2i

61.

5 136 1

32i

5 136 1

96i

5 s86 1

56d 1 s2

6 176di

s43 1

13id 1 s5

6 176id 5 s4

3 156d 1 s1

3 176di 63.

5 23 1 49i

5 23 1 s15 1 25 1 9di

15i 2 s3 2 25id 1 !281 5 15i 2 3 1 25i 1 9i

65.

5 7 1 s3!7 2 5di 8 2 s5 2 !263d 1 s4 2 5id 5 8 2 5 1 3i!7 1 4 2 5i

67.

5 236

s3ids12id 5 36i2 69.

5 24

5 224s21d

s3ids28id 5 224i2

71. s26ids2ids6id 5 36i3 5 236i 73. s23id3 5 227i3 5 27i

75.

5 29

5 9s21d

s23id2 5 9i2 77. 25s13 1 2id 5 265 2 10i

79. 4is23 2 5id 5 212i 2 20i2 5 20 2 12i 81.

5 4 1 18i

5 18i 1 4

5 18i 2 4i2

s9 2 2ids!24d 5 s9 2 2ids2id

83.

5 220 1 12i!5

5 12i!5 1 4i2s5d

!220s6 1 2!5id 5 2i!5s6 1 2!5id 85.

5 240 2 5i

5 228 2 12 2 5i

s4 1 3ids27 1 4id 5 228 1 16i 2 21i 1 12i2


87.

5 214 1 42i

5 228 1 42i 1 14

s27 1 7ids4 2 2id 5 228 1 14i 1 28i 2 14i2

89.

5 9

5 4 1 5

5 4 1 2i!5 2 2i!5 2 5i2

s22 1 !25ds22 2 !25d 5 s22 1 i!5ds22 2 i!5d

91.

5 27 2 24i

5 9 2 16 2 24i

5 9 2 24i 1 16i2

s3 2 4id2 5 32 2 2s3ds4id 1 s4id2 93.

5 221 1 20i

5 4 2 25 1 20i

5 4 1 20i 1 25i2

s2 1 5id2 5 22 1 2s2ds5id 1 s5id2

95.

5 2 1 11i

5 6 1 3i 1 8i 2 4

5 6 1 3i 1 8i 1 4i2

5 s3 1 4ids2 1 id

5 s4 1 4i 2 1ds2 1 id

5 s4 1 2s2di 1 i2ds2 1 id

s2 1 id3 5 s2 1 ids2 1 ids2 1 id 97. conjugate

product

5 5

5 4 1 1

5 22 2 i2

5 s2 1 ids2 2 id

5 2 2 i2 1 i,

99. conjugate

product

5 4 2 64i2 5 4 1 64 5 68

5 s22d2 2 s8id2

5 s22 2 8ids22 1 8id

5 22 1 8i22 2 8i, 101. conjugate

product

5 25 2 6i2 5 25 1 6 5 31

5 52 2 s!6id2

5 s5 2 !6ids5 1 !6id5 5 1 !6i5 2 !6i,

103. conjugate

product

5 100

5 2100i2

5 2s10id2

5 s10ids210id

5 210i10i,

105. conjugate

product

5 4

5 1 1 3

5 1 2 3i2

5 12 2 si!3d2

5 s1 1 i!3ds1 2 i!3d5 1 2 i!31 1 !23 5 1 1 i!3,


107. conjugate

product

5 2.5

5 2.25 1 0.25

5 1.52 2 s0.5id2

5 s1.5 1 0.5ids1.5 2 0.5id

5 1.5 2 !20.251.5 1 !20.25, 109.

5 0 2 10i

202i

510i

?2i2i

5210i

1

111.

5 2 1 2i

5 2s1 1 id

54s1 1 id

2

4

1 2 i5

41 2 i

?1 1 i1 1 i

54s1 1 id1 1 1

113.

522453

18453

i

5224 1 84i

53

5212s2 2 7id

53

212

2 1 7i5

2122 1 7i

?2 2 7i2 2 7i

5212s2 2 7id

4 1 49

115.

5 265

125

i

52s23 1 id

55

26 1 2i5

5212 1 4i

105

4s23 1 id10

54i 1 12i2

10

4i1 2 3i

54i

1 2 3i?

1 1 3i1 1 3i

54is1 1 3id

1 1 9117.

585

215

i

58 2 i

5

52 1 6 2 i

5

2 1 3i1 1 2i

52 1 3i1 1 2i

?1 2 2i1 2 2i

52 2 4i 1 3i 2 6i2

1 1 4

119.

5 1 265

i

5s1 1 4d 1 s2 2 8di

55

5 2 6i5

51 1 2i1 1 4

14 2 8i1 1 4

51 1 2i

51

4 2 8i5

1

1 2 2i1

41 1 2i

51

1 2 2i?

1 1 2i1 1 2i

14

1 1 2i?

1 2 2i1 2 2i

121.

5253 1 29i

255

25325

12925

i

523 1 4i

252

50 2 25i25

5s23 2 50d 1 s4 1 25di

25

54i 1 3i2

16 1 92

10 2 5i4 1 1

523 1 4i

252

10 2 5i5

?55

i

4 2 3i2

52 1 i

5i

4 2 3i?

4 1 3i4 1 3i

25

2 1 i?

2 2 i2 2 i


123. (a)

Solution 0 5 0

1 2 4 2 2 1 5 5?

0

1 2 4i 1 4i2 2 2 1 4i 1 5 5?

0

s21 1 2id2 1 2s21 1 2id 1 5 5?

0

x 5 21 1 2i (b)

Solution 0 5 0

1 2 4 2 2 1 5 5?

0

1 1 4i 1 4i2 2 2 2 4i 1 5 5?

0

s21 2 2id2 1 2s21 2 2id 1 5 5?

0

x 5 21 2 2i

125. (a)

Solution 0 5 0

264 1 64 2 36 1 36 5?

0

s24d3 1 4s24d2 1 9s24d 1 36 5?

0

x 5 24 (b)

Solution 0 5 0

27i 2 36 2 27i 1 36 5?

0

227i3 1 36i2 2 27i 1 36 5?

0

s23id3 1 4s23id2 1 9s23id 1 36 5?

0

x 5 23i

127. (a)

(b) use same method as part (a)

55004

5 125

51254

13754

5125

42

1254!3i 1

1254!3i 2

1254

s3di2

5 12252

2252!3i2125

21

52!3i2

5 12504

2252!3i2125

21

52!3i2

5 1254

2252!3i 2

754 2125

21

52!3i2

5 1254

2252!3i 1

254

s3di221252

152!3i2

125 1 5!3i2 23

5 1252

152!3i2

2

1252

152!3i2

129. (a)

(b)

(c) 4, 24 1 4!3i

25 22 1 2!3i,

24 2 4!3i2

5 22 2 2!3i

2, 22 1 2!3i

25 21 1 !3i,

22 2 2!3i2

5 21 2 !3i

1, 21 1 !3i

2,

21 2 !3i2

131.

5 2a 1 0i

sa 1 bid 1 sa 2 bid 5 sa 1 ad 1 sb 2 bdi 133.

5 0 1 2bi

sa 1 bid 2 sa 2 bid 5 sa 2 ad 1 sb 1 bdi

135. i 5 !21



137.

The product rule for radicals does not hold if both radicands are negative.

!23!23 5 i!3 ? i!3 5 i2s3d 5 23

!23!23 5 !s23ds23d 5 !9 5 3 139. conjugate

product

5 13

5 9 1 4

5 32 2 s2id2

5 s3 2 2ids3 1 2id

5 3 1 2i3 2 2i,

1. because 7 ? 7 5 49!49 5 7 3. because 9 ? 9 5 812!81 5 29

5. because 22 ? 22 ? 22 5 283!28 5 22 7. because 4 ? 4 ? 4 5 642 3!64 5 24

9. (inverse property of powers and roots)!s1.2d2 5 1.2 11. (inverse property of powers and roots)!s56d2

556

13. (inverse property of powers and roots)3!2s15d3

5 215 15. !222 5 2i

17. 491y2 5 7 19. 3!216 5 6 21. 274y3 5 s 3!27d45 34 5 81

23. 2s52d3y2 5 2s!25d35 253 5 2125 25. 824y3 5

184y3 5

1

s 3!8d4 5124 5

116

27.

5 29

16

5 2s34d2

2s2764d2y3

5 2s 3!2764 d2

29.

5 x7y12

5 x9y121s22dy12

x3y4 ? x21y6 5 x3y41s21y6d 31.

5 z5y3

5 z112y3

z 3!z2 5 z ? z2y3

33.4!x3

!x 45

x3y4

x4y2 5 x3y422 5 x3y428y4 5 x25y4 51

x5y4 35. 3!a3b2 5 a 3!b2

37. 4!!x 5 4!x1y2 5 sx1y2d1y4 5 x1y8 5 8!x 39.

5 3!3x 1 2

5 s3x 1 2d1y3

5 s3x 1 2d2y321y3

s3x 1 2d2y3

3!3x 1 25

s3x 1 2d2y3

s3x 1 2d1y3

41. 7523y4 5 0.0392377 < 0.04 43. !132 2 4s2ds7d 5 10.630146 < 10.63

45. Keystrokes:

3 2

Domain5 s2`, `d

47. Keystrokes:

4 .75

Domain5 f0, `d

−5

−10

50

100

−15

−10

15

10

Y5 3! GRAPH GRAPHY5 >X,T,u X,T,u


49.

5 6!10

!360 5 !36 ? 10 51.

5 5u2v2!3u

!75u5v 4 5 !25 ? 3 ? u 4 ? u ? v 4 53.

5 0.5x2!y

5 5 3 1021x2!y

!0.25x 4y 5 !25 3 1022x4y

55.

5 2b 4!4a2b

4!64a2b5 5 4!16 ? 4 ? a2 ? b4 ? b 57.

5 2ab 3!6b

3!48a3b4 5 3!8 ? 6a3b3b 59. !56

5!56

?!6!6

5!30

6

61.3

!12x5

3

!4 ? 3x5

3

2!3x?!3x

!3x5

3!3x6x

5!3x2x

63.2

3!2x5

23!2x

?3!22x2

3!22x25

2 3!4x2

3!8x35

2 3!4x2

2x5

3!4x2

x

65.

5 !7

2!7 2 5!7 1 4!7 5 !7s2 2 5 1 4d 67.

5 224!10

5 6!10 2 30!10

3!40 2 10!90 5 3!4 ? 10 2 10!9 ? 10

69.

5 14!x 2 9 3!x

5 s5 1 9d!x 1 s21 2 8d 3!x

5!x 2 3!x 1 9!x 2 8 3!x 5 5!x 1 9!x 2 3!x 2 8 3!x

71.

5 7 4!y 1 3

10 4!y 1 3 2 3 4!y 1 3 5 s10 2 3d 4!y 1 3 73.

5 12!x 2 2 3!x

!25x 1 !49x 2 3!8x 5 5!x 1 7!x 2 2 3!x

75.

52!5

5

55!5

52

3!55

5 !5 ?55

23!5

5

5 !5 23!5

5

!5 23

!55 !5 2

3

!5?!5

!577.

5 10!3

5 !100 ? 3

5 !300

!15 ? !20 5 !15 ? 20

79.

5 5!2 1 3!5

5 !25 ? 2 1 3!5

5 !50 1 3!5

!5s!10 1 3d 5 !5!10 1 !5 ? 3 81.

5 2!5 1 5!2

5 !4 ? 5 1 !25 ? 2

5 !20 1 !50

!10s!2 1 !5d 5 !10!2 1 !10!5

83.

5 6!2 2 4!3 1 7!6 2 14

5 2!18 2 4!3 1 7!6 2 14

s2!3 1 7ds!6 2 2d 5 2!3!6 2 4!3 1 7!6 2 14

85. s!5 1 6d25 s!5 d2

1 2s6d!5 1 62 5 5 1 12!5 1 36 5 41 1 12!5

87. s!3 2 !x ds!3 1 !x d 5 3 2 !3x 1 !3x 2 x 5 3 2 x


89.

5 23s1 1 !2 d

53s1 1 !2d

21

53s1 1 !2d

1 2 2

3

1 2 !2?

1 1 !2

1 1 !25

3s1 1 !2d12 2 s!2d2 91.

524 2 6!6

5

524 2 6!6

8 2 3

3!8

2!2 1 !3?

2!2 2 !3

2!2 2 !35

6!16 2 3!24

s2!2 d22 s!3 d2

93.

5 2!6 1 4!2 2 !3 2 4

13

5!6 1 4!2 2 !3 2 4

213

5!6 1 4!2 2 !3 2 4

3 2 16

5!6 1 4!2 2 !3 2 4

s!3 d22 42

!2 2 1

!3 2 45

!2 2 1

!3 2 4?!3 1 4

!3 1 495.

5x 1 20!x 1 100

x 2 100

5x 1 10!x 1 10!x 1 100

s!x d22 102

!x 1 10

!x 2 105

!x 1 10

!x 2 10?!x 1 10

!x 1 10

97. Keystrokes:

5 2

10 2

! 52x

5! 52x

?!2x

!2x5

!10x2x

y2

y1 Y5 !

! x

x

x

4

GRAPH

ENTERcc

c 4 c

x

−1

−1

6

6

99. Keystrokes:

5 2

3

5!x 2 2!x 5 s5 2 2d!x 5 3!x

y2

y1 ! !

! GRAPH

ENTERY5 2

−1

−1

6

8

101. Check:

y 5 225

15 5 15 s!y d25 s15d2

!225 5?

15 !y 5 15 103. Check:

No real solution

x 5 27

x 5813

18 Þ 0 3x 5 81

9 1 9 5?

0 s!3x d25 s29d2

!81 1 9 5?

0 !3 5 29

!3 ? 27 1 9 5?

0 !3x 1 9 5 0

105. Check:

a 5 105

2a 5 210

2a 2 14 5 196

14 5 14 2sa 2 7d 5 196

!196 5?

14 s!2sa 2 7d d25 s14d2

!2s105 2 7d 5?

14 !2sa 2 7d 5 14 107.

x 5 3

5x 5 15

5x 2 7 5 8

s 3!5x 2 7 d35 23

3!5x 2 7 5 2

3!5x 2 7 2 3 5 21

X,T,u

X,T,uX,T,u

X,T,u X,T,u

X,T,u

Check:

21 5 21

2 2 3 5?

21

3!8 2 3 5?

21

3!5s3d 2 7 2 3 5?

21


109.

5 5 x

10 5 2x

5x 1 2 5 7x 2 8

s 3!5x 1 2 d35 s 3!7x 2 8 d3

3!5x 1 2 5 3!7x 2 8

3!5x 1 2 2 3!7x 2 8 5 0 Check:

0 5 0

3!27 2 3!27 5?

0

3!5s5d 1 2 2 3!7s5d 2 8 5?

0

111.

, x 5 23 25 5 x

0 5 sx 1 5dsx 1 3d

0 5 x2 1 8x 1 15

2x 1 10 5 x2 1 10x 1 25

2sx 1 5d 5 x2 1 10x 1 25

s!2sx 1 5d d25 sx 1 5d2

!2sx 1 5d 5 x 1 5 Check:

2 5 2

!4 5?

2

!2s23 1 5d 5?

23 1 5

0 5 0

!0 5?

0

!2s25 1 5d 5?

25 1 5

113.

, v 5 7 v 5 6

0 5 sv 2 6dsv 2 7d

0 5 v2 2 13v 1 42

v 2 6 5 36 2 12v 1 v2

s!v 2 6 d25 s6 2 vd2

!v 2 6 5 6 2 v Check:

not a solution

1 Þ 21

!7 2 6 5?

6 2 7

0 5 0

!6 2 6 5?

6 2 6

115.

332

5 x

996

5 x

9 5 16s6xd

s3d2 5 s4!6x d2

23 5 24!6x

1 5 4 2 4!6x

1 1 6x 5 4 2 4!6x 1 6x

s!1 1 6x d25 s2 2 !6x d2

!1 1 6x 5 2 2 !6xCheck:

54

554

54

5? 8

42

34

54

5?

2 234

!2516

5?

2 2! 916

!25 ? 216 ? 2

5?

2 2! 9 ? 216 ? 2

!5032

5?

2 2! 9 ? 216 ? 2

!3232

11832

5?

2 2!1832

!1 1 61 3322 5

?2 2!61 3

322

117. !248 5 !16 ? 3 ? 21 5 4i!3 119.

5 10 2 9i!3

5 10 2 3!21 ? !9 ? !3

10 2 3!227 5 10 2 3!21 ? 9 ? 3


121.

5 34 2 i!3

5 34 2

55i!3

34 2 5!2325 5

34 2 5! 3

25 ? 21123.

5 15i

!281 1 !236 5 9i 1 6i

125. !2121 2 !284 5 11i 2 2i!21 127. !25!25 5 i!5 ? i!5 5 i2 ? 5 5 25

129.

5 22!10 1 !70

5 2i2!10 2 i2!70

!210 s!24 2 !27 d 5 i!10 s2i 2 i!7 d 131.

3 5 y x 5 2

26 5 22y 4x 5 8

4x 2 6i 5 8 2 2yi

4x 2 !236 5 8 2 2yi

133.

y 5 125

5y 5 625 4 5 x

!5y 5 25 24 5 6x

24 1 i!5y 5 6x 1 25i

24 1 !25y 5 6x 1 25i 135.

5 8 2 3i

s24 1 5id 2 s212 1 8id 5 s24 1 12d 1 s5 2 8di

137.

5 8 1 4i

5 s3 1 5d 1 s28 1 12di

s3 2 8id 1 s5 1 12id 5 3 2 8i 1 5 1 12i 139.

5 25

5 16 1 9

s4 2 3ids4 1 3id 5 42 2 s3id2

141.

5 11 2 60i

5 36 2 60i 2 25

s6 2 5id2 5 62 2 2s6ds5id 1 s5id2 143.73i

573i

?2i2i

527i23i2

527i

3

145.

52817

1217

i

528 1 2i

17

58i 2 32

68

58i 2 324 1 64

58i 1 32i2

22 2 s8id2

4i

2 2 8i5

4i2 2 8i

?2 1 8i2 1 8i

147.

51337

23337

i

513 2 33i

37

518 2 33i 2 5

36 1 1

518 2 3i 2 30i 1 5i2

62 2 i2

3 2 5i6 1 i

53 2 5i6 1 i

?6 2 i6 2 i



14 in.

3in

.3

in.

3 in. 3 in.

8 in.12

149.

Equation:

5 21 1 12!2 inches

5 16 1 5 1 12!2

P 5 2s8d 1 2s212d 1 4s!18 d

c 5 !32 1 32 5 !9 1 9 5 !18

151.

1.3698624 5 L < 1.37 feet

1.694p2 s32d 5 L

1.694p2 5

L32

11.32p2

2

5 1! L322

2

1.32p

5! L32

1.3 5 2p! L32

153.

500 watts 5 P

25 5P20

52 5 1! P202

2

5 5! P20

I 5!PR

155.

9000 watts 5 P

225 5P40

152 5 1! P402

2

15 5! P40

I 5!PR

157.

9.77 feet < h

625 5 2s32dh

252 5 s!2s32dh d2

25 5 !2s32dh

v 5 !2gh

1. (a)

5 64

5 43

163y2 5 s!16 d3

2. (a)

519

2722y3 51

272y3

(b)

5 10

5 !100

!5!20 5 !5 ? 20

(b)

5 6

5 !36

!2!18 5 !2 ? 18

3. (a)

5 x122y3 5 x1y3

1x1y2

x1y322

5x

x2y3 (b)

5 58y4 5 52 5 25

51y4 ? 57y4 5 51y417y4

4. (a) !329

5!16 ? 29

543!2 (b) 3!24 5 3!8 ? 3 5 2 3!3


5. (a)

5 2x!6x

!24x3 5 !4 ? 6 ? x2 ? x (b)

5 2xy2 4!x

4!16x5y8 5 4!16x4xy8

6.

Multiply the numerator and denominator of a fraction by afactor such that no radical contains a fraction and nodenominator of a fraction contains a radical.

3

!65

3

!6?!6

!65

3!66

5!62

7.

5 210!3x

5 5!3x 2 15!3x

5!3x 2 3!75x 5 5!3x 2 3!25 ? 3x

8.

5 5!3x 1 3!5

5 !25 ? 3x 1 3!5

!5s!15x 1 3d 5 !75x 1 3!5 9. s4 2 !2x d25 16 2 8!2x 1 2x

10.

5 7!3s3 1 4yd

5 21!3 1 28y!3

7!27 1 14y!12 5 7!9 ? 3 1 14y!4 ? 3 11. Check:

y 5 27

3 5 3 3y 5 81

9 2 6 5?

3 s!3y d25 92

!81 2 6 5?

3 !3y 5 9

!3s27d 2 6 5?

3 !3y 2 6 5 3

12.

No solution

x 554

4x 5 5

x2 2 1 5 x2 2 4x 1 4

s!x2 2 1 d25 sx 2 2d2

!x2 2 1 5 x 2 2 Check:

34 Þ 234

! 916 5

?2

34

!2516 2

1616 5

? 54 2

84

!s54d2

2 1 5? 5

4 2 2

13.

Not a solution

4 5 x9 5 x

0 5 x 2 40 5 x 2 9

0 5 sx 2 9dsx 2 4d

0 5 x2 2 13x 1 36

x 5 x2 2 12x 1 36

s!xd2 5 sx 2 6d2

!x 2 x 1 6 5 0 Check:

4 Þ 0

2 2 4 1 6 5?

0

!4 2 4 1 6 5?

0

0 5 0

3 2 9 1 6 5?

0

!9 2 9 1 6 5?

0

14.

y 5 400

!y 5 20x 5 4

2!y 5 403x 5 12

3x 1 2!yi 5 12 1 40i

3x 1 !24y 5 12 1 40i 15.

y 5 1

!y 5 1 3 5 x

24!y 5 24 27 5 9x

27 2 4!yi 5 9x 2 4i

27 2 !216y 5 9x 2 4i

16. s2 1 3id 2 !225 5 2 1 3i 2 5i 5 2 2 2i


17.

5 25 2 12i

5 4 2 12i 2 9

5 4 2 6i 2 6i 1 9i2

s2 2 3id2 5 s2 2 3ids2 2 3id 18.

5 28 1 4i

5 4i 1 8i2

!216 s1 1 !4 d 5 4is1 1 2id

19.

5 13 1 13i

5 3 1 13i 1 10

s3 2 2ids1 1 5id 5 3 1 13i 2 10i2 20.5 2 2i

i5

5 2 2ii

?2i2i

525i 1 2i2

2i25 22 2 5i

21.

100 feet 5 h

6400 5 64h

802 5 s!64h d2

80 5 !64h

80 5 !2s32dh

v 5 !2gh


5.

y 5 23x 1 4

3x 1 y 5 4 6.

y 5 223

x 123

3y 5 22x 1 2

2x 1 3y 5 2 7.

y 54 2 x2

35

13

s4 2 x2d

3y 5 4 2 x2

x2 1 3y 5 4

8.

y 5 2x2 1 4

x2 1 y 2 4 5 0 9.

y 523!x 2 5

23y 5 22!x 1 15

2!x 2 3y 5 15 10.

y 565|x| 1 2

25y 5 26|x| 2 10

6|x| 2 5y 1 10 5 0

11.

Labels:

Equation:

x 5125

hours

5x 5 12

3x 1 2x 5 12

12x114

1162 5 11

x212x

14

116

51x

Time together 5 x hours

Friend’s time 5 6 hours

Your time 5 4 hours

Rateperson 1

VerbalModel:

1 5Rateperson 2

Ratetogether 12.

Labels:

Equation:

Rate 5 47.25 mph

Rate 5180

9054 1

9042

Time 2 59042

Time 1 59054

Totaltime

VerbalModel:

15 Time 1 Time 2

SECTION 5.2 Simplifying Radical Expressions

1. Graph with a dotted line since the inequality is Test one point in each half-plane formed by the line. Shade thehalf-plane that satisfies the inequality.

>.x 2 y 5 23

2. and

The first inequality includes the points on the line and the second does not.3x 1 4y 5 4

3x 1 4y < 43x 1 4y ≤ 4

3.

5 2sx 2 3dsx2 1 1d

5 sx 2 3ds2x2 2 1d

2x3 1 3x2 2 x 1 3 5 2x2sx 2 3d 2 1sx 2 3d 4. 4t 2 2 169 5 s2t 2 13ds2t 1 13d

5. x2 2 3x 1 2 5 sx 2 2dsx 2 1d 6. 2x2 1 5x 2 7 5 s2x 1 7dsx 2 1d

7. 11x2 1 6x 2 5 5 s11x 2 5dsx 1 1d 8.

5 s2x 2 7d2

4x2 2 28x 1 49 5 s2x 2 7ds2x 2 7d

DistanceTime

5Rate


9.

Labels:

System:

Solve by substitution:

y 5 1200 2 816 5 384 students

x 5 816 adults

7.5x 5 6120

20x 1 15,000 2 12.50x 5 21,120

20x 1 12.50s1200 2 xd 5 21,120

y 5 1200 2 x

x

20x

1

1

y

12.50y

5

5

1200

21,120

Student tickets 5 y

Adult tickets 5 x

Price adulttickets

1 5 21,120

Adulttickets

VerbalModel:

1 5 1200Studenttickets

?Adulttickets

Price studenttickets

?Studenttickets

10.

Labels:

Equation:

x < 267 units

x 52s10,000d

75

275

5x

10,000

Number defective units s2d 5 x

Number defective units 1Total number units 1

VerbalModel:

5Number defective units 2

Total number units 2

SECTION 5.3 Multiplying and Dividing Radical Expressions

1.

mn 5 c

x2 1 bx 1 c 5 sx 1 mdsx 1 nd 2.

If the signs of and must be the same.nmc > 0,

x2 1 bx 1 c 5 sx 1 mdsx 1 nd

3. If the signs of and must be different.nmc < 0, 4. If and have like signs, then m 1 n 5 b.nm

5.

0 5 2x 2 y

y 5 2x

y 2 6 5 2x 2 6

y 2 6 5 2sx 2 3d

y 2 y1 5 msx 2 x1d

m 5y2 2 y1

x2 2 x15

6 2 s22d3 2 s21d 5

6 1 23 1 1

584

5 2

s21, 22d, s3, 6d 6.

x 1 y 2 6 5 0

y 5 2x 1 6

y 2 0 5 21sx 2 6d

y 2 y1 5 msx 2 x1d

m 5y2 2 y1

x2 2 x15

0 2 56 2 1

5255

5 21

s1, 5d, s6, 0d


7.

y 2 3 5 0

y 2 3 5 0sx 2 10d

y 2 y1 5 msx 2 x1d

m 5y2 2 y1

x2 2 x15

3 2 310 2 6

504

5 0

s6, 3d, s10, 3d 8.

x 2 4 5 0

x 5 4

m 5y2 2 y1

x2 2 x15

5 2 s22d4 2 4

570

5 undefined

s4, 22d, s4, 5d

9.

0 5 6x 1 11y 2 96

11y 2 66 5 26x 1 30

y 2 6 5 26

11sx 2 5d

y 2 y1 5 msx 2 x1d

m 5y2 2 y1

x2 2 x15

6 2 8

5 243

522113

5 2611

143

, 82, s5, 6d 10.

x 1 y 2 11 5 0

y 2 1 5 2x 1 10

y 2 1 5 21sx 2 10d

y 2 y1 5 msx 2 x1d

m 5y2 2 y1

x2 2 x15

1 2 410 2 7

5233

5 21

s7, 4d, s10, 1d

11.

Labels:

Equation:360

r5 t

Rate 5 r

Distance 5 360

Time 5 t


5 ?Rate Time

DistanceRate

5 Time

12.

Labels:

Equation:

P 583

L

P 5 2L 1 21L32

Width 5L3

Length 5 L

Perimeter 5 P

PerimeterVerbalModel:

5 2 ? Length 1 2 ? Width

SECTION 5.4 Solving Radical Equations

1.

The function is undefined when the denominator is zero.Set the denominator equal to zero and solve for

The domain is all real numbers such that and x Þ 3.x Þ 22x

x 5 22 x 5 3

x 1 2 5 0 x 2 3 5 0

sx 1 2dsx 2 3d 5 0

x.

f sxd 54

sx 1 2dsx 2 3d 2.

Undefined

00

576

18 2 15 2 3

9 2 95

2726

2s23d2 1 5s23d 2 3

s23d2 2 95

2s23d 2 123 2 3

2x2 1 5x 2 3

x2 2 95

2x 2 1x 2 3

, x Þ 23


3. s23x2y3d2 ? s4xy2d 5 9x 4y6 ? 4xy2 5 36x5y8

7.

52sx 1 13d

5x2

x 1 13

x3s3 2 xd ?xsx 2 3d

55

x 1 13x3s3 2 xd ?

2xs3 2 xd5

8.

5x2 2 4

25sx2 2 9d

x 1 2

5x 1 15?

x 2 25sx 2 3d 5

sx 1 2dsx 2 2d5sx 1 3d5sx 2 3d

9.2x

x 2 52

55 2 x

52x

x 2 51

5x 2 5

52x 1 5x 2 5

10.

5 25x 2 8x 2 1

58 2 5xx 2 1

53 2 5x 1 5

x 2 1

53

x 2 12

5sx 2 1dx 2 1

3

x 2 12 5 5

3x 2 1

2 51x 2 1x 2 12

11. intercept:

intercept:

32 5 x

3 5 2x

0 5 2x 2 3

x-

y 5 23

y 5 2s0d 2 3y

x2 3 4−1−2

−2

−1

−3

2

1

(0, 3)−

32, 0( (

y-y 5 2x 2 3 12. intercept:

intercept:

x 583

x 5 2 ? 43

34 x 5 2

0 5 234 x 1 2

x-

y 5 234s0d 1 2 5 2y

x21 4−1−2

−2

−1

3

4

1 83, 0( (

(0, 2)

y-y 5 234 x 1 2

4. sx2 2 3xyd0 5 1

5.64r2s 4

16rs2 5 4r221s 422 5 4rs2 6. 1 3x4y32

2

5 1 3x4y321 3x

4y32 59x2

16y6

SECTION 5.5 Complex Numbers

1.

Multiply numerators. Multiply denominators.

3t5

?8t 2

155

s3tds8t 2ds5ds15d 5

24t3

752.

Multiply by the reciprocal of the divisor.

3t5

48t 2

155

3t5

?158t 2 5

s3tds15ds5ds8t 2d 5

98t

3.

Change each fraction into an equivalent fraction with thelowest common denominator as the denominator. Add thenumerators and put over the lowest common denominator.

3t5

18t 2

155

9t15

18t 2

155

9t 1 8t 2

154.

t 2 55 2 t

5t 2 5

21st 2 5d 5 21


5.

5x5

5x2s2x 1 3ds2x 1 3d5x

x2

2x 1 34

5x2x 1 3

5x2

2x 1 3?

2x 1 35x

6.

5x

5sx 1 yd

5sx 2 ydx2

5xsx 2 ydsx 1 yd

x 2 y

5x4

x2 2 y2

x2 5x 2 y

5x?

x2

sx 2 ydsx 1 yd

7.

9x

16x

1 22?

xx

59

6 1 2x8.

11 12x2

1x 24x2

?xx

5x 1 2x2 2 4

5x 1 2

sx 2 2dsx 1 2d 51

x 2 2

9.

5x2 1 2x 2 13

xsx 2 2d

52sx2 1 2x 2 13d

2xsx 2 2d

52x2 1 4x 2 26

2x2 2 4x

54x 2 8 1 2x2 2 18

x2 2 5x 1 6 1 x2 1 x 2 6

4x2 2 9

12

x 2 21

x 1 31

1x 2 3

?sx 2 3dsx 1 3dsx 2 2dsx 2 3dsx 1 3dsx 2 2d 5

4sx 2 2d 1 2sx2 2 9dsx 2 3dsx 2 2d 1 sx 1 3dsx 2 2d

10.

5sx 1 1dsx 1 3d

3

5x2 1 4x 1 3

3

52x 1 2 1 x2 1 2x 1 1

3

52sx 1 1d 1 sx 1 1d2

3

1 1

x 1 11

122

1 32x2 1 4x 1 22

51 1

x 1 11

122

1 32sx2 1 2x 1 1d2

?2sx 1 1d2

2sx 1 2d2 11.

1st number:

2nd number:14x18

15x18

519x18

x2

15x18

59x18

15x18

514x18

57x9

4x3

2x2

3?

66

58x 2 3x

185

5x18

12.1

1 1c1

11c22

51

1c1

11c2

?c1c2

c1c25

c1c2

c2 1 c1

C H A P T E R 6Quadratic Equations and Inequalities

Section 6.1 Factoring and Extracting Square Roots . . . . . . . . . .307

Section 6.2 Completing the Square . . . . . . . . . . . . . . . . . . .313

Section 6.3 The Quadratic Formula . . . . . . . . . . . . . . . . . . .321

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329

Section 6.4 Applications of Quadratic Equations . . . . . . . . . . . .332

Section 6.5 Quadratic and Rational Inequalities . . . . . . . . . . . .340


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354

Cumulative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .357

307

C H A P T E R 6Quadratic Equations and Inequalities

Section 6.1 Factoring and Extracting Square RootsSolutions to Odd-Numbered Exercises

1.

x 5 7x 5 5

sx 2 5dsx 2 7d 5 0

x2 2 12x 1 35 5 0 3.

x 5 8x 5 29

sx 1 9dsx 2 8d 5 0

x2 1 x 2 72 5 0 5.

x 5 5x 5 29

sx 1 9dsx 2 5d 5 0

x2 1 4x 2 45 5 0

x2 1 4x 5 45

7.

x 5 6 x 5 6

x 2 6 5 0 x 2 6 5 0

sx 2 6dsx 2 6d 5 0

x2 2 12x 1 36 5 0 9.

x 5 243 x 5 2

43

3x 5 24 3x 5 24

3x 1 4 5 0 3x 1 4 5 0

s3x 1 4ds3x 1 4d 5 0

9x2 1 24x 1 16 5 0 11.

x 5 3 x 5 0

x 2 3 5 0 4x 5 0

4xsx 2 3d 5 0

4x2 2 12x 5 0

13.

u 5 12 u 5 9

u 2 12 5 0 u 2 9 5 0

su 2 9dsu 2 12d 5 0

usu 2 9d 2 12su 2 9d 5 0 15.

x 553 x 5 6

3x 2 5 5 0 x 2 6 5 0

sx 2 6ds3x 2 5d 5 0

3xsx 2 6d 2 5sx 2 6d 5 0 17.

y 5 1 y 5 6

y 2 1 5 0 y 2 6 5 0

sy 2 6dsy 2 1d 5 0

y2 2 7y 1 6 5 0

y2 2 7y 1 12 2 6 5 0

sy 2 4dsy 2 3d 5 6

19.

x 512 x 5 2

56

2x 2 1 5 0 6x 1 5 5 0

s6x 1 5ds2x 2 1d 5 0

12x2 1 4x 2 5 5 0

6x2 1 4x 5 5 2 6x2

2xs3x 1 2d 5 5 2 6x2 21.

x 5 ±8

x 5 ±!64

x2 5 64 23.

x 5 ±3

x 5 ±!9

x2 5 9

6x2 5 54

25.

x 5 ±45

x 5 ±!1625

x2 51625

25x2 5 16 27.

y 5 ±8

y 5 ±!64

y2 5 64

12y2 5 32 29.

x 5 ±52

x 5 ±!254

x2 5254

4x2 5 25

4x2 2 25 5 0

31.

u 5 ±152

u 5 ±!2254

u2 52254

4u2 2 225 5 0 33.

x 5 9, 217

x 5 24 ± 13

x 1 4 5 ±!169

sx 1 4d2 5 169 35.

x 5 3.5, 2.5

x 5 3 ± 0.5

x 2 3 5 ±!0.25

sx 2 3d2 5 0.25

308 Chapter 6 Quadratic Equations and Inequalities

37.

x 5 2 ± !7

x 2 2 5 ±!7

sx 2 2d2 5 7 39.

x 521 ± 5!2

2

2x 5 21 ± 5!2

2x 1 1 5 ±!50

s2x 1 1d2 5 50 41.

x 53 ± 7!2

4

4x 5 3 ± 7!2

4x 2 3 5 ±!98

s4x 2 3d2 5 98

s4x 2 3d2 2 98 5 0

43.

z 5 ±6i

z 5 ±!236

z2 5 236 45.

x 5 ±2i

x 5 ±!24

x2 5 24

x2 1 4 5 0 47.

5 ± i!17

3

u 5 ±!2179

9u2 5 217

9u2 1 17 5 0

49.

t 5 3 ± 5i

t 2 3 5 ±!225

st 2 3d2 5 225 51.

z 5 243

± 4i

z 524 ± 12i

3

3z 5 24 ± 12i

3z 1 4 5 ±12i

3z 1 4 5 ±!2144

s3z 1 4d2 5 2144

s3z 1 4d2 1 144 5 0 53.

x 5 232

±3i!6

2

2x 5 23 ± 3i!6

2x 1 3 5 ±!254

s2x 1 3d2 5 254

55.

x 5 26 ±113

i

x 1 6 5 ±!21219

sx 1 6d2 52121

9

9sx 1 6d2 5 2121 57.

x 5 1 ± 3i!3

x 2 1 5 ±!227

sx 2 1d2 5 227 59.

x 5 21 ± 0.2i

x 1 1 5 ±!20.04

sx 1 1d2 5 20.04

sx 1 1d2 1 0.04 5 0

61.

c 523

±13

i

c 223

5 ±!219

1c 2232

2

5 219

1c 2232

2

119

5 0 63.

x 5 273

±i3!38

x 173

5 ±!2389

1x 1732

2

5 2389

65.

x 552

2x 2 5 5 0x 5 0

xs2x 2 5d 5 0

2x2 2 5x 5 0

67.

x 5 24x 532

s2x 2 3dsx 1 4d 5 0

2x2 1 5x 2 12 5 0 69.

x 5 ±30

x2 5 900

x2 2 900 5 0 71.

x 5 ±30i

x 5 ±!2900

x2 5 2900

x2 1 900 5 0

Section 6.1 Factoring and Extracting Square Roots309

73.

x 5 ±3

x2 5 9

32

?23

x2 5 6 ?32

23

x2 5 6 75.

x 5 15, 25

x 2 5 5 ±10

sx 2 5d2 5 100

sx 2 5d2 2 100 5 0

77.

x 5 5 1 10i x 5 5 2 10i

x 2 5 5 10i x 2 5 5 210i

u 5 10i u 5 210i

u 2 10i 5 0 u 1 10i 5 0

su 1 10idsu 2 10id 5 0

u2 1 100 5 0

let u 5 sx 2 5d

sx 2 5d2 1 100 5 0 79.

x 5 22 ± 3i!2

x 1 2 5 ±!218

sx 1 2d2 5 218

sx 1 2d2 1 18 5 0

81. Keystrokes:

9

x-intercepts are and 3.

x 5 23 x 5 3

x 1 3 5 0 x 2 3 5 0

5 sx 2 3dsx 1 3d

0 5 x2 2 9

23

83. Keystrokes:

2 15


x 5 23 x 5 5

x 1 3 5 0 x 2 5 5 0

0 5 sx 2 5dsx 1 3d

0 5 x2 2 2x 2 15

23

85. Keystrokes:

4 3

x-intercepts are 1 and 5.

x 5 5, 1

x 2 3 5 ±2

sx 2 3d2 5 4

0 5 4 2 sx 2 3d2

−15

−10

15

10

−12

−14

18

6

−18

−18

18

6

2x 2X,T,u5Y GRAPH

2 2x 2X,T,u X,T,u5Y GRAPH

2 2 x 2X,T,u5Y GRAPHx x


87. Keystrokes:

2 6


x 5 2x 5 232

0 5 s2x 1 3dsx 2 2d

0 5 2x2 2 x 2 6

232

−9

−7

5

9

89. Keystrokes:

3 8 16


x 5 4x 5 243

0 5 s3x 1 4dsx 2 14d

0 5 3x2 2 8x 2 16

243

−6

−24

6

10

91. Keystrokes:

7

The equation has complex roots.

± i!7 5 x

±!27 5 x

27 5 x2

0 5 x2 1 7

−2

16

−12 12

93. Keystrokes:

1 1


1 ± i 5 x

± i 5 x 2 1

21 5 sx 2 1d2

0 5 sx 2 1d2 1 1

−4

−1

8

7

95. Keystrokes:

3 5


23 ± !5i 5 x

±!5i 5 x 1 3

±!25 5 x 1 3

25 5 sx 1 3d2

0 5 sx 1 3d2 1 5−18

−2

12

18

97.

y 5 ±!4 2 x2

y2 5 4 2 x2

x2 1 y2 5 4

−3

−2

3

2

x 2X,T,u5Y GRAPHx x

2 x 2X,T,u5Y GRAPHx x

22x 2X,T,u X,T,u5Y GRAPH

x 2X,T,u5Y GRAPH


1

1

1 1

Section 6.1 Factoring and Extracting Square Roots311

99.

y 5 ±!4 2 x2

45 ±

!4 2 x2

2

y2 54 2 x2

4

4y2 5 4 2 x2

x2 1 4y2 5 4

−3

−2

3

2

101.

x 5 21 x 5 1 x 5 22 x 5 2

x 1 1 5 0 x 2 1 5 0 x 1 2 5 0 x 2 2 5 0

sx 2 2dsx 1 2dsx 2 1dsx 1 1d 5 0

sx2 2 4dsx2 2 1d 5 0

x4 2 5x2 1 4 5 0

103.

x 5 ±!2 x 5 ±!3

x2 5 2 x2 5 3

x2 2 2 5 0 x2 2 3 5 0

sx2 2 3dsx2 2 2d 5 0

x4 2 5x2 1 6 5 0 105.

x 5 ± i x 5 ±2

x2 5 21 x2 5 4

x2 1 1 5 0 x2 2 4 5 0

sx2 2 4dsx2 1 1d 5 0

x4 2 3x2 2 4 5 0

107.

x 5 ±!5 x 5 ±1

x2 5 5 x2 5 1

x2 2 5 5 0 x2 2 1 5 0

sx2 2 1dsx2 2 5d 5 0

fsx2 2 4d 1 3g fsx2 2 4d 2 1g 5 0

sx2 2 4d2 1 2sx2 2 4d 2 3 5 0

109.

let

x 5 4 x 5 25

!x 5 2 !x 5 5

u 5 2 u 5 5

su 2 5dsu 2 2d 5 0

u2 2 7u 1 10 5 0

s!x d22 7s!x d 1 10 5 0

u 5 !x

x 2 7!x 1 10 5 0 Check:

Check:

0 5 0

4 2 14 1 10 5?

0

4 2 7!4 1 10 5?

0

0 5 0

25 2 35 1 10 5?

0

25 2 7!25 1 10 5?

0

111.

x 5 28 x 5 27

x1y3 5 22 x1y3 5 3

x1y3 1 2 5 0 x1y3 2 3 5 0

sx1y3 2 3dsx1y3 1 2d 5 0

x2y3 2 x1y3 2 6 5 0


117.

x 5 243 x 51

32

x 5 35 x 5 1122

5

x1y5 5 3 x1y5 512

s2x1y5 2 1dsx1y5 2 3d 5 0

2x2y5 2 7x1y5 1 3 5 0 119.

x 5 1 x 512

x 2 1 5 0 2x 2 1 5 0

s2x 2 1dsx 2 1d 5 0

2x2 2 3x 1 1 5 0

1 2 3x 1 2x2 5 0

1x2 2

3x

1 2 5 0

113.

let

x 51258

x 5 1 sx1y3d 5 s52d3

sx1y3d 5 13 x1y3 552

x1y3 5 1 u 552

u 5 1 2u 5 5

u 2 1 5 0 2u 2 5 5 0

s2u 2 5dsu 2 1d 5 0

2u2 2 7u 1 5 5 0

2sx1y3d2 2 7x1y3 1 5 5 0

u 5 x1y3

2x2y3 2 7x1y3 1 5 5 0 115.

x 5 1 x 5 32

x 5 15 x 5 25

x1y5 5 1 x1y5 5 2

sx1y5 2 2dsx1y5 2 1d 5 0

x2y5 2 3x1y5 1 2 5 0

121.

x2 2 3x 2 10 5 0

x2 2 5x 1 2x 2 10 5 0

sx 2 5dsx 1 2d 5 0

sx 2 5)sx 2 s22dd 5 0 123.

x2 2 2x 2 1 5 0

x2 2 2x 1 1 2 2 5 0

sx 2 1d2 2 s!2 d25 0

fsx 2 1d 2 !2 dg fsx 2 1d 1 !2g 5 0

fx 2 s1 1 !2 dg fx 2 s1 2 !2 dg 5 0

125.

x2 1 25 5 0

x2 2 25i2 5 0

sx 2 5idsx 1 5id 5 0

sx 2 5idsx 2 s25idd 5 0 127.

t 5 4 seconds

t2 5 16

16t2 5 256

0 5 216t2 1 256 129.

seconds t 5 2!2 < 2.828

t 5 ±2!2

t 5 ±!8

t2 5 8

16t2 5 128

0 5 216t2 1 128

131.

seconds t 5 21 t 5 9

t 1 1 5 0 t 2 9 5 0

0 5 216st 2 9dst 1 1d

0 5 216st2 2 8t 2 9d

0 5 216t2 1 128t 1 144

0 5 144 1 128 2 162 133.

6% 5 r

.06 5 r

1.06 5 1 1 r

1.1236 5 s1 1 rd2

1685.40 5 1500s1 1 rd2

Section 6.2 Completing the Square 313

Section 6.2 Completing the Square

135.

Year 1993

3 < t

!892 2 26.6 5 t

!892 5 26.6 1 t

892 5 s26.6 1 td2

137. (a)

seconds

Extracting the roots method was used because thequadratic equation did not have a linear term.

t 5 2.5

t 5 !6.25

t2 5 6.25

16t2 5 100

0 5 16t2 1 0 ? t 1 100

h 5 0v0 5 0 feetysech0 5 100 feet (b)

Factoring method was used because the quadratic equationdid not have a constant term.

t 5 2 seconds t 5 0 seconds

t 2 2 5 0 216t 5 0

0 5 216tst 2 2d

0 5 216t2 1 32t

100 5 216t2 1 32t 1 100

h 5 100 feetv0 5 32 feetysech0 5 100 feet

139. Factoring and the Zero-Factor Property allow you tosolve a quadratic equation by converting it into two linear equations that you already know how to solve.

141. False. The solutions are and x 5 25.x 5 5

143. To solve an equation of quadratic form, determine analgebraic expression u such that substitution yields thequadratic equation Solve thisquadratic equation for u and then, through back-substitu-tion, find the solution of the original equation.

au2 1 bu 1 c 5 0.

1. 316 5 1822

2

4x2 1 8x 1 16 3. 3100 5 12202 22

4y2 2 20y 1 100

5. 364 5 12162 22

4x2 2 16x 1 64 7. 3254

5 1522

2

4t2 1 5t 1254

9. 3814

5 12922

2

4x2 2 9x 1814

11. 3 136

5 3121321

122

2

4a2 213

a 11

36

13. 3 9100

5 3123521

122

2

4y2 235

y 19

10015. 30.04 5 120.4

2 22

4r2 2 0.4r 1 0.04

17. (a)

(b)

x 5 20x 5 0

xsx 2 20d 5 0

x2 2 20x 5 0

x 5 20, 0

x 5 10 ± 10

x 2 10 5 ±10

sx 2 10d2 5 100

x2 2 20x 1 100 5 100 19. (a)

(b)

x 5 26, 0

x 1 6 5 0x 5 0

xsx 1 6d 5 0

x2 1 6x 5 0

x 5 26, 0

x 5 23 ± 3

x 1 3 5 ±3

sx 1 3d2 5 9

x2 1 6x 1 9 5 0 1 9


21. (a)

(b)

y 5 5

y 2 5 5 0y 5 0

ysy 2 5d 5 0

y2 2 5y 5 0

5 0, 5

y 552 ± 5

2

y 252 5 ±5

2

sy 252d2

5254

y2 2 5y 1254 5

254

y2 2 5y 5 0 23. (a)

(b)

t 5 1t 5 7

st 2 7dst 2 1d 5 0

t2 2 8t 1 7 5 0

t 5 7, 1

t 5 4 ± 3

t 2 4 5 ±3

st 2 4d2 5 9

t2 2 8t 1 16 5 27 1 16

25. (a)

(b)

x 5 4x 5 26

sx 1 6dsx 2 4d 5 0

x2 1 2x 2 24 5 0

x 5 4, 26

x 5 21 ± 5

x 1 1 5 ±5

sx 1 1d2 5 25

x2 1 2x 1 1 5 24 1 1 27. (a)

(b)

x 5 23x 5 24

sx 1 4dsx 1 3d 5 0

x2 1 7x 1 12 5 0

x 5 23, 24

x 5 262, 28

2

x 5 272 ± 1

2

x 172 5 ±1

2

sx 172d2

514

x2 1 7x 1494 5 212 1

494

29. (a)

(b)

x 5 23x 5 6

sx 2 6dsx 1 3d 5 0

x2 2 3x 2 18 5 0

x 5 6, 23

x 5122 , 26

2

x 532 ± 9

2

x 232 5 ±9

2

sx 232d2

5814

x2 2 3x 194 5 18 1

94 31. (a)

(b)

x 5 1x 5 6

sx 2 6dsx 2 1d 5 0

x2 2 7x 1 6 5 0

2x2 2 14x 1 12 5 0

x 5 6, 1

x 5122 , 22

x 572 ± 5

2

x 272 5 ±5

2

sx 272d2

5254

sx 272d2

5 2244 1

494

x2 2 7x 1494 5 26 1

494

x2 2 7x 5 26

x2 2 7x 1 6 5 0

2x2 2 14x 1 12 5 0


35.

x < 4.65, 20.65

x 5 2 ± !7

x 2 2 5 ±!7

sx 2 2d2 5 7

x2 2 4x 1 4 5 3 1 4

x2 2 4x 2 3 5 0

39.

u < 3.73, 0.27

u 5 2 ± !3

u 2 2 5 ±!3

su 2 2d2 5 3

u2 2 4u 1 4 5 21 1 4

u2 2 4u 1 1 5 0

37.

x < 0.65, 24.65

x 5 22 ± !7

x 1 2 5 ±!7

sx 1 2d2 5 7

x2 1 4x 1 4 5 3 1 4

x2 1 4x 2 3 5 0

41.

x < 21 2 1.41i

x < 21 1 1.41i

x 5 21 ± i!2

x 1 1 5 ±!22

sx 1 1d2 5 22

x2 1 2x 1 1 5 23 1 1

x2 1 2x 1 3 5 0

43.

x < 10.20, 20.20

x 5 5 ± 3!3

x 2 5 5 ±!27

sx 2 5d2 5 27

x2 2 10x 1 25 5 2 1 25

x2 2 10x 2 2 5 0 45.

y < 20.51, 219.49

y 5 210 ± 3!10

y 1 10 5 ±!90

sy 1 10d2 5 90

y2 1 20y 1 100 5 210 1 100

y2 1 20y 1 10 5 0

47.

t < 20.70, 24.30

t 525 ± !13

2

t 5 252

±!13

2

t 152

5 ±!134

1t 1522

2

5134

t2 1 5t 1254

5 23 1254

t2 1 5t 1 3 5 0 49.

v < 0.56, 23.56

v 523 ± !17

2

v 5 232

±!17

2

v 5 232

± !174

v 132

5 ±!174

1v 1322

2

5174

v2 1 3v 194

5 2 194

v2 1 3v 2 2 5 0

33. (a)

x 532, 25

2

x 5 212 ± 2

x 112 5 ±!4

sx 112d2

5164

x2 1 x 114 5

154 1

14

x2 1 x 5154

x2 1 x 2154 5 0

4x2 1 4x 2 15 5 0 (b)

x 5 252x 5

32

s2x 2 3ds2x 1 5d 5 0

4x2 1 4x 2 15 5 0


51.

x < 0.5 2 0.87i

x < 0.5 1 0.87i

x 51 ± i!3

2

x 512

±i!3

2

x 212

5 ±!234

1x 2122

2

5 234

x2 2 x 114

5 21 114

x2 2 x 1 1 5 0

2x2 1 x 2 1 5 0 53.

x 5 4, 3

x 572

±12

x 272

5 ±!14

1x 2722

2

514

1x 2722

2

5248

41

494

x2 2 7x 1494

5 212 1494

x2 2 7x 1 12 5 0

55.

x < 2.10, 21.43

x 51 ± 2!7

3

x 513

±23!7

x 213

5 ±!289

1x 2132

2

5289

x2 223

x 119

5 3 119

x2 223

x 2 3 5 0 57.

v < 1.09, 21.84

v 5 238

±!137

8

v 138

5 ±!13764

1v 1382

2

513764

1v 1382

2

512864

19

64

v2 134

v 19

645 2 1

964

v2 134

v 2 2 5 0

59.

x < 20.42, 23.58

x 5 22 ±!10

2

x 1 2 5 ±!52

?!2

!2

sx 1 2d2 552

x2 1 4x 1 4 5 232

1 4

2x2 1 8x 1 3 5 0 61.

x < 20.74, 22.26

x 529 ± !21

6

x 5 232

±!21

6

x 132

5 ±! 712

?!3

!3

1x 1322

2

57

12

1x 1322

2

5220 1 27

12

x2 1 3x 194

5 253

194

3x2 1 9x 1 5 5 0


63.

y < 1.08, 22.08

y 521 ± !10

2

y 5 212

±!10

2

y 112

5 ±!104

1y 1122

2

5104

y2 1 y 114

594

114

4y2 1 4y 2 9 5 0 65.

x < 0.30 1 1.38i, 0.30 2 1.38i

x 53

10±!191

10i

x 23

105 ±!2

191100

1x 23

1022

5 2191100

1x 23

1022

52200100

19

100

x2 235

x 19

1005 22 1

9100

x2 235

x 5 22

5x2 2 3x 1 10 5 0

67.

x < 7.27, 20.27

x 57 ± !57

2

x 572

±!57

2

x 272

5 ±!574

1x 2722

2

5574

1x 2722

2

58 1 49

4

x2 2 7x 1494

5 2 1494

xsx 2 7d 5 2 69.

t < 21 1 1.73i, 21 2 1.73i

t 5 21 ± !3i

t 1 1 5 ±!3i

t 1 1 5 ±!23

st 1 1d2 5 23

t2 1 2t 1 1 5 24 1 1

t2 1 2t 5 24

0.5t2 1 t 1 2 5 0

71.

x 5 21 ± 2i

x 1 1 5 ±!24

sx 1 1d2 5 24

x2 1 2x 1 1 5 25 1 1

x2 1 2x 1 5 5 0

0.1x2 1 0.2x 1 0.5 5 0 73.

x 5 1 ± !3

x 2 1 5 ±!3

sx 2 1d2 5 3

x2 2 2x 1 1 5 2 1 1

x2 2 2 5 2x

2x1x2

21x2 5 s1d2x

x2

21x

5 1


75.

x 5 1 ± !3

x 2 1 5 ±!3

sx 2 1d2 5 3

x2 2 2x 1 1 5 2 1 1

x2 2 2x 2 2 5 0

2x2 2 4x 2 4 5 0

2x2 5 4x 1 4

x2

45

x 1 12

79. Keystrokes:

4 1

−14

−8

10

8

x < 24.236

x < .236

22 ± !5 5 x

±!5 5 x 1 2

5 5 sx 1 2d2

1 1 4 5 x2 1 4x 1 4

1 5 x2 1 4x

0 5 x2 1 4x 2 1

77.

4 ± 2!2 5 x

4 ± !8 5 x

±!8 5 x 2 4

8 5 sx 2 4d2

116 2 8 5 x2 2 8x 1 16

0 5 x2 2 8x 1 8

2x 1 1 5 x2 2 6x 1 9

s!2x 1 1 d25 sx 2 3d2

!2x 1 1 5 x 2 3

81. Keystrokes:

2 5

−12

−8

12

8

x < 21.449

x < 3.449

1 ± !6 5 x

±!6 5 x 2 1

6 5 sx 2 1d2

1 1 5 5 x2 2 2x 1 1

5 5 x2 2 2x

0 5 x2 2 2x 2 5

83. Keystrokes:

1 3 2 6

x < 28.20

x < 2.20

23 ± 3!3 5 x

±!27 5 x 1 3

27 5 sx 1 3d2

9 1 18 5 x2 1 6x 1 9

18 5 x2 1 6x

0 5 x2 1 6x 2 18

0 513x2 1 2x 2 6 8

10

−12

−10

2 2x 2X,T,u X,T,u5Y GRAPH

x x1

2x 2X,T,u X,T,u5Y GRAPH1


4


85. Keystrokes:

3

−8 8

−6

6

x < 1.30, 22.30

x 5 212

±!13

2

x 112

5 ±!13

2

1x 1122

2

5134

x2 1 x 114

5 3 114

x2 1 x 5 3

x2 1 x 2 3 5 0

2x2 2 x 1 3 5 0

87. (a) Area of square

Area of vertical rectangle

Area of horizontal rectangle

(b) Area of small square

Total area

(c) sx 1 4dsx 1 4d 5 x2 1 8x 1 16

5 x2 1 8x 1 16

5 4 ? 4 5 16

Total area 5 x2 1 4x 1 4x 5 x2 1 8x

5 4 ? x 5 4x

5 4 ? x 5 4x

5 x ? x 5 x2

91. Verbal model:

Labels:

Equation:

200 2 4x

35 46

23 ft.

200 2 4x3

5 20 ft.

x 5 15 ft. x 5 35 ft.

x 2 15 5 0 x 2 35 5 0

sx 2 35dsx 2 15d 5 0

x2 2 50x 1 525 5 0

8x2 2 400x 1 4200 5 0

4200 5 400x 2 8x2

1400 5400x

32

8x2

3

1400 5 232003

x 24x2

3 4

1400 5 23x ? 1200 2 4x3 24

Width 5200 2 4x

3

Length 5 x

Width?Length5Area89. Verbal model:

Labels:

Equation:

base

not a solution heightx 1 2 5 6 cm

x 5 4 cmx 5 26

0 5 sx 1 6dsx 2 4d

0 5 x2 1 2x 2 24

24 5 x2 1 2x

12 512xsx 1 2d

Height 5 x 1 2

Base 5 x

Height?Base512 ?Area

2x 2X,T,u X,T,u5Y GRAPH1x2c


93. Verbal model:

Labels:

Equation:

meters and 129.29 meters x < 270.71

x 5 200 ± 50!2 meters

x 5 200 ± 50!2

x 2 200 5 ±50!2

x 2 200 5 ±!5000

sx 2 200d2 5 5000

x2 2 400x 1 40,000 5 235,000 1 40,000

x2 2 400x 5 235,000

x2 2 400x 1 35,000 5 0

2x2 2 800x 1 70,000 5 0

x2 1 160,000 2 800x 1 x2 5 90,000

x2 1 160,000 2 2s400dx 1 x2 5 90,000

x2 1 s400 2 xd2 5 3002

side 2 5 400 2 x

side 1 5 x

2Hypotenuse

25side 2

21side 1

95. Equation:

Thus, 139 or 861 units must be sold.

x < 860.56, 139.44

x 5 500 ± 100!13

x 2 500 5 ±100!13

x 2 500 5 ±!130,000

sx 2 500d2 5 130,000

x2 2 1000x 1 250,000 5 2120,000 1 250,000

x2 2 1000x 5 2120,000

x2 2 1000x 1 120,000 5 0

120,000 5 1000x 2 x2

12,000 5 100x 2110 x2

12,000 5 xs100 21

10 xd

97. Divide the coefficient of the first-degree term by 2, and square the result to obtain s52d2

5254 .25

4 .

99. Yes. x2 1 1 5 0

101. True. Given the solutions and the quadratic equation can be written as sx 2 r1dsx 2 r2d 5 0.x 5 r2,x 5 r1

Section 6.3 The Quadratic Formula 321

Section 6.3 The Quadratic Formula

1.

2x2 1 2x 2 7 5 0

2x2 5 7 2 2x 3.

x2 2 10x 1 5 5 0

2x2 1 10x 2 5 5 0

10x 2 x2 5 5

xs10 2 xd 5 5

5. (a)

(b)

x 5 4 x 5 7

x 2 4 5 0 x 2 7 5 0

sx 2 7dsx 2 4d 5 0

x 5 7, 4

x 511 ± 3

2

x 511 ± !9

2

x 511 ± !121 2 112

2

x 511 ± !112 2 4s1ds28d

2s1d 7. (a)

(b)

x 5 22 x 5 24

x 1 2 5 0 x 1 4 5 0

sx 1 4dsx 1 2d 5 0

x 5 22, 24 x 526 ± 2

2

x 526 ± !4

2

x 56 ± !36 2 32

2

x 526 ± !62 2 4s1ds8d

2s1d

9. (a)

(b)

x 5 212

x 5 212

2x 1 1 5 0 2x 1 1 5 0

s2x 1 1ds2x 1 1d 5 0

x 5248

5212

x 524 ± !16 2 16

8

x 524 ± !42 2 4s4ds1d

2s4d 11. (a)

(b)

x 5 232

x 5 232

2x 1 3 5 0 2x 1 3 5 0

s2x 1 3ds2x 1 3d 5 0

x 5 2128

5 232

x 5212 ± 0

8

x 5212 ± !144 2 144

8

x 5212 ± !122 2 4s4ds9d

2s4d

13. (a)

x 58

12, 2

612

523

, 212

x 51 ± 7

12

x 51 ± !49

12

x 51 ± !1 1 48

12

x 51 ± !s21d2 2 4s6ds22d

2s6d (b)

x 5 212

x 523

2x 1 1 5 0 3x 2 2 5 0

s3x 2 2ds2x 1 1d 5 0


15. (a)

(b)

x 5 215 x 5 20

x 1 15 5 0 x 2 20 5 0

sx 2 20dsx 1 15d 5 0

x 5 20, 215

x 55 ± 35

2

x 55 ± !1225

2

x 55 ± !25 1 1200

2

x 52s25d ± !s25d2 2 4s1ds2300d

2s1d 17.

x 5 1 ± !5

x 52s1 ± !5d

2

x 52 ± 2!5

2

x 52 ± !20

2

x 52 ± !4 1 16

2

x 52s22d ± !s22d2 2 4s1ds24d

2s1d

19.

t 5 22 ± !3

t 52s22 ± !3d

2

t 524 ± 2!3

2

t 54 ± !12

2

t 524 ± !16 2 4

2

t 524 ± !42 2 4s1ds1d

2s1d 21.

x 5 23 ± 2!3

x 52s23 ± 2!3d

2

x 526 ± 4!3

2

x 526 ± !48

2

x 526 ± !36 1 12

2

x 526 ± !62 2 4s1ds23d

2s1d

23.

x 5 5 ± !2

x 52s5 ± !2d

2

x 510 ± 2!2

2

x 510 ± !8

2

x 510 ± !100 2 92

2

x 52s210d ± !s210d2 2 4s1ds23d

2s1d 25.

x 5234

±!15

4i

x 523 ± i!15

4

x 523 ± !215

4

x 523 ± !9 2 24

4

x 523 ± !32 2 4s2ds3d

2s2d


29.

x 522 ± !10

2

x 52s22 ± !10d

4

x 524 ± 2!10

4

x 524 ± !40

4

x 524 ± !16 1 24

4

x 524 ± !42 2 4s2ds23d

2s2d

31.

z 521 ± !5

3

z 56s21 ± !5d

18

z 526 ± 6!5

18

z 526 ± !180

18

z 526 ± !36 1 144

18

z 526 ± !62 2 4s9ds24d

2s9d 33.

x 523 ± !21

4

x 56 ± 2!21

28

x 56 ± !84

28

x 56 ± !36 1 48

28

x 52s26d ± !s26d2 2 4s24ds3d

2s24d

35.

x 53 ± !27

85

38

±!78

i

x 53 ± !9 2 16

8

x 52s23d ± !s23d2 2 4s4ds1d

2s4d

4x2 2 3x 1 1 5 0 37.

x 55 ± !73

4

x 55 ± !25 1 48

4

x 52s25d ± !s25d2 2 4s2ds26d

2s2d

2x2 2 5x 2 6 5 0

39.

or 3 ± !13

6x 5

12

±!13

6

x 59

18±

3!1318

x 59 ± !117

18

x 59 ± !81 1 36

18

x 52s29d ± !s29d2 2 4s9ds21d

2s9d

9x2 2 9x 2 1 5 0

9x2 5 1 1 9x 41.

x 523 ± !57

6

x 523 ± !9 1 48

6

x 523 ± !32 2 4s3ds24d

2s3d

3x2 1 3x 2 4 5 0

3x 2 2x2 2 4 1 5x2 5 0

27.

v 5 1, 213

v 566

, 226

v 52 ± 4

6

v 52 ± !16

6

v 52 ± !4 1 12

6

v 52s22d ± !s22d2 2 4s3ds21d

2s3d


43.

or 1 ± !5

5x 5 0.2 ± !0.2

x 50.4 ± 2!0.2

2

x 50.4 ± !0.80

2

x 50.4 ± !0.16 1 0.64

2

x 52s20.4d ± !s20.4d2 2 4s1ds20.16d

2s1d 45.

x 521 ± !10

5

x 521 ± !1 1 9

5

x 521 ± !12 2 4s2.5ds20.9d

2s2.5d

47.

2 distinct imaginary solutions

5 23

5 1 2 4

b2 2 4ac 5 12 2 4s1ds1d 49.

2 distinct irrational solutions

5 57

5 25 1 32

b2 2 4ac 5 s25d2 2 4s2ds24d

51.


5 211

5 49 2 60

b2 2 4ac 5 72 2 4s5ds3d 53.

1 rational repeated solution

5 0

5 144 2 144

b2 2 4ac 5 s212d2 2 4s4ds9d

55.


5 223

5 1 2 24

b2 2 4ac 5 s21d2 2 4s3ds2d 57.

z 5 ± 13

z2 5 169

z2 2 169 5 0

59.

y 5 23 y 5 0

y 1 3 5 0 5y 5 0

5ysy 1 3d 5 0

5y2 1 15y 5 0 61.

x 5215

, 95

x 5155

±65

x 5 3 ±65

x 2 3 5 ±!3625

sx 2 3d2 53625

25sx 2 3d2 2 36 5 0

63.

y 5 232

2y 5 23 y 5 18

2y 1 3 5 0 y 2 18 5 0

sy 2 18ds2y 1 3d 5 0

2ysy 2 18d 1 3sy 2 18d 5 0 65.

x 5 24 ± 3i

x 1 4 5 ±!29

sx 1 4d2 5 29

x2 1 8x 1 16 5 225 1 16

x2 1 8x 1 25 5 0


67.

x 5 16, 8

x 5 12 ± 4

x 2 12 5 ±!16

sx 2 12d2 5 16

x2 2 24x 1 144 5 2128 1 144

x2 2 24x 1 128 5 0 69.

x 5136

±13!11

6i

x 513 ± !21859

6

x 513 ± !169 2 2028

6

x 52s213d ± !s213d2 2 4s3ds169d

2s3d

71.

x 525 ± 5!17

12

x 5215 ± 15!17

36

x 5215 ± !3825

36

x 5215 ± !225 1 3600

36

x 5215 ± !152 2 4s18ds250d

2s18d 73.

x 552 x 5 2

116

2x 5 5 6x 5 211

2x 2 5 5 0 6x 1 11 5 0

s6x 1 11ds2x 2 5d 5 0

12x2 2 8x 2 55 5 0

1.2x2 2 0.8x 2 5.5 5 0

77. Keystrokes:

4 20 25

−2

−30

7

5

x 5208

552

5 2.50

x 520 ± !400 2 400

8

x 52s220d ± !s220d2 2 4s4ds25d

2s4d

5 4x2 2 20x 1 25

0 5 2s4x2 2 20x 1 25d

75. Keystrokes:

3 6 1

−5

−3

7

5

x < 1.82, 0.18

x 56 ± !24

6

x 56 ± !36 2 12

6

x 52s26d ± !s26d2 2 4s3ds1d

2s3d

0 5 3x2 2 6x 1 1

2x 2X,T,u X,T,u5Y GRAPH1 x x



79. Keystrokes:

5 18 6

−1

−12

5

5

x < 3.23, 0.37

x 59 ± !51

5

x 52s9 ± !51d

10

x 518 ± 2!51

10

x 518 ± !204

10

x 518 ± !324 2 120

10

x 52s218d ± !s218d2 2 4s5ds6d

2s5d


81. Keystrokes:

.04 4 .8

−20

120

120

−10

x < 0.20, 99.80

x 521 ± 4!0.062

20.02

x 54s21 ± 4!0.062d

20.08

x 524 ± 16!0.062

20.08

x 524 ± !15.872

20.08

x 524 ± !16 2 0.128

20.08

x 524 ± !42 2 4s20.04ds20.8d

2s20.04d


83. Keystrokes:

2 5 5

No real solutions

−1

8

7

−4

5 215

5 25 2 40

b2 2 4ac 5 s25d2 2 4s2ds5d

85. Keystrokes:

6 40

Two real solutions

10

−50

6−12

5 196

5 36 1 160

b2 2 4ac 5 62 2 4s1ds240d

2x 2X,T,u X,T,u5Y GRAPH1 2x 2X,T,u X,T,u5Y GRAPH1


91. (a)

c < 9

24c > 236

36 2 4c > 0

s26d2 2 4s1dc > 0

b2 2 4ac > 0 (b)

c 5 9

24c 5 236

36 2 4c 5 0

s26d2 2 4s1dc 5 0

b2 2 4ac 5 0 (c)

c > 9

24c < 236

36 2 4c < 0

s26d2 2 4s1dc < 0

b2 2 4ac < 0

93. (a)

c < 16

24c > 264

64 2 4c > 0

82 2 4s1dc > 0

b2 2 4ac > 0 (b)

c 5 16

24c 5 264

64 2 4c 5 0

82 2 4s1dc 5 0

b2 2 4ac 5 0 (c)

c > 16

24c < 264

64 2 4c < 0

82 2 4s1dc < 0

b2 2 4ac < 0

95. Verbal model:

Labels:

Equation:

x 1 6.3 < 11.4 inches

x < 5.1 inches

x 526.3 ± !272.25

2

x 526.3 ± !39.69 1 232.56

2

x 526.3 ± !6.32 2 4s1ds258.14d

2s1d

0 5 x2 1 6.3x 2 58.14

58.14 5 x2 1 6.3x

58.14 5 sx 1 6.3d ? x

Width 5 x

Length 5 x 1 6.3

Width?Length5Area

89.

does not check.x 53 2 !17

2

x 53 1 !17

2

x 53 ± !17

2

x 53 ± !9 1 8

2

x 52s23d ± !s23d2 2 4s1ds22d

2s1d

0 5 x2 2 3x 2 2

x 1 3 5 x2 2 2x 1 1

s!x 1 3d2 5 sx 2 1d2

!x 1 3 5 x 2 187.

x 55 ± !185

8

x 55 ± !25 1 160

8

x 52s25d ± !s25d2 2 4s4ds210d

2s4d

4x2 2 5x 2 10 5 0

4x2 2 5x 5 10

1012x2

52

x22 5 s1d10

2x2

52

x2

5 1


97. (a)

(b)

reject

t < 3.4 seconds

t 55 1 5!3

4,

5 2 5!34

t 54s5 ± 5!3 d

16

t 520 ± 20!3

16

t 520 ± !1200

16

t 520 ± !400 1 800

16

t 52s220d ± !s220d2 2 4s8ds225d

2s8d

0 5 22s8t2 2 20t 2 25d

0 5 216t2 1 40t 1 50

t 552

5 2.5 seconds

2t 2 5 5 00 5 t

0 5 ts2t 2 5d

0 5 2t2 2 5t

0 5 28s2t2 2 5td

0 5 216t2 1 40t

50 5 216t2 1 40t 1 50 99. (a) Keystrokes:

831.3 85.71 3.452

(b)

year 1991

(c)

00

6

1000

y 5 400,500

y 5 831.3 2 85.71s7d 1 3.452s7d2

t < .9879

t 585.71 ± !6223.6137

6.9045

85.71 ± 78.896.904

t 52s285.71d ± !s285.71d2 2 4s3.452ds81.3d

2s3.452d

0 5 3.452t2 2 85.71t 1 81.3

750 5 831.3 2 85.71t 1 3.452t2

101. (a) 1

(b)

(c) 0

(d) 10 34

x 5 5 ± 3i

x 2 5 5 ±!29

sx 2 5d2 5 29

x2 2 10x 1 25 5 234 1 25

5 1 3i, 5 2 3ix2 2 10x 1 34 5 0

x 5 232x 5

32

s2x 2 3ds2x 1 3d

294

32, 23

24x2 2 9 5 0

x 5 23x 512

s2x 2 1dsx 1 3d 5 0

2322

52

12, 232x2 1 5x 2 3 5 0

x 5 22x 5 3

sx 2 3dsx 1 2d 5 0

263, 22x2 2 x 2 6 5 0

x1 ? x2x1 1 x2x1, x2

2 x 2X,T,u X,T,u5Y GRAPH1



1.

x 5 26 x 5 6

x 1 6 5 0x 2 6 5 0

2sx 2 6dsx 1 6d 5 0

2sx2 2 36d 5 0

2x2 2 72 5 0 2.

x 5 24 x 552

x 1 4 5 02x 2 5 5 0

s2x 2 5dsx 1 4d 5 0

2x2 1 3x 2 20 5 0 3.

t 5 ±2!3

t 5 ±!12

t2 5 12

4.

u 5 3 ± 4 5 7, 21

u 2 3 5 ±4

su 2 3d2 5 16

su 2 3d2 2 16 5 0 5.

s 5 25 ± 2!6

s 1 5 5 ±!24

ss 1 5d2 5 24

s2 1 10s 1 25 5 21 1 25

s2 1 10s 5 21

s2 1 10s 1 1 5 0 6.

y 5 232

±!19

2

y 132

5 ±!19

2

1y 1322

2

5194

1y 1322

2

5104

194

y2 1 3y 194

552

194

y2 1 3y 552

2y2 1 6y 2 5 5 0

103. (c) feet/sec

feet

(Divide by )

seconds

Quadratic formula method was used because the numbers were large and equation would not factor.

(d)

t < 3.5 secondst < 3.7 seconds

t 58 ± !400

8t 5

8 ± !4648

t 58 ± !64 1 336

8t 5

8 ± !64 1 4008

t 52s28d ± !s28d2 2 4s4ds221d

2s4dt 52s28d ± !s28d2 2 4s4ds225d

2s4d

0 5 4t2 2 8t 2 210 5 4t2 2 8t 2 25

0 5 216t2 1 32t 1 840 5 216t2 1 32t 1 100

t 516 ± !1056

165

16 ± 4!6616

54 ± !66

4< 3.0

t 516 ± !256 1 800

16

t 52s216d ± !s216d2 2 4s8ds225d

2s8d

22 0 5 8t2 2 16t 2 25

0 5 216t2 1 32t 1 50

h 5 50 50 5 216t2 1 32t 1 100

v0 5 32 h 5 216t2 1 v0t 1 h0

105. If the discriminant is positive, the quadraticequation has two real solutions; if it is zero, the equationhas one (repeated) real solution; and if it is negative, theequation has no real solutions.

b2 2 4ac. 107. The four methods are factoring, extracting square roots,completing the square, and the Quadratic Formula.


13.

b 532

b 532

2b 2 3 5 02b 2 3 5 0

s2b 2 3ds2b 2 3d 5 0 14.

m 525 ± !10

3

m 5210 ± 2!10

6

m 5210 ± !40

6

m 5210 ± !100 2 60

6

m 5210 ± !102 2 4s3ds5d

2s3d

15.

Check:

0 5 0

36 2 12 2 24 5?

0

36 2 2!36 2 24 5?

0

x 5 16 x 5 36

x 5 s24d2 x 5 62

!x 5 24 !x 5 6

u 5 24 u 5 6

su 2 6dsu 1 4d 5 0

u2 2 2u 2 24 5 0

s!xd22 2!x 2 24 5 0

let u 5 !x

x 2 2!x 2 24 5 0 16.

x 5 ±!3i x 5 ±2i

x 5 ±!23 x 5 ±!24

x2 5 23 x2 5 24

sx2 1 4dsx2 1 3d 5 0

x4 1 7x2 1 12 5 0

Not a solution

216 Þ 0

16 2 8 2 24 5?

0

16 2 2!16 2 24 5?

0

10.

10, 22 5 t

4 ± 6 5 t

±6 5 t 2 4

36 5 st 2 4d2 11.

x 5 23 x 5 10

x 1 3 5 0sx 2 10d 5 0

sx 2 10dsx 1 3d 5 0 12.

x 5 22 x 5 5

x 1 2 5 0x 2 5 5 0

sx 2 5dsx 1 2d 5 0

x2 2 3x 2 10 5 0

7.

x 524 ± 2!10

25 22 ± !10

x 524 ± !40

2

x 524 ± !16 1 24

2

x 524 ± !42 2 4s1ds26d

2s1d 8.

v 53 ± !105

12

v 53 ± !9 1 96

12

v 52s23d ± !s23d2 2 4s6ds24d

2s6d 9.

x 525 ± i!3

25 2

52

±!32

i

x 525 ± !23

2

x 525 ± !25 2 28

2

x 525 ± !52 2 4s1ds7d

2s1d


20. Verbal model:

Equation:

35 meters 65 meters3

x 5 65 meters x 5 35 meters

x 2 65 5 0x 2 35 5 0

0 5 sx 2 35dsx 2 65d

0 5 x2 2 100x 1 2275

2275 5 100x 2 x2

2275 5 x ? s100 2 xd

Width?Length5Area

18. Keystrokes:

.045 4

−6

−5

6

3

x < 1.79 and 22.24

x 520.45 ± !16.2025

2

x 520.45 ± !0.2025 1 16

2

x 520.45 ± !s0.45d2 2 4s1ds24d

2s1d

0 5 x2 1 0.45x 2 4

19.

x 5100 ± !0

25 50 units

x 5100 ± !10,000 2 10,000

2

x 52s2100d ± !s2100d2 2 4s1ds2500d

2s1d

x2 2 100x 1 2500 5 0

0.2x2 2 20x 1 500 5 0

5 20x 2 0.2x2

500 5 xs20 2 0.2xd

17. Keystrokes:

.5 3 1

−6

−6

12

6

x < 6.32 and 20.32

x 5 3 ± !11

x 56 ± 2!11

2

x 56 ± !44

2

x 56 ± !36 1 8

2

x 52s26d ± !s26d2 2 4s1ds22d

2s1d

0 5 x2 2 6x 2 2

0 5 .5x2 2 3x 2 1

2 2x 2X,T,u X,T,u5Y GRAPH GRAPH2x 2X,T,u X,T,u5Y 1


Section 6.4 Applications of Quadratic Equations

1. Verbal model:

Equation:

Labels: Number eggs soldNumber eggs purchased

dozen

Selling price per dozen521.60

185 $1.20

x 5 18x 5 224

0 5 sx 1 24dsx 2 18d

0 5 x2 1 6x 2 432

0 5 3x2 1 18x 2 1296

0 5 0.3x2 1 1.8x 2 129.6

21.6x 1 129.6 5 21.6x 1 0.3x2 1 1.8x

21.60sx 1 6d 5 21.60x 1 0.30xsx 1 6d

5 x 1 65 x

21.60

x5

21.60x 1 6

1 0.30

Profit perdoz eggs

1Cost perdoz eggs

5Selling priceper doz eggs

3. Verbal model:

Labels: Number videos soldNumber videos purchased

Equation:

videos

Selling price548016

5 $30

x 5 16x 5 224

0 5 sx 1 24dsx 2 16d

0 5 x2 1 8x 2 384

0 5 10x2 1 80x 2 3840

480x 1 3840 5 480x 1 10x2 1 80x

480sx 1 8d 5 480x 1 10xsx 1 8d

480

x5

480x 1 8

1 10

5 x 1 85 x

Profit pervideo

1Cost per

video5

Selling priceper video

5. Verbal model:2

Labels: Length

Width

Equation:

inches

inches

Verbal model:

Equation:

108 square inches 5 A

12 ? 9 5 A

Area5Width?Length

w 5 0.75 l 5 9

l 5 12

3.5l 5 42

2l 1 1.5l 5 42

2l 1 2s0.75ld 5 42

5 0.75 l

5 l

Perimeter5Width1 2Length 7. Verbal model:

Labels: LengthWidth

Equation:

Verbal model:2

Equation:

70 feet 5 P

2s25d 1 2s10d 5 P

Perimeter5Width1 2Length

25 5 2.5w

10 5 w

100 5 w2

250 5 2.5w2

250 5 2.5w ? w

5 w5 2.5w

Width?Length5Area

Section 6.4 Applications of Quadratic Equations333

9. Verbal model:

Labels: Length

Width

Equation:

inches

inches

Verbal model:2

Equation:

64 inches 5 P

48 1 16 5 P

2s24d 1 2s8d 5 P


w 513 l 5 8

l 5 24

l2 5 576

13 l2 5 192

l ? 13l 5 192

513 l

5 l

Area5Width?Length 11. Verbal model:2

Labels: Length

Width

Equation:

km

km

Verbal model:

Equation: 15 ? 12 5 180 square kilometers 5 A

Area5Width?Length

l 5 w 1 3 5 15

w 5 12

4w 5 48

2w 1 6 1 2w 5 54

2sw 1 3d 1 2w 5 54

5 w

5 w 1 3


13. Verbal model:

Labels: Length

Width

Equation:

meters

Verbal model:2

Equation: 2s120d 1 2s100d 5 440 meters 5 P


w 5 l 2 20 5 100 meters

l 5 10 1 110 5 120

l 2 10 5 ±!12,100

sl 2 10d2 5 12,100

l2 2 20l 1 100 5 12,000 1 100

l2 2 20l 5 12,000

l ? sl 2 20d 5 12,000

5 l 2 20

5 l

Area5Width?Length 15. Verbal model:

Labels: Length

Width

Equation:

inches

inchesx 1 4 5 16

x 5 12x 5 216

0 5 sx 1 16dsx 2 12d

0 5 x2 1 4x 2 192

192 5 x2 1 4x

192 5 sx 1 4dx

5 x

5 x 1 4

Width?Length5Area

19. Verbal model:

Labels: LengthWidth

Equation:

or 50 ft 3 250 ft.100 ft 3 125 ft.

350 2 2x 5 100, 250

x 5 125, 50

x 5175 ± !5625

25

175 ± 752

x 5175 ± !1752 2 4s1ds6, 250d

2s1d

x2 2 175x 1 6,250 5 0

2x2 2 350x 1 12,500 5 0

350x 2 2x2 5 12,500

s350 2 2xd ? x 5 12,500

5 x5 350 2 2x

Area5Width?Length17. Verbal model:

Labels: Height

Base

Equation:

reject

x 2 8 5 16 inches

x 5 216x 5 24 inches

0 5 sx 2 24dsx 1 16d

0 5 x2 2 8x 2 384

384 5 x2 2 8x

192 512

sx 2 8dx

5 x

5 x 2 8

Base?Height512

?Area


21. Verbal model:

Equation:

Verbal model:

Labels: Height

Base 1

Base 2

Equation:

This has no real solution, so it would be impossible to have an area of 43,560 square feet.

0 5 x2 2 550x 1 87,120

2x2 1 550x 5 87,120

212 x2 1 275x 5 43,560

12 xs2x 1 550d 5 43,560

12 xsx 1 550 2 2xd 5 43,560

12 xsx 1 bd 5 43,560

5 6

5 x

5 x

Area5)Base 21Base 1(Height?

b 5 550 2 2x

2x 1 b 5 550

x 1 x 1 b 5 550

5 550Side 31Side 21Side 1

12

23. Verbal model:

Labels: Height

Width

Equation:

5 48 2 24 5 24 inches

width 5 48 2 2s12d

height 5 12 inches

x 5 12

sx 2 12dsx 2 12d 5 0

x2 2 24x 1 144 5 0

2x2 2 48x 1 288 5 0

x ? s48 2 2xd 5 288

5 48 2 2x

5 x

Area5Width?Height

25.

0.08 5 r or 8%

1.08 5 1 1 r

1.1664 5 s1 1 rd2

3499.20 5 3000s1 1 rd2

A 5 Ps1 1 rd2 27.

6% 5 r

.06 5 r

1.06 5 1 1 r

1.1236 5 s1 1 rd2

280.90250.00

5 s1 1 rd2

280.90 5 250.00s1 1 rd2

A 5 Ps1 1 rd2 29.

.0259 < r or 2.59%

1.0259 < 1 1 r

1.052525 5 s1 1 rd2

8420.20 5 8000.00s1 1 rd2

A 5 Ps1 1 rd2


33. Verbal model:

Labels: Number in current group

Number in new group

Equation:

investors x 5 5 x 5 28

x 2 5 5 0x 1 8 5 0

0 5 sx 1 8dsx 2 5d

0 5 x2 1 3x 2 40

0 5 6000x2 1 18,000x 2 240,000

80,000x 1 240,000 2 80,000x 5 6000x2 1 18,000x

80,000sx 1 3d 2 80,000x 5 6000sx2 1 3xd

xsx 1 3d180,000x

280,000x 1 3 2 5 s6000dxsx 1 3d

80,000

x2

80,000x 1 3

5 6000

5 x 1 3

5 x

60005Investment per

person; new group2

Investment per person; current group

31. Verbal model:

Labels: Number of members

Number going to game

Equation:

x 1 8 5 48

x 5 248 x 5 40

sx 1 48dsx 2 40d 5 0

x2 1 8x 2 1920 5 0

2x2 2 8x 1 1920 5 0

240x 1 1920 2 x2 2 8x 5 240x

s240 2 xdsx 1 8d 5 240x

1240 2 xx 2 sx 1 8d 5 240

1240x

2 12 ? sx 1 8d 5 240

5 x 1 8

5 x

$2405Number ofmembers?

Cost permember

35. Common Formula:

Equation:

reject 2.1443454

miles < 15.86

x 5 15.855655,

x 518 ± !324 2 136

25

18 ± !1882

x 518 ± !182 2 4s1ds34d

2s1d

x2 2 18x 1 34 5 0

2x2 2 36x 1 68 5 0

x2 1 324 2 36x 1 x2 5 256

x2 1 s18 2 xd2 5 162

a2 1 b2 5 c2


37. (a)

Keystrokes:

3 4

Approximate value of when

(b)

x 527 ± !199

2< 3.55 meters

x 514 ± 2!199

4

x 5214 ± !796

4

x 5214 ± !196 1 600

4

x 5214 ± !142 2 4s2ds275d

2s2d

0 5 2x2 1 14x 2 75

5 9 1 6x 1 x2 1 16 1 8x 1 x2

100 5 s3 1 xd2 1 s4 1 xd2

00

30

60 10 5 !3 1 xd2 1 s4 1 xd2

d 5 10.x < 3.55

d 5 !s3 1 xd2 1 s4 1 xd2

x xx x x xx 2 x 2X,T,u X,T,u5Y GRAPH1 1 1!

39. Verbal model:

Labels: Time to do job by Person 1

Time to do job by Person 2

Equation:

reject

x 1 2 < 11.1 hours

21.1 x < 9.1 hours,

x 58 ± !104

2

x 58 ± !64 1 40

2

x 58 ± !s28d2 2 4s1ds210d

2s1d

x2 2 8x 2 10 5 0

2x2 1 8x 1 10 5 0

5x 1 10 1 5x 5 x2 1 2x

5sx 1 2d 1 5x 5 xsx 1 2d

xsx 1 2d3s5d11x

11

x 1 22 5 14xsx 1 2d

1x

s5d 11

x 1 2s5d 5 1

5 x 1 2

5 x

One complete job5Work done by

Person 21

Work done byPerson 1


41. Verbal model:

Labels: Time Company A

Time Company B

Equation:

x 1 3 < 9.8

x 5 21.8 x < 6.8 days

x 55 ± !73

2

x 55 ± !25 1 48

2

x 52s25d ± !s25d2 2 4s1ds212d

2s1d

0 5 x2 2 5x 2 12

4x 1 4x 1 12 5 x2 1 3x

4x 1 4sx 1 3d 5 xsx 1 3d

4xsx 1 3d1 1x 1 3

11x2 5 11

424xsx 1 3d

1

x 1 31

1x

514

5 x

5 x 1 3

Rate together5Rate Company1Rate Company

43.

t 5 3 seconds

t2 5 9

16t2 5 144

0 5 144 2 16t2

h 5 h0 2 16t2 45.

t 5 9.532838 seconds < 9.5 seconds

t2 5 90.875

16t2 5 1454

0 5 1454 2 16t2

h 5 h0 2 16t2

49. (a)

at 3 seconds and at 7 seconds

(b)

after 10 seconds.

t 5 0, 10

0 5 216tst 2 10d

0 5 216t2 1 160t

0 5 st 2 7dst 2 3d

0 5 t2 2 10t 1 21

0 5 216t2 1 160t 2 336

336 5 216t2 1 160t47.

reject

< 4.7 seconds

20.0396644 t 5 4.7271644,

t 575 ± 76.26926

32

t 575 ± !5817

32

t 575 ± !5625 1 192

32

t 575 ± !s275d2 2 4s16ds23d

2s16d

0 5 16t2 2 75t 2 3

0 5 3 1 75t 2 16t2

h 5 3 1 75t 2 16t2


51. Verbal model:

Labels: First integer

Second integer

Equation:

n 1 1 5 2 2 15 n 1 1 5 16

n 5 216 n 5 15

n 521 ± 31

2

n 5 212

±!961

2

n 112

5 ±!9614

1n 1122

2

5960 1 1

4

n2 1 n 114

5 240 114

n ? sn 1 1d 5 240

5 n 1 1

5 n

Product5Integer?Integer

reject6

53. Verbal model:

Labels: First even integer

Second even integer

Equation:

2n 1 2 5 16 2n 1 2 5 214

2n 5 14 2n 5 216

n 5 7 n 5 28

n 2 7 5 0 n 1 8 5 0

sn 1 8dsn 2 7d 5 0

n2 1 n 2 56 5 0

n2 1 n 5 56

4n2 1 4n 5 224

2n ? s2n 1 2d 5 224

5 2n 1 2

5 2n

Product5Even

integer?Even

integer

55. Verbal model:

Labels: First odd integer

Second odd integer

Equation:

2n 1 3 5 23

2n 1 1 5 21

n 5 10 n 5 212

n 2 10 5 0n 1 12 5 0

sn 1 12dsn 2 10d 5 0

n2 1 2n 2 120 5 0

4n2 1 8n 2 480 5 0

4n2 1 8n 1 3 5 483

s2n 1 1d ? s2n 1 3d 5 483

5 2n 1 3

5 2n 1 1

Product5Odd

integer?Odd

integer

57. Verbal model:

Labels: SpeedIncreased speed

Equation:

miles per hour x 1 40 5 400

x 5 360, 2400

x 5240 ± 760

2

x 540 ± !1600 1 576,000

2

x 5240 ± !402 2 4s1ds2144,000d

2s1d

0 5 x2 1 40x 2 144,000

3600x 1 144,000 5 3600x 1 x2 1 40x

720s5dsx 1 40d 5 720s5xd 1 xsx 1 40d

720

x5

720x 1 40

115

5 x 1 405 x

1New time5Original time15

reject5


59. Verbal model:

Label: Time

Equation:

or

v 51101.67

< 65 miyhr

v 51102.39

< 46 miyhr

x < 2.39, 1.67

x 5122.34 ± !477.0756

60

x 52s2122.34d ± !s2122.34d2 2 4s30ds121d

2s30d

0 5 30x2 2 122.34x 1 121

122.34x 5 30x2 1 121

20.39 5 5x 11216x

20.39 5 5x 1 x531110x 2

2

600 45 x

Fuel Cost1Wage Cost5Total Cost

61. (a)

(b)

(c)

a < 12.1, 7.9

a 520p ± !177.9305761

2p

a 52s220pd ± !s220pd2 2 4spds300d

2spd

0 5 pa2 2 20pa 1 300

0 5 20pa 2 pa2 2 300

300 5 pas20 2 ad

< 201.1 < 285.9

5 64p 5 91p

5 ps16ds4d 5 ps13ds7d

A 5 ps16ds20 2 16dA 5 ps13ds20 2 13d

< 314.2 < 285.9 < 201.1

5 100p 5 91p 5 64p

5 ps10ds10d 5 ps7ds13d 5 ps4ds16d

A 5 ps10ds20 2 10dA 5 ps7ds20 2 7dA 5 ps4ds20 2 4d

A 5 pas20 2 ad b 5 20 2 a

A 5 paba 1 b 5 20

a 4 7 10 13 16

A 201.1 285.9 314.2 285.9 201.1

(d)

Keystrokes:

20

00

20

400

A 5 pas20 2 ad

x x

2X,T,u X,T,u5Y GRAPHp


63. Guidelines for solving word problems:

(a) Write a verbal model that will describe what you need to know.

(b) Assign labels to each part of the verbal model—numbers to the known quantities and letters to the variable quantities.

(c) Use the labels to write an algebraic model based on the verbal model.

(d) Solve the resulting algebraic equation and check your solution.

65. Unit Analysis

9 dollarshour

? s20 hoursd 5 180 dollars

67. An example of a quadratic equation that has only one repeated solution is Any equation of the formwhere c is a constant will have only one repeated solution.sx 2 cd2 5 0,

sx 1 4d2 5 0.

Section 6.5 Quadratic and Rational Inequalities

1.

Critical numbers5 0, 52

x 552

2x 2 5 5 0x 5 0

xs2x 2 5d 5 0 3.

Critical numbers:92, 292

x 5 ±92

x2 5814

4x2 2 81 5 0 5.

Critical numbers: 5, 23

x 5 23x 5 5

sx 2 5dsx 1 3d 5 0

xsx 1 3d 2 5sx 1 3d 5 0

7.

Critical numbers5 3, 1

x 5 1x 5 3

sx 2 3dsx 2 1d 5 0

x2 2 4x 1 3 5 0 9.

Critical number:52

x 552

2x 2 5 5 0

s2x 2 5d2 5 0

4x2 2 20x 1 25 5 0

11. Negative:

Positive:

Choose a test value from each interval.

s4, `d ⇒ x 5 5 ⇒ 5 2 4 5 1 > 0

s2`, 4d ⇒ x 5 0 ⇒ 0 2 4 5 24 < 0

4

x

+−

s4, `d

s2`, 4d 13. Negative:

Positive:


s6, `d ⇒ x 5 8 ⇒ 3 212s8d 5 21 < 0

s2`, 6d ⇒ x 5 0 ⇒ 3 212s0d 5 3 > 0

+ −

6

x

s2`, 6d

s6, `d

15. Positive:

Negative:

Positive:


s4, `d ⇒ x 5 5 ⇒ 2s5ds5 2 4d 5 10 > 0

s0, 4d ⇒ x 5 1 ⇒ 2s1ds1 2 4d 5 26 < 0

s2`, 0d ⇒ x 5 21 ⇒ 2s21ds21 2 4d 5 10 > 0

++

0

x

4

−

s4, `d

s0, 4d

s2`, 0d

Section 6.5 Quadratic and Rational Inequalities 341

21.

Critical number:

Test intervals:

Negative:

Positive:

Solution:

0

x

124 3

f23, `d

f23, `d

s2`, 23g

x 5 23

2sx 1 3d ≥ 023.

Critical number:

Test intervals:

Negative:

Positive:

Solution:

10

x

7 8 9

s8, `d

s2`, 8d

s8, `d

x 5 8

234

x 1 6 < 0 25.

Critical number:

Test intervals:

Positive:

Negative:

Positive:

Solution:

3

x

2101

s0, 2d

s2, `d

s0, 2d

s2`, 0d

x 5 0, 2

3xsx 2 2d < 0

27.

Critical numbers:

Test intervals:

Negative:

Positive:

Negative:

Solution:

−1 0 1 2 3

x

f0, 2g

f2, `d

f0, 2g

s2`, 0g

x 5 0, 2

3xs2 2 xd ≥ 0 29.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

−4 −2 0 2 4

x

s2`, 22d < s2, `d

s2, `d

s22, 2d

s2`, 2d

x 5 2, 22

sx 2 2dsx 1 2d > 0

x2 2 4 > 0

x2 > 4 31.

Critical number:

Test intervals:

Positive:

Negative:

Positive:

Solution:

4

x

22 046

5

f25, 2g

f2, `d

f25, 2g

s2`, 25g

x 5 25, 2

sx 1 5dsx 2 2d ≤ 0

x2 1 3x 2 10 ≤ 0

33.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

−1 0 1 2

u

−4 −3 −2

s2`, 23d < s1, `ds1, `ds23, 1d

s2`, 23d

u 5 23, 1

su 1 3dsu 2 1d > 0

u2 1 2u 2 3 > 0

u2 1 2u 2 2 > 135.

No critical numbers

is not less than zerofor any value of x.

Solution: none

x2 1 4x 1 5

x 524 ± !16 2 20

2

x2 1 4x 1 5 < 037.

for all real numbers

Solution:

0 1 2 3

x

−3 −2 −1

s2`, `d

sx 1 1d2 ≥ 0

sx 1 1d2 ≥ 0

17.

Negative:

Positive:


s2, `d ⇒ x 5 3 ⇒ s2 2 3ds2 1 3d 5 25 < 0

s22, 2d ⇒ x 5 0 ⇒ s2 2 0ds2 1 0d 5 4 > 0

s2`, 22d ⇒ x 5 23 ⇒ s223ds2 1 23d 5 25 < 0

+ −−

−2

x

2

s22, 2d

s2`, 22d < s2, `d

4 2 x2 5 s2 2 xds2 1 xd 19.

Positive:

Negative:

Positive:


s5, `d ⇒ x 5 6 ⇒ s6 2 5ds6 1 1d 5 7 > 0

s21, 5d ⇒ x 5 0 ⇒ s0 2 5ds0 1 1d 5 25 < 0

s2`, 21d ⇒ x 5 22 ⇒ s22 2 5ds22 1 1d 5 7 > 0

+−+

−1

x

5

s5, `d

s21, 5d

s2`, 21d

sx 2 5dsx 1 1d


39.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

5

x

2

431 2

2 2

1 0

2

s2`, 2 2 !2d < s2 1 !2, `ds2 1 !2, `ds2 2 !2, 2 1 !2 d

s2`, 2 2 !2 d

2 2 !2x 5 2 1 !2,

5 2 ± !2

54 ± !8

25

4 ± 2!22

x 54 ± !16 2 8

2

x2 2 4x 1 2 > 041.

for all real numbers

0 1 2 3

x

−3 −2 −1

sx 2 3d2 ≥ 0

sx 2 3d2 ≥ 0

x2 2 6x 1 9 ≥ 0

43.

Critical number:

Test intervals:

Positive:

Positive:

Solution: none

s5, `d

s2`, 5d

u 5 5

su 2 5dsu 2 5d < 0

u2 2 10u 1 25 < 0 45.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

x

43

−3 −2 −1 0 1 2

f22, 43gf43, `df22, 43g

s2`, 22g

x 543, 22

s3x 2 4dsx 1 2d ≤ 0

3x2 1 2x 2 8 ≤ 0

47.

Multiply by

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

3

u

0 1 2

2 53 2

s23, 52d

s52, `ds2

3, 52ds2`, 23d

u 523, 52

s3u 2 2ds2u 2 5d < 0

21ds 6u2 2 19u 1 10 < 0

26u2 1 19u 2 10 > 0 49.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

6

u

1

2 40

2

4 2

s2`, 212d < s4, `d

s4, `d

s212, 4d

s2`, 212d

u 5 212, 4

s2u 1 1dsu 2 4d > 0

2u2 2 7u 2 4 > 0


55.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

10

x

6 84

6 5

0

5

2

6

s2`, 5 2 !6d < s5 1 !6, `ds5 1 !6, `ds5 2 !6, 5 1 !6d

s2`, 5 2 !6d

x 5 5 1 !6, 5 2 !6

5 5 ± !6

510 ± !24

25

10 ± 2!62

x 510 ± !100 2 76

2

x2 2 10x 1 19 > 0

6 2 x2 1 10x 2 25 < 0

6 2 sx2 2 10x 1 25d < 0 57.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

0

u

−10 −8 −6 −4 −2

−9 −1

s2`, 29g < f21, `d

f21, `d

s29, 21g

s2`, 29g

x 5 29, 21

su 1 9dsu 1 1d ≥ 0

u2 1 10u 1 9 ≥ 0

u2 1 10u 1 25 2 16 ≥ 0

su 1 5d2 ≥ 16

16 ≤ su 1 5d2

51.

Critical number:

Test intervals:

Positive:

Positive:

Solution:

x

72

−5 −4 −3

−

−1−2 0

272

s272, `d

s2`, 272d

x 5272

s2x 1 7ds2x 1 7d ≤ 0

4x2 1 28x 1 49 ≤ 0 53. for all real numbers except 5.

Solution: none

sx 2 5d2 > 0

59.

Critical numbers:

Test intervals:

Negative:

Positive:

Negative:

Positive:

Solution:

3

x

1 2013 2

s22, 0d < s2, `d

s2, `d

s0, 2d

s22, 0d

s2`, 22d

x 5 0, 2, 22

xsx 2 2dsx 1 2d > 0 61. Keystrokes:

6

−2

−10

8

10

s0, 6d

2X,T,u X,T,u5Y GRAPHx 2


63. Keystrokes:

0.5 1.25 3

−7

−5

5

5

s2`, 24d < s32, `d

65. Keystrokes:

4 4

9

−8

−2

4

14

s2`, 25g < f1, `d

y2

y1

67. Keystrokes:

9 0.2 2

4

s2`, 23d < s7, `d

y2

y1

69. Critical number:

1 2 3 4 5

x

x 5 3 71. Critical numbers:

−2 −1 0 1

x

−5 −4 −3

x 5 0, 25

73.

Critical number:

Test intervals:

Negative:

Positive:

Solution:

4

x

320 1

− +

s3, `d

s3, `d

s2`, 3d

x 5 3

5x 2 3

> 0

2

2 2

1X,T,u X,T,u5Y GRAPHx 2 1 1X,T,u X,T,u5Y

GRAPH

ENTERx 2

X,T,u5Y

GRAPH

x 2xx

75.

Critical number:

Test intervals:

Positive:

Negative:

Solution:

3 4 5 6

x

0 1 2

+ −

s2`, 3d

s3, `d

s2`, 3d

x 5 3

25x 2 3

> 0

77.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

x

−1 0 1 2 3 4

++ −

s0, 3d

s3, `d

s0, 3d

s2`, 0d

x 5 0, 3

xx 2 3

< 0 79.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

0 1 2 43

x

−3−4 −2 −1

++ −

f23, 4d

s4, `d

f23, 4d

s2`, 23g

x 5 23, 4

x 1 3x 2 4

≤ 0

−6

−3

10

12


81.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

y

0 2 4−8 −6 −4 −2 6

+ +−

s26, 4d

s4, `d

s26, 4d

s2`, 26d

y 5 4, 26

y 2 4y 1 6

< 083.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

8

y

11

62 4

3 2

0

+ +−

s2`, 3g < 1112

, `2111

2, `2

33, 112 2

s2`, 3g

y 5 3, 112

y 2 3y 2 11

≥ 0 85.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

0

x

−3 −2

−

−1

32

++ −

322, 2322

1232

, `2322, 2

322

s2`, 22g

x 5 22, 232

x 1 24x 1 6

≤ 0

87.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

u

−2 −1 0 1 2 3 4

++ −

s21, 3d

s3, `d

s21, 3d

s2`, 21d

u 5 3, 21

3su 2 3du 1 1

< 0 89.

22s27 1 xd

x 2 4> 0

14 2 2x

x 2 4> 0

6 2 2x 1 8

x 2 4> 0

6 2 2sx 2 4d

x 2 4> 0

6

x 2 42 2 > 0

6

x 2 4> 2 Critical numbers:

Test intervals:

Negative:

Positive:

Negative:

Solution:

8

x

64

7

0 2

− −+

s4, 7d

s7, `d

s4, 7d

s2`, 4d

x 5 7, 4

91.

5x 1 2x 1 2

< 0

4x 1 sx 1 2dx 1 2

< 0

4x

x 1 21 1 < 0

4x

x 1 2< 21 93.

2x 1 5x 2 3

≤ 0

x 2 1 2 2x 1 6

x 2 3≤ 0

x 2 1 2 2sx 2 3d

x 2 3≤ 0

x 2 1x 2 3

2 2 ≤ 0

x 2 1x 2 3

≤ 2Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

0

x

−3 −2

−

−1

25

+ +−

122, 2252

1225

, `2

122, 2252

s2`, 22d

x 5 225

, 22 Critical numbers:

Test intervals:

Negative:

Positive:

Negative:

Solution:

3 4 5 6

x

0 1 2

+− −

s2`, 3d < f5, `d

f5, `d

s3, 5g

s2`, 3d

x 5 5, 3


95. Keystrokes:

1

Solution:

4

6

−4

−6

s2`, 21d < s0, 1d

97. Keystrokes:

6 1 2

Solution:

4

7

−6

−8

s2`, 21d < s4, `d

99. Keystrokes:

6 3 5

2

Solution:

12

9

−6

−18

s25, 3.25d

y2

y1

101. Keystrokes:

1

3

Solution:

8

8

−8

−8

s0, 0.382d < s2.618, `d

y2

y1

X,T,u X,T,u5Y GRAPH4 2 x xx xX,T,u X,T,u5Y GRAPH1 14 2

x xx xX,T,u X,T,u5Y ENTER

GRAPH GRAPH

14 X,T,u X,T,u5Y ENTER1 42

103. Keystrokes:

3 2

(a) Solution

Look at x-axis and vertical asymptote

(b)

(Graph as and find the intersection.)

−5

−8

12

12

y2y 5 6

s2, 4g

x 5 2ds

f0, 2d

4

105. Keystrokes:

2 4

(a) Solution:

(Graph as and find the intersection.)

(b) Solution

(Notice graph stays below line )

−4

−1

3

4

y 5 2.

s2`, `d

y2y 5 1

s2`, 22g < f2, `d

X,T,u X,T,u5Y x 2 GRAPH X,T,u X,T,u5Y 14 x GRAPHx 2 x 2 x

107.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution: s3, 5d

s5, `d

s3, 5d

s2`, 3d

x 5 3, 5

st 2 3dst 2 5d < 0

t2 2 8t 1 15 < 0

216t2 1 128t 2 240 > 0

216t2 1 128t > 240

height > 240


113.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution: s12, 20ds20, `ds12, 20d

s2`, 12d

l 5 20, 12

sl 2 20dsl 2 12d < 0

l2 2 32l 1 240 < 0

2l2 1 32l 2 240 > 0

32l 2 l2 > 240

ls32 2 ld > 240

Area > 240

111. Verbal model:

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

units90,000 ≤ x ≤ 100,000

s90,000, 100,000ds100,000, `ds90,000, 100,000d

s0, 90,000d

90,000, 100,000

0 > s0.0002x 2 20dsx 2 90,000d

0 > 0.0002x2 2 38x 1 1,800,000

20.0002x2 1 38x 2 150,000 > 1,650,000

50x 2 0.0002x2 2 12x 2 150,000 > 1,650,000

xs50 2 0.0002xd 2 f12x 1 150,000g > 1,650,000

Profit > 1,650,0005Cost2Revenue

> 1,650,000Profit


244.20 13.23 1 .13

.005

(b) Let and find the intersection of the graphs.

Solution:

−10

18−1

600

f5.7, 13.7g, 5.7 ≤ t ≤ 13 ? 7

y2 5 400

109.

Critical numbers:

r cannot be negative.

Test intervals:

Negative:

Positive:

Solution:

s0.0724, `d, r > 7.24%

1240 1 !184040

, `21240 1 !1840

40, `2

10, 240 1 !1840

40 2

r 5240 1 !1840

40,

240 2 !184040

20r2 1 40r 2 3 > 0

1000r2 1 2000r 2 150 > 0

1000 1 2000r 1 1000r2 > 1150

1000s1 1 2r 1 r2d > 1150

1000s1 1 rd2 > 1150

X,T,u

X,T,u X,T,u

5Y

1

4x x

GRAPHx 2

x

x

2 2


117. The direction of the inequality is reversed, when both sides are multiplied by a negative real number.

119. A polynomial can change signs only at the x-values that make the polynomial zero. The zeros of the polynomial are called theciritical numbers, and they are used ro determine the test intervals in solving polynomial inequalities.

121. is one example of a quadratic inequality that has no real solution. Any inequality of the form c anypositive constant or c any positive constant will not have a real solution.2x2 2 c > 0,

x2 1 c < 0,x2 1 1 < 0


1.

x 5 212x 5 0

x 1 12 5 0x 5 0

xsx 1 12d 5 0

x2 1 12x 5 0 3.

y 5 212 y 5

12

2y 1 1 5 02y 2 1 5 0

s2y 2 1ds2y 1 1d 5 0

4y2 2 1 5 0 5.

y 5 252 y 5 2

52

2y 5 25 2y 5 25

2y 1 5 5 02y 1 5 5 0

s2y 1 5ds2y 1 5d 5 0

4y2 1 20y 1 25 5 0

7.

x 5 29 x 5 10

x 1 9 5 0 x 2 10 5 0

2sx 2 10dsx 1 9d 5 0

2sx2 2 x 2 90d 5 0

2x2 2 2x 2 180 5 0 9.

x 5 6x 5 232

s2x 1 3dsx 2 6d 5 0

2x2 2 9x 2 18 5 0

6x2 2 12x 5 4x2 2 3x 1 18 11.

x 5 ±50

x 5 ±!2500

x2 5 2500

4x2 5 10,000

13.

y 5 ±2!3

y 5 ±!12

y2 5 12

y2 2 12 5 0 15.

x 5 36, 24

x 5 16 ± 20

x 2 16 5 ±!400

sx 2 16d2 5 400 17.

z 5 ±11i

z 5 ±!2121

z2 5 2121

19.

y 5 ±5!2i

y 5 ±!250

y2 5 250

y2 1 50 5 0 21.

y 5 24 ± 3!2i

y 1 4 5 ±!218

sy 1 4d2 5 218

sy 1 4d2 1 18 5 0

23.

x 5 ± i x 5 ±!5

x 5 ±!21 x2 5 5

x2 5 21 x2 2 5 5 0

x2 1 1 5 0

sx2 2 5dsx2 1 1d 5 0

x4 2 4x2 2 5 5 0 25.

Check: Check:

0 5 0 0 5 0

1 2 4 1 3 5?

0 9 2 12 1 3 5?

0

1 2 4!1 1 3 5?

09 2 4!9 1 3 5?

0

x 5 1 x 5 9

s!xd25 12 s!xd2

5 32

!x 5 1 !x 5 3

s!x 2 1d 5 0s!x 2 3d 5 0

s!x 2 3ds!x 2 1d 5 0

x 2 4!x 1 3 5 0


27.

x 5 1 x 5 1 ± !6

sx 2 1d2 5 0 x 52 ± 2!6

2

x 52 ± !24

2

x 52 ± !4 1 20

2

x 52s22d ± !s22d2 2 4s1ds25d

2s1d

sx2 2 2x 2 5dsx2 2 2x 1 1d 5 0

fsx2 2 2xd 2 5gfsx2 2 2xd 1 1g 5 0

sx2 2 2xd2 2 4sx2 2 2xd 2 5 5 0 29.

x 5 64 x 5 2343

s 3!xd35 43 s 3!xd3

5 s27d3

3!x 5 4 3!x 5 27

x1y3 5 4 x1y3 5 27

x1y3 2 4 5 0x1y3 1 7 5 0

sx1y3 1 7dsx1y3 2 4d 5 0

x2y3 1 3x1y3 2 28 5 0

31.

x 5 3 ± 2!3

x 2 3 5 ±!12

sx 2 3d2 5 12

x2 2 6x 1 9 5 3 1 9

x2 2 6x 2 3 5 0 33.

x 532

±i!3

2

x 232

5 ±!234

1x 2322

2

5 234

1x 2322

2

5212 1 9

4

x2 2 3x 194

5 23 194

x2 2 3x 1 3 5 0

35.

y 513

±!17i

3

y 213

5 ±!2179

1y 2132

2

5217

9

y2 223

y 119

5 22 119

y2 223

y 5 22

y2 223

y 1 2 5 0 37.

y 5 252

±!19

2

y 152

5 ±!194

1y 1522

2

5194

1y 1522

2

526 1 25

4

y2 1 5y 1254

5 232

1254

2y2 1 10y 1 3 5 0

55.

Critical numbers:

Test intervals:

Negative:

Positive:

Negative:

Solution: s0, 7d

s7, `d

s0, 7d

s2`, 0d

x 5 0, 7 8

x

62 42 0

75xs7 2 xd > 0


39.

y 5 5, 26

y 521 ± 11

2

y 521 ± !121

2

y 521 ± !1 1 120

2

y 521 ± !12 2 4s1ds230d

2s1d

y2 1 y 2 30 5 0 41.

y 5 3, 272

y 521 ± 13

4

y 521 ± !169

4

y 521 ± !1 1 168

4

y 521 ± !12 2 4s2ds221d

2s2d

2y2 1 y 2 21 5 0

43.

x 58 ± 3!6

5

x 516 ± 6!6

10

x 516 ± !216

10

x 516 ± !256 2 40

10

x 52s216d ± !s216d2 2 4s5ds2d

2s5d

5x2 2 16x 1 2 5 0 45.

t 520 ± 10!2i

65

103

±5!2i

3

t 52 ± i!2

0.6

t 52 ± !22

0.6

t 52 ± !4 2 6

0.6

t 52s22d ± !s22d2 2 4s0.3ds5d

2s0.3d

0.3t2 2 2t 1 5 5 0

47.

One repeated rational solution.

5 0

5 16 2 16

b2 2 4ac 5 42 2 4s1ds4d

x2 1 4x 1 4 5 0 49.

Two distinct rational solutions.

5 81

5 1 1 80

b2 2 4ac 5 s21d2 2 4s1ds220d

s2 2 s 2 20 5 10

51.

Two distinct rational solutions.

5 169

5 289 2 120

b2 2 4ac 5 172 2 4s3ds10d

3t2 1 17t 1 10 5 0 53.

Two distinct imaginary solutions.

5 248

5 36 2 84

b2 2 4ac 5 s26d2 2 4s1ds21d

v2 2 6v 1 21 5 0


57.

Critical numbers:

Test intervals:

Negative:

Positive:

Negative:

Solution:

8

x

4 60 24 2

s2`, 22g < f6, `d

f6, `d

f22, 6g

s2`, 2g

x 5 22, 6

s6 2 xds2 1 xd ≤ 0

s4 2 x 1 2ds4 1 x 2 2d ≤ 0

16 2 sx 2 2d2 ≤ 0 59.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

4

x

5

22 0

2

6 4

s24, 52ds5

2, `ds24, 52d

s2`, 24d

x 5 24, 52

s2x 2 5dsx 1 4d < 0

2x2 1 3x 2 20 < 0

61.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

0 1 2 43

x

−3−4 −2 −1

72

f2`, 23g < s72, `d

s72, `df23, 72g

s2`, 23g

x 5 23, 72

x 1 32x 2 7

≥ 0 63.

4x 1 10

x 1 6< 0

2x 2 2 1 2x 1 12x 1 6

< 0

2x 2 2 1 2sx 1 6d

x 1 6< 0

2x 2 2x 1 6

1 2 < 0

65. Verbal model:

Labels: Number cars sold

Number cars purchased

Equation:

reject

Average price per car580,000

165 $5,000

x 5 16 carsx 5 220

0 5 sx 1 20dsx 2 16d

0 5 x2 1 4x 2 320

0 5 1,000x2 1 4,000x 2 320,000

80,000x 1 320,000 5 80,000x 1 1,000x2 1 4,000x

80,000sx 1 4d 5 80,000x 1 1,000xsx 1 4d

xsx 1 4d180,000x 2 5 180,000

x 1 41 1,0002xsx 1 4d

80,000

x5

80,000x 1 4

1 1,000

5 x 1 4

5 x

Profit per car

1Cost per

car5

Selling price per car

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

x

5− 2

−7 −6 −5 −4 −3 −2

s26, 252d

s252, `d

s26, 252d

s2`, 26d

x 5 26, 252


67. Verbal model:

Labels: Width

Length

Equation:

reject inches

inchesx 1 12 5 18

x 5 6x 5 218

0 5 sx 1 18dsx 2 6d

0 5 x2 1 12x 2 108

108 5 sx 1 12dx

5 x 1 12

5 x

Width?Length5Area 69. Formula:

or 3.5% .035 5 r

1.035 5 1 1 r

1.071225 5 s1 1 rd2

21,424.50 5 20,000s1 1 rd2

A 5 Ps1 1 rd2

71. Verbal model:

Labels: Number in Current Group

Number in New Group

Equation:

x 1 8 5 48

x 5 40 x 5 248

x 2 40 5 0x 1 48 5 0

0 5 sx 1 48dsx 2 40d

0 5 x2 1 8x 2 1920

0 5 1.5x2 1 12x 2 2880

360x 1 2880 2 360x 5 1.50x2 1 12x

360sx 1 8d 2 360x 5 1.50sx2 1 8xd

fxsx 1 8dg1360x

2360

x 1 82 5 s1.50dfxsx 1 8dg

360

x2

360x 1 8

5 1.50

5 x 1 8

5 x

$1.505Cost per person

New Group2

Cost per person Current Group

73. Verbal model:

Labels: Number in team

Number going to game

Equation:

x 1 3 5 15

x 5 215 reject x 5 12

x 1 15 5 0x 2 12 5 0

sx 2 12)sx 1 15d 5 0

x2 1 3x 2 180 5 0

1.6x2 1 4.8x 2 288 5 0

96x 2 1.6x2 2 4.8x 1 288 5 96x

s96 2 1.6xdsx 1 3d 5 96x

196 2 1.60xx 2sx 1 3d 5 96

196x

2 1.602sx 1 3d 5 96

5 x 1 3

5 x

$965Number of

tickets?Cost per

ticket75. Formula:

Labels:

Equation:

60 feet and 80 feet

140 2 x 5 60 140 2 x 5 80

x 5 80 x 5 60

0 5 sx 2 60dsx 2 80d

0 5 x2 2 140x 1 4800

0 5 2x2 2 280x 1 9,600

10,000 5 x2 1 19,600 2 280x 1 x2

1002 5 x2 1 s140 2 xd2

b 5 140 2 x

a 5 x

c 5 100

c2 5 a2 1 b2

b 5 140 2 x

x 1 b 5 140

a 1 b 5 140


77. Verbal model:

Labels: Time Person 1

Labels: Time Person 2

Equation:

19 hours, 21 hours

x 1 2 < 21

x < 21 x < 19

x 5 9 ± !101

x 518 ± 2!101

2

x 518 ± !404

2

x 518 ± !324 1 80

2

x 52s218d ± !s218d2 2 4s1ds220d

2s1d

0 5 x2 2 18x 2 20

10x 1 20 1 10x 5 x2 1 2x

10sx 1 2d 1 10x 5 xsx 1 2d

xsx 1 2d31011x

11

x 1 224 5 f1gxsx 1 2d

1x

s10d 11

x 1 2s10d 5 1

5 x 1 2

5 x

One complete job5Work doneby Person 2

1Work done by Person 1

79. (a)

secondst 5 2

0 5 st 2 2d2

0 5 t2 2 4t 1 4

0 5 216t2 1 64t 2 64

256 5 216t2 1 64t 1 192 (b)

discard seconds t 5 6t 5 22

t 2 6 5 0t 1 2 5 0

0 5 216st 1 2dst 2 6d

0 5 216st2 2 4t 2 12d

0 5 216t2 1 64t 1 192

81.

Critical numbers:

Test intervals:

x must be positive

Positive:

Negative:

Solution: s13,158, `d

s13,158, `d

s0, 13,158d

x 5 0, 13158

50,000 2 3.8xx

< 0

50,000x

2 3.8 < 0

50,000x

1 1.2 < 5

C < 5

C 5Cx

550,000 1 1.2x

x5

50,000x

1 1.2


83.

Divide by

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

5.3 < t < 14.2

s5.3, 14.2d

s14.2, `d

s5.3, 14.2d

s2`, 5.3d

t 5 14.2, 5.3

t < 14.2, 5.3

t 519.5 ± !80.25

2

t 52s219.5d ± !s219.5d2 2 4s1ds75d

2s1d

t2 2 19.5t 1 75 < 0

216ds216t2 1 312t 2 1200 > 0

216t2 1 312 t > 1200

h 5 216t2 1 312t

4.

x 5 23 ± 9i

x 1 3 5 ±!281

sx 1 3d2 5 281

sx 1 3d2 1 81 5 0


1.

x 5 10 x 5 25

x 2 10 5 0x 1 5 5 0

sx 1 5dsx 2 10d 5 0

xsx 1 5d 2 10sx 1 5d 5 0 2.

x 5 3 x 5 238

x 2 3 5 08x 1 3 5 0

s8x 1 3dsx 2 3d 5 0

8x2 2 21x 2 9 5 0

3.

x 5 2.3, 1.7

x 5 2 ± 0.3

x 2 2 5 ±0.3

sx 2 2d2 5 0.09

5.

x 532

±!32

x 232

5 ±!34

1x 2322

2

534

1x 2322

2

526 1 9

4

x2 2 3x 194

5 232

194

2x2 2 6x 1 3 5 0 6.

and 20.41y 52 ± 3!2

2< 7.41

y 54 ± 6!2

4

y 54 ± !72

4

y 54 ± !16 1 56

4

y 52s24d ± !s24d2 2 4s2ds27d

2s2d

2y2 2 4y 2 7 5 0

2ysy 2 2d 5 7


9.

2 imaginary solutions.

5 256

5 144 2 200

b2 2 4ac 5 s212d2 2 4s5ds10d 10.

x2 2 x 2 20 5 0

sx 1 4dsx 2 5d 5 0

sx 2 s24ddsx 2 5d 5 0

7.

Check: Check:

5x2 2 12x 1 10 5 0

0 5 0 0 5 0

1 2 5 1 4 5?

016 2 20 1 4 5?

0

1 2 5!1 1 4 5?

016 2 5!16 1 4 5?

0

x 5 1 x 5 16

s!xd25 12 s!xd2

5 42

!x 5 1 !x 5 4

!x 2 1 5 0!x 2 4 5 0

s!x 2 4ds!x 2 1d 5 0

x 2 5!x 1 4 5 0 8.

x 5 ±2!2i

x 5 ±!2 x 5 ±!28

x2 5 2 x2 5 28

x2 2 2 5 0 x2 1 8 5 0

sx2 1 8dsx2 2 2d 5 0

x4 1 6x2 2 16 5 0

11.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

2 4 6 8

x

−4 −2 0

s2`, 22g < f6, `d

f6, `d

f22, 6g

s2`, 22g

x 5 22, 6

sx 2 6dsx 1 2d ≥ 0

x2 2 4x 2 12 ≥ 0

x2 2 4x 1 4 ≥ 16

sx 2 2d2 ≥ 16

16 ≤ sx 2 2d2 12.

Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Solution:

x

−1 0 1 2 3 4

s0, 3d

s3, `d

s0, 3d

s2`, 0d

x 5 0, 3

2xsx 2 3d < 0


15. Verbal model:

Labels: Length

Width

Equation:

reject

12 feet 5 l 2 8

212 5 l20 feet 5 l

0 5 l 1 120 5 l 2 20

0 5 sl 2 20dsl 1 12d

0 5 l2 2 8l 2 240

240 5 l ? sl 2 8d

5 l 2 8

5 l

Width?Length5Area

16. Verbal model:

Labels: Number Current Group

Number New Group

Equation:

reject x 5 40 club membersx 5 250

0 5 sx 1 50dsx 2 40d

0 5 x2 1 10x 2 2000

0 5 6.25x2 1 62.5x 2 12500

1250x 1 12500 2 1250x 5 6.25x2 1 62.5x

1250sx 1 10d 2 1250x 5 6.25xsx 1 10d

xsx 1 10d11250x

21250

x 1 102 5 s6.25dxsx 1 10d

1250

x2

1250x 1 10

5 6.25

5 x 1 10

5 x

5 6.25Cost per person

New Group2

Cost per person Current Group

13.

u 1 8u 2 3

≤ 0

3u 1 2 2 2u 1 6

u 2 3≤ 0

3u 1 2u 2 3

22su 2 3d

u 2 3≤ 0

3u 1 2u 2 3

≤ 2 14.

21s4x 2 11d

x 2 2> 0

11 2 4x

x 2 2> 0

3 2 4x 1 8

x 2 2> 0

3 2 4sx 2 2d

x 2 2> 0

3

x 2 22 4 > 0

3

x 2 2> 4 Critical numbers:

Test intervals:

Negative:

Positive:

Negative:

Soluton:

4

x

1 2 3

114

12, 114 2

1114

, `2

12, 114 2

s2`, 2d

x 5114

, 2Critical numbers:

Test intervals:

Positive:

Negative:

Positive:

Soluton:

3u

−10 −8 −6 −4 −2 0 2 4

f28, 3d

s3, `d

f28, 3d

s2`, 28g

u 5 28, 3

Cumulative Test for Chapters 4–6 357

17.

seconds t < 1.58

t 5!10

2< 1.5811388

t 5!52

t2 54016

552

16t2 5 40

35 5 216t2 1 75 18.

passengers will produce a maximum revenuen 5 120

R 5 21

20sn 2 120d2 1 720

R 5 21

20sn2 2 240n 1 14,400d 1 720

R 5 21

20sn2 2 240nd, 80 ≤ n ≤ 160

19.

(Divide by )

st 2 5dst 2 13d < 0

t2 2 18t 1 65 < 0

216 216t2 1 288t 2 1040 > 0

216t2 1 288t > 1040

h 5 216t2 1 288t Critical numbers:

Test intervals:

t must be positive

Positive:

Negative:

Positive:

Solution:

seconds5 < t < 13

s5, 13d

s13, `d

s5, 13d

s0, 5d

t 5 5, 13

Cumulative Test for Chapters 4–6

1.

59x18

4y12

5 12x29y6

3 222

5 12y6

3x9222

5 13x9

2y622

1 2x24y3

3x5y23z0222

5 12xs24d1s25dy31s3d

3 222 2. s4 3 103d2 5 42 3 106 5 16 3 106 5 1.6 3 107

3.

2 4

2 2x2 1 x

2 2x2 1 x

2 4x3 1 2x2

2 4x3 1 0x2

4x4 2 2x3

2x 2 1 ) 4x4 2 6x3 1 0x2 1 x 2 4

2x3 2 2x2 2 x 1 24

2x 2 1

10.

5 35!5x

5 20!5x 1 15!5x

10!20x 1 3!125x 5 10!4 ? 5x 1 3!25 ? 5x 11. s!2x 2 3d25 2x 2 6!2x 1 9

12.

5 !10 1 2

56s!10 1 2d

6

56s!10 1 2d

10 2 4

6

!10 2 25

6

!10 2 2?!10 1 2

!10 1 213.

52 2 9i

175

217

29

17i

54 2 i 2 8i 1 2i2

16 2 i25

4 2 9i 2 216 1 1

1 2 2i4 1 i

51 2 2i4 1 i

?4 2 i4 2 i


4.x2 1 8x 1 16

18x2 ?2x4 1 4x3

x2 2 165

sx 1 4d2

18x2 ?2x3sx 1 2d

sx 2 4dsx 1 4d 5xsx 1 4dsx 1 2d

9sx 2 4d

5.

53x 1 5

xsx 1 3d

52x 1 6 2 1 1 x

xsx 1 3d

52x 1 6

xsx 1 3d 21

xsx 1 3d 1x

xsx 1 3d

52x1

x 1 3x 1 32 2

1xsx 1 3d 1

112 1

1x 1 3 1

xx2

52x

21

xsx 1 3d 11

x 1 3

2x

2x

x3 1 3x2 11

x 1 35

2x

2x

x2sx 1 3d 11

x 1 3

6.

5 x 1 y

5sx 2 ydsx 1 yd

x 2 y

5x2 2 y2

x 2 y

1x

y2

yx2

1x 2 yxy 2

51x

y2

yx2

1x 2 yxy 2

?xyxy

7.

5 24 1 3i!2

5 2i2 ? 2 1 3i!2

!22s!28 1 3d 5 i!2s2i!2 1 3d

8.

5 27 2 24i

5 9 2 16 2 24i

5 9 2 24i 1 16i2

s3 2 4id2 5 32 1 2s3ds24id 1 s4id2 9.

5 t1y2

1t1y2

t1y422

5t

t1y2 5 t121y2


14.

x 5 2x 5 5

sx 2 5dsx 2 2d 5 0

x2 2 7x 1 10 5 0

10 2 x 1 4x 5 10x 2 x2

xs10 2 xd11x

14

10 2 x2 5 s1dx s10 2 xd

1x

14

10 2 x5 1 Check:

1 5 1 1 5 1

12

112

5?

1 55

5?

1

12

148

5?

1 15

145

5?

1

12

14

10 2 25?

1 15

14

10 2 55?

1

15.

x 5 2x 5 9

sx 2 9dsx 2 2d 5 0

x2 2 11x 1 18 5 0

x2 2 9x 1 18 1 x2 2 6x 5 x2 2 4x

sx 2 6dsx 2 3d 1 xsx 2 6d 5 xsx 2 4d

xsx 2 6d1x 2 3x

1 12 5 1x 2 4x 2 62x sx 2 6d

x 2 3

x1 1 5

x 2 4x 2 6

16.

Not a solution

x 5 9x 5 16

0 5 sx 2 16dsx 2 9d

0 5 x2 2 25x 1 144

x 5 x2 2 24x 1 144

s!x d25 sx 2 12d2

!x 5 x 2 12

!x 2 x 1 12 5 0

Check:

53

553

12

512

23

133

5? 5

3

212

122

5? 22

24 69

1 1 5? 5

3

2 2 3

21 1 5

? 2 2 42 2 6

9 2 3

91 1 5

? 9 2 49 2 6

Check:

6 Þ 0 0 5 0

3 2 9 1 12 5?

0 4 2 16 1 12 5?

0

!9 2 9 1 12 5?

0 !16 2 16 1 12 5?

0

17.

x 5 4

2x 5 24

5 2 x 5 1

s!5 2 x d25 12

!5 2 x 5 1

!5 2 x 1 10 5 11 Check:

11 5 11

!1 1 10 5?

11

!5 2 4 1 10 5?

11


20.

r2 5 r1

!15 5

r2 5!15r1

2

5

r2 5!3r12

5

r22 5

35

r12

pr22s5d 5 pr1

2s3d 21.

inchesP < 38.6

P 5 16 1 16!2 inches

P 5 4s4d 1 4s!32d 5 !32

4

4 c

c 5 !42 1 42

18.

x 5 5 ± 5i!2

x 2 5 5 ±!250

sx 2 5d2 5 250

sx 2 5d2 1 50 5 0 19.

x 5 21±!33

523 ± !3

3

x 1 1 5 ±!13

sx 1 1d2 513

x2 1 2x 1 1 5 223

1 1

3x2 1 6x 1 2 5 0

22.

Keystrokes:

6 8

Estimate of x-intercepts

x < 21.12 and 7.12

x 56 ± !68

2

x 56 ± !36 1 32

2

x 52s26d ± !s26d2 2 4s1ds28d

2s1d

x2 2 6x 2 8 5 0

< 21.12 and 7.12

−4

−18

10

6

y 5 x2 2 6x 2 8 23.

x2 2 4x 2 12 5 0

x2 2 6x 1 2x 2 12 5 0

sx 1 2dsx 2 6d 5 0

sx 2 6d 5 0sx 1 2d 5 0

x 5 22 and x 5 6

Y5 X,T,u x2 2 X,T,u 2 GRAPH


CHAPTER 6 Quadratic Equations and Inequalities

SECTION 6.1 Factoring and Extracting Square Roots

1. The leading coefficient is because is the term ofhighest degree.

23t323 2.

Degree: 5 (the highest power)

sy2 2 2dsy3 1 7d 5 y5 1 7y2 2 2y3 2 14

3.

For some values of there correspond two values of y.x

x

y

−2 1 3 42

2

3

−2

−3

4.

For each value of there corresponds exactly one value of y.

x

x

y

−2−3 2 3

2

1

3

4

−2

5. sx3 ? x22d23 5 sx31s22dd23 5 sx1d23 5 x23 51x3 6.

5 215x22y4 5215y4

x2

s5x24y5ds23x2y21d 5 215x2412y51s21d

7. 12x3y2

22

5 13y2x2

2

59y2

4x2 8.

542u2421v22s22d

425 u25v 4 5

v 4

u5

17u24

3v222114u6v2 2

21

5 17u24

3v22216v2

14u2

9.6u2v23

27uv3 52u221v2323

95

2u1v26

95

2u9v6

10.214r 4s2

298rs2 51r 421s222

75

r3s0

75

r3

7

11.

N 5 100 prey

N 5300

!8 1 1

300 5 k

300 5k

!0 1 1

N 5k

!t 1 112.

measures the distance traveled in hours at miles perhour.

rtk

t 52918

< 1.6 hours

t 511672

116 5 k

2 5k

58

t 5kr


SECTION 6.2 Completing the Square

1. sabd4 5 a 4b4 2. sard8 5 ar?8 5 a8r 3. 1ab2

2r

5 1ba2

r

5br

ar, a Þ 0, b Þ 0

4. a2r 51ar , a Þ 0 5.

x 5 6

22x 5 212

12 2 2x 5 0

s3xd14x

2232 5 s0ds3xd

4x

223

5 0 6.

x 5 3

5x 5 15

2x 2 15 1 3x 5 0

2x 2 3f5 2 xg 5 0

2x 2 3f1 1 4 2 xg 5 0

2x 2 3f1 1 s4 2 xdg 5 0

7.

x 5 223 x 5 5

3x 1 2 5 0 x 2 5 5 0

s3x 1 2dsx 2 5d 5 0

3x2 2 13x 2 10 5 0 8.

x 5 8 x 5 25

x 2 8 5 0 x 1 5 5 0

sx 2 8dsx 1 5d 5 0

x2 2 3x 2 40 5 0

xsx 2 3d 5 40

9.

intercept:

intercept:

7.5 5 x

152 5 x

15 5 2x

0 5 2x 2 15

0 523 x 2 5

x-

gs0d 523s0d 2 5 5 25

y-

y

x4 102 8−2

−4

−2

−8

4

2

gsxd 523 x 2 5 10.

intercept:

intercept:

x 5 52 5 25

!x 5 5

0 5 5 2 !x

x-

hs0d 5 5 2 !0 5 5

y-

y

x4 102 6 8−2

−4

−2

4

6

8

2

hsxd 5 5 2 !x

11.

intercept:

intercept:

Vertical asymptote:

Horizontal asymptote: since the degree of thenumerator is less than the degree of the denominator.

y 5 0

x 5 22

x 1 2 5 0

0 5 4, none

0 54

x 1 2

x-

f s0d 54

0 1 25 2

y-

x

y

−4 2 4

6

2

−4

−2

−6

f sxd 54

x 1 212.

intercept:

intercept:

Check:

223 Þ

23

22s13d 5

? |13 2 1|

x 5 21 x 513

2x 5 1 23x 5 21

22x 5 2x 1 1 22x 5 x 2 1

22x 5 |x 2 1| 0 5 2x 1 |x 2 1|

x-

f s0d 5 2s0d 1 |0 2 1| 5 1

y-

x

y

−2−3 21 3

2

1

3

5

4

−1

f sxd 5 2x 1 |x 2 1|

Check:

2 5 2

2 5 |22|22s21d 5 |21 2 1|


SECTION 6.3 The Quadratic Formula

1. Multiplication Property:!ab 5 !a!b 2. Division Property:!ab

5!a!b

, b Þ 0

3. is not in simplest form. A factor of 72 is a perfect square.

!72 5 !36 ? 2 5 6!2

s36d!72 4. is not in simplest form. There is a radical in thedenominator which needs to be rationalized.

10!5

510!5

?!5!5

510!5

55 2!5

10y!5

5.

5 23!2

5 8!2 1 15!2

!128 1 3!50 5 !64 ? 2 1 3!25 ? 2 6. 3!5!500 5 3!5 ? 500 5 3!2500 5 3 ? 50 5 150

7. s3 1 !2 ds3 2 !2 d 5 32 2 s!2 d25 9 2 2 5 7 8.

5 11 1 6!2

5 9 1 6!2 1 2

s3 1 !2 d25 32 1 2s3d!2 1 s!2 d2

9.8

!105

8!10

?!10!10

58!10

105

4!105

10.

55s!3 1 1d

4

510s!3 1 1 d

8

55s2!3 1 2d

8

55s!12 1 2d

12 2 4

55s!12 1 2ds!12 d2

2 22

5

!12 2 25

5!12 2 2

?!12 1 2!12 1 2

11.

Common Formula:

Equation:


25 2 w 5 15 25 2 w 5 10

w 5 10 w 5 15

sw 2 10dsw 2 15d 5 0

w2 2 25w 1 150 5 0

2w2 2 50w 1 300 5 0

625 2 50w 1 w2 1 w2 5 s25ds13d

s25 2 wd2 1 w2 5 s5!13 d2

a2 1 b2 5 c2

25 2 w 5 l

25 5 l 1 w

50 5 2l 1 2w

PerimeterVerbal Model: 5 2 ? Length 1 2 ? Width 12.

200 units < x

190.00 < x 2 10

228.01 5 1.2sx 2 10d

s215.10d2 5 s2!1.2sx 2 10d d2

215.10 5 2!1.2sx 2 10d

59.90 5 75 2 !1.2sx 2 10d

p 5 75 2 !1.2sx 2 10d


9.

y 2 8 5 0

y 2 8 5 0sx 2 0d

m 58 2 85 2 0

505

5 0

s0, 8d, s5, 8d 10.

x 1 3 5 0

x 5 23

m 55 2 2

s23d 2 s23d 530

5 undefined

s23, 2d, s23, 5d

SECTION 6.4 Applications of Quadratic Equations

1. m 5y2 2 y1

x2 2 x1

2. (a) Slope-intercept form:

(b) Point-slope form:

(c) General form:

(d) Horizontal line: y 2 b 5 0

Ax 1 By 1 C 5 0

y 2 y1 5 msx 2 x1d

y 5 mx 1 b

3.

x 1 2y 5 0

2y 5 2x

y 5 212

x

m 522 2 04 2 0

5224

5212

s0, 0d, s4, 22d 4.

0 5 3x 2 4y

4y 5 3x

y 534

x

y 2 0 534

sx 2 0d

m 575 2 0

100 2 05

75100

534

s0, 0d, s100, 75d 5.

0 5 2x 2 y

y 2 6 5 2x 2 6

y 2 6 5 2sx 2 3d

m 56 2 s22d3 2 s21d 5

6 1 23 1 1

584

5 2

s21, 22d, s3, 6d

6.

x 1 y 2 6 5 0

y 5 2x 1 6

y 2 0 5 21sx 2 6d

m 50 2 56 2 1

5255

5 21

s1, 5d, s6, 0d7.

22x 1 16y 2 161 5 0

16y 2 128 5 222x 1 33

y 2 8 5211

8x 1

3316

y 2 8 5211

8 1x 2322

m 5

52 2 8112 2

32

?22

55 2 1611 2 3

5211

8

132

, 82, 1112

, 522

8.

0 5 134x 2 73y 1 146

73y 2 146 5 134x

y 2 2 513473

x

y 2 2 513473

sx 2 0d

m 515.4 2 27.3 2 0

513.47.3

513473

s0, 2d, s7.3, 15.4d


11.

Labels: Number current group

Number new group

Equation:

Reject

x 5 210 x 5 8 people

0 5 sx 1 10dsx 2 8d

0 5 x2 1 2x 2 80

0 5 6250x2 1 12,500x 2 500,000

250,000x 1 500,000 2 250,000x 5 6250x2 1 12,500x

250,000sx 1 2d 2 250,000x 5 6250sx2 1 2xd

xsx 1 2d1250,000x

2250,000x 1 2 2 5 s6250dxsx 1 2d

250,000

x2

250,000x 1 2

5 6250

5 x 1 2

5 x

Cost per personcurrent group

VerbalModel:

2 5 6250Cost per personnew group

12.

Labels: Speed of the current

Equation:

Reject

miles per hourx 5 3

23

x 5 ±3

x2 5 9

4sx2 2 9d 5 0

4x2 2 36 5 0

630 1 35x 1 630 2 35x 5 1296 2 4x2

35s18 1 xd 1 35s18 2 xd 5 4s324 2 x2d

s18 2 xds18 1 xd1 3518 2 x

135

18 1 x2 5 s4ds18 2 xds18 1 xd

35

18 2 x1

3518 1 x

5 4

5 x

Timeupstream

VerbalModel:

1Timedownstream5

Totaltime


SECTION 6.5 Quadratic and Rational Inequalities

1. is not written in scientific notation. The number must be between 1 and 10 such as 3.682 3 109.36.82 3 108 2.

and

106 ≤ sn1 3 102dsn2 3 104d < 108

f1 ? 1 ≤ n1 ? n2 < 10 ? 10g106

1 ≤ n2 < 101 ≤ n1 < 10

5 n1 ? n2 ? 106

sn1 3 102dsn2 3 104d 5 n1 ? n2 ? 10214

3. 6u2v 2 192v2 5 6vsu2 2 32vd 4. 5x2y3 2 10x1y3 5 5x1y3sx1y3 2 2d

5. xsx 2 10d 2 4sx 2 10d 5 sx 2 4dsx 2 10d 6.

5 sx 1 3dsx 2 2dsx 1 2d

5 sx 1 3dsx2 2 4d

5 x2sx 1 3d 2 4sx 1 3d

x3 1 3x2 2 4x 2 12 5 sx3 1 3x2d 1 s24x 2 12d

7. 16x2 2 121 5 s4x 2 11ds4x 1 11d 8. 4x3 2 12x2 1 16x 5 4xsx2 2 3x 1 4d

9.

5 32 h2

A 532 h ? h

h

h32

Area 5 Length ? Width 10.

5 13b2

5 12 ? b ? 2

3b

b

b23

Area 512 ? Base ? Height

12.

5 x2 1 8x

5 x2 1 6x 1 2x

5 x ? sx 1 6d 112 ? x ? 4

x + 6

x x

x + 104 Area 5 Area of rectangle 1 Area of triangle

11. Divide figure into 5 con-gruent squares, each with side length

5 5 ? x2

Area 5 5 ? Area of square

x.

x

x

x

x

x

x

x

x

x

x

x

x

C H A P T E R 7Linear Models and Graphs of Nonlinear Models

Section 7.1 Variation . . . . . . . . . . . . . . . . . . . . . . . . . .362

Section 7.2 Graphs of Linear Inequalities . . . . . . . . . . . . . . .365

Section 7.3 Graphs of Quadratic Functions . . . . . . . . . . . . . . .370

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .376

Section 7.4 Conic Sections . . . . . . . . . . . . . . . . . . . . . . .379

Section 7.5 Graphs of Rational Functions . . . . . . . . . . . . . . .386


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .403

362

C H A P T E R 7Linear Models and Graphs of Nonlinear Models

Section 7.1 VariationSolutions to Odd-Numbered Exercises

1. I 5 kV 3. V 5 kt 5. u 5 kv2 7. p 5kd

9. P 5k

!1 1 r11. A 5 klw 13. P 5

kV

15. The area of a trianglevaries jointly as thebase and height.

17. The area of a rectangle variesjointly as the length and the width.

19. The volume of a right circularcylinder varies jointly as thesquare of the radius and the height.

21. The average speed varies directlyas the distance and inversely asthe time.

23.

s 5 5t

5 5 k

20 5 ks4d

s 5 kt 25.

F 55

16 x2

516 5 k

5001600 5 k

500 5 ks40d2

F 5 kx2 27.

H 552 u

52 5 k

10040 5 k

100 5 ks40d

H 5 ku

29.

n 548m

48 5 k

32 5k

1.5

n 5km

31.

g 54!z

4 5 k

45

5k

!25

g 5k!z

33.

F 5256

xy

256

5 k

500120

5 k

500 5 ks15ds8d

F 5 kxy

35.

d 5120x2

r

120 5 k

3000 5 ks25d

3000 5 k1102

4 2

d 5 k1x2

r 2 37. (a) (b) Price per unit

R 5 $4921.25

R 5 7.75s635d

R 5 7.75x

7.75 5 k

3875 5 ks500d

R 5 kx

Section 7.1 Variation 363

39. (a)

(b)

15 pounds 5 F

1.5 51

10F

d 51

10F

110

5 k

550

5 k

5 5 ks50d

d 5 kF 41.

18 pounds 5 F

12 523

F

23

5 k

70105

5 k

7

10.55 k

7 5 ks10.5d

d 5 kF

d 5 2 inches

d 51

10s20d

d 51

10F

43.

acceleration 5 32 ftysec2

32 5 k

96 5 ks3d

v 5 kt 45.

d 5 208.3 feet

d 51

12s50d2

112 5 k

75900 5 k

75 5 ks30d2

d 5 ks2 47.

will change by a factor of 4.F

F 5 4sks2d

F 5 4ks2

F 5 ks2sd2

F 5 ks2

49.

0.106575 < k

6.78

20.25p5 k

6.78 5 20.25pk

9-inch: 6.78 5 kspds4.5d2

p 5 kA

0.0864745 < k

9.7836p

5 k

9.78 5 36pk

12-inch: 9.78 5 kspds6d2

0.068923 < k

12.18

56.25p5 k

12.18 5 56.25pk

15-inch: 12.18 5 kspds7.5d2

No, the price of the pizza is not directly proportional to its area. The 15-inch pizza at $12.18 is the best buy.

51.

x 5 666.6 < 667 units

x 54000

6

4000 5 k

800 5k5

x 5kp

53.

x 5 324 pounds k 516

54 516

? x 60 5 k ? 360

Wm 516

? We W m 5 k ? We

364 Chapter 7 Linear Models and Graphs of Nonlinear Models

55.

will change by a factor of or 14.324

1296I

I 5k

324 I 5

k1296

I 5k

182 I 5k

362

I 5kd2 57.

So,

p 5 17.5%

p 51146.5

p 5114

t.

114 5 k

38 5k3

p 5kt

59. (a)

(b) is the principal or the amount of investment.K

I 5 $270 750 5 K

I 5 750s0.09ds4d 202.50 5 Ks0.27d

I 5 750rt 202.50 5 Ks0.09ds3d

I 5 Krt 61.

10x

y

100

80

864

40

60

2

20

2 4 6 8 10

4 16 36 64 100y 5 kx2

x

63.

10x

y

100

80

864

40

60

2

20

2 4 6 8 10

2 8 18 32 50y 5 kx2

x65.

10x

y

21

84 6

8

41

1

2

8

3

2 4 6 8 10

150

132

118

18

12

y 5kx2

x

67.

10x

y

3

2

84 6

1

2

2 4 6 8 10

110

532

518

58

52

y 5kx2

x69.

Using any two pairs of numbers, is 4.k

4 5 k 4 5 k

15

5k

20 25

5k

10

10 20 30 40 50

225

110

215

15

25

y

x

Section 7.2 Graphs of Linear Inequalities 365

71.

Using any two pairs of numbers, is 2310.k

2310 5 k

2620 5 k2

310 5 k

26 5 k ? 20 23 5 k ? 10

x 10 20 30 40 50

y 215212292623

75.

will quadruple.y

y 5 4kx2

y 5 ks4x2d

y 5 ks2xd2

y 5 kx2

Section 7.2 Graphs of Linear Inequalities

1.

(a)

is a solution.

(c)

is a solution.s3, 4d

25 < 4

3 2 8 < 4

3 2 2s4d <?

4

s0, 0d

0 < 4

0 2 2s0d <?

4

x 2 2y < 4

(b)

is not a solution.

(d)


3 < 4

5 2 2 < 4

5 2 2s1d <?

4

s2, 21d

4 </ 4

2 1 2 < 4

2 2 2s21d <?

4

3.

(a)

is not a solution.

(c)

is a solution. s3, 1d

10 ≥ 10

3s3d 1 1 ≥? 10

s1, 3d

9 ≥/ 10

3s1d 1 3 ≥? 10

3x 1 y ≥ 10

(b)

is not a solution.

(d)


21 ≥ 10

3s2d 1 15 ≥? 10

s23, 1d

28 ≥/ 10

3s23d 1 1 ≥? 10


7.

(a)

is not a solution.

(c)

is not a solution.s6, 0d

0 ≤/ 23

0 ≤ 3 2 6

0 ≤? 3 2 |6|

s21, 4d

4 ≤/ 3 2 1

4 ≤? 3 2 |21|y ≤ 3 2 |x|

(b)

is a solution.

(d)


22 ≤ 22

22 ≤ 3 2 5

22 ≤? 3 2 |5|

s2, 22d

22 ≤ 3 2 2

22 ≤? 3 2 |2|

5.

(a)

is a solution.

(c)


21 >/ 20.2

21 >?

0.2s4d 2 1

s0, 2d

2 > 21

2 >?

0.2s0d 2 1

y > 0.2x 2 1

(b)

is not a solution.

(d)


7 > 21.4

7 >?

0.2s22d 2 1

s6, 0d

0 > 0.2

0 >?

0.2s6d 2 1

9. (b)y ≥ 22 11. (d)3x 2 2y < 0 13. (f)x 1 y < 4

15.

x

3

y

1

2

431

1

2

1

x ≥ 2 17.

x

8

y

66

4

2

6422

46 2

y < 5 19.

x21

y

2

1

2

2

1

y > 12x

21.

y

x4−2 2

−2

2

4

y ≥ 5 2 x 23.

x3−3 1

y

−2

1

2

2−2 −1−1

3

4

y ≤ x 1 2 25.

y

x4−2 2

−2

2

4

6

6

y ≥ 2x 1 4

x 1 y ≥ 4


27.

x62

y

2

4

2

2

6

x 2 2y ≥ 6 29.

y

x

2−1−1

1

3

3

4−2

−2

−3

y ≥ 232x 1 1

2y ≥ 23x 1 2

3x 1 2y ≥ 2 31.

2−2−3 −1−1

3

1

2

3

y

x

y ≤ 32x 2 2

22y ≥ 23x 1 4

3x 2 2y ≥ 4

33. or

10x

y

10

8

8642

4

2

22

66

y < 223x 1

2030.2x 1 0.3y < 2 35.

x4

4

y

1

3

2

321

1

1

y 2 1 > 212sx 2 2d

37. or

x5

y

5

1

3

2

4

43211

1

y ≤ 243

x 1 4x3

1y4

≤ 1

39.

Keystrokes:

.75 1

7 1 1 10

y ≥ 34x 2 1

X,T,u 2−10

−10

10

10

41.

Keystrokes:

2 3 6

7 10 1 1

y ≤ 223x 1 6

Y5 x x2c 4 d X,T,u 1

DRAW x2c , Y-VARS d ENTER

−10

−10

10

10

Y5

DRAW x Y-VARS , d ENTER

x

368 Chapter 7 Linear Models and Graphs of Nonlinear Models368 Chapter 7 Linear Models and Graphs of Nonlinear Models

43.

Keystrokes:

.5 2

7 10 1 1

y ≤ 12x 2 2

22y ≥ 2x 1 4

x 2 2y 2 4 ≥ 0

Y5 X,T,u 2

DRAW x x2c , Y-VARS d ENTER

−10

−10

10

10

45.

Keystrokes:

2 3 4

7 10 1 1

y ≤ 223x 1 4

3y ≤ 22x 1 12

2x 1 3y 2 12 ≤ 0

DRAW x x2c

4 d X,T,u 1

d ENTER

−10

−10

10

10

47.

3x 1 4y > 17

4y 2 8 > 23x 1 9

y 2 2 > 234

sx 2 3dm 52 2 53 1 1

5 234

49. y < 2

51.

x 2 2y < 0

2x 1 2y > 0

2y > x

y >12

xm 51 2 02 2 0

512

53.

or

or

(Note: xand y cannot be negative.)

Keystrokes: 250

7 0 1 1

y ≤ 2x 1 250

0 ≤ x 1 y ≤ 250

2x 1 2y ≤ 500

P 5 2x 1 2y

Y5 x2c X,T,u 1

DRAW x dY-VARS ENTER,

00

300

300

55. (Note: xand y cannot be negative.)


y ≤ 223x 1

2003

15y ≤ 210x 1 1000

20 40 60 80 100

20

40

60

80

100

x

y 10x 1 15y ≤ 1000

Y5 x x2c

Y-VARS d


57. yes

2 4 6 8 10 12

50

100

150

200

250

300

t

p

s12, 220d

59. Verbal model: 48

Labels: Cost of cheese pizzas

Cost for extra toppings (dollars)

Cost for drinks (dollars)

Inequality:


5 10 20 25x

20

y

5

10

15

25

15

x 1 1.5y ≤ 21

1.00x 1 1.50y ≤ 21

27 1 1.00x 1 1.50y ≤ 48

5 1.50y

5 1.00x

5 3s9d 5 $27

≤Cost fordrinks1

Cost forextra

toppings1

Cost ofcheesepizzas

yes 18 ≤ 21

6 1 12 ≤? 21

6 1 1.5s8d ≤? 21

s6, 8d

61. (Note: x and y cannot be negative.)

Here are some examples of ordered pairs that are solutions.Note that there are other correct answers.

s22, 0ds12, 7d

s20, 1ds10, 10d

s4, 21ds2, 22d

y ≥ 232x 1 25

6y ≥ 29x 1 150

9x 1 6y ≥ 150

5 10 15 20 25 30

5

10

15

20

25

30

x

y


67. The solution of does not include the points on the line The solution of does include thepoints on the line x 2 y 5 1.

x 2 y ≥ 1x 2 y 5 1.x 2 y > 1

69. On the real number line, the solution of is an unbounded interval.

On a rectangular coordinate system, the solution of is a half-plane.x ≤ 3

x ≤ 3

63.

r

A

240

192

144

96

48

20 40 60 80 100

r A= 0.75(220 )−

r A= 220−

r 5 0.75s220 2 Ad 65. is a solution of a linear inequality in x and y meansthe inequality is true when and are substituted for xand y respectively.

y1x1

sx1, y1d

Section 7.3 Graphs of Quadratic Functions

1. (e)y 5 4 2 2x 3. (b)y 5 x2 2 3 5. (d)y 5 sx 2 2d2

7.

vertex s0, 2d

y 5 x2 1 2 5 sx 2 0d2 1 2 9.

vertex5 s2, 3d

5 sx 2 2d2 1 3

5 sx2 2 4x 1 4d 1 7 2 4

y 5 x2 2 4x 1 7 11.

vertex5 s23, 24d

y 5 sx 1 3d2 2 4

y 5 (x2 1 6x 1 9d 1 5 2 9

y 5 x2 1 6x 1 5

13.

vertex s3, 21d

y 5 21sx 2 3d2 2 1

y 5 21sx2 2 6x 1 9d 2 10 1 9

y 5 21sx2 2 6xd 2 10

y 5 2x2 1 6x 2 10 15.

vertex5 s1, 26d

5 21sx 2 1d2 2 6

5 21sx2 2 2x 1 1d 2 7 1 1

y 5 2x2 1 2x 2 7 17.

vertex5 1232

, 2522

5 21x 1322

2

252

5 21x2 1 3x 1942 1 2 2

92

y 5 2x2 1 6x 1 2

19.

vertex5 s4, 21d

5 21

5 16 2 32 1 15

f12b

2a2 5 42 2 8s4d 1 15

x 52b2a

52s28d

2s1d 5 4

b 5 28a 5 1

f sxd 5 x2 2 8x 1 15 21.

vertex5 s21, 2d

5 2

5 21 1 2 1 1

g12b2a 2 5 2s21d2 2 2s21d 1 1

x 52b2a

52s22d2s21d 5 21

b 5 22a 5 21

g sxd 5 2x2 2 2x 1 1 23.

vertex5 1212

, 32 5 3

5 1 2 2 1 4

5 41142 2 2 1 4

y 5 412122

2

1 412122 1 4

x 52b2a

5242s4d 5

212

b 5 4a 5 4

y 5 4x2 1 4x 1 4

7.3 Graphs of Quadratic Functions 371

25. opens upward

vertex5 s0, 2d

2 > 0 27. opens downward

vertex5 s10, 4d

21 < 0 29. opens upward

vertex5 s0, 26d

1 > 0

31. opens downward

vertex5 s3, 0d

21 < 0 33.

s0, 25d

y 5 25

y 5 25 2 02

y 5 25 2 x2

s5, 0d, s25, 0d

x 5 ±5

x2 5 25

0 5 25 2 x2

y 5 25 2 x2 35.

s0, 0d

y 5 0

y 5 02 2 9s0d

y 5 x2 2 9x

s0, 0d, s9, 0d

0 5 xsx 2 9d

0 5 x2 2 9x

y 5 x2 2 9x

37.

s0, 9d

y 5 9

y 5 4s0d2 2 12s0d 1 9

y 5 4x2 2 12x 1 9

132

, 02

32

5 x

0 5 2x 2 3

0 5 s2x 2 3d2

0 5 4x2 2 12x 1 9

y 5 4x2 2 12x 1 9 39.

no x-intercepts

s0, 3d

y 5 3

y 5 02 2 3s0d 1 3

y 5 x2 2 3x 1 3

53 ± !23

2

x 53 ± !9 2 12

2

0 5 x2 2 3x 1 3

y 5 x2 2 3x 1 3

41.

x-intercepts

vertex

s0, 24d

gsxd 5 sx 2 0d2 2 4

x 5 22 x 5 2

0 5 sx 2 2dsx 1 2d

0 5 x2 2 42 0

3

),x

y

1

0

1

, )

3

( 2

2

3

4),0(

5

1

(

g sxd 5 x2 2 4 43.

x-intercepts

vertex

s0, 4dfsxd 5 2sx 2 0d2 1 4

x 5 ±2

x2 5 4

0 5 2x2 1 4

0, )

3x

y

, )45

(0

(

2

1

1, 0)(3 2

3

2

f sxd 5 2x2 1 4


49.

x-intercepts

vertex

s4, 0d

y 5 sx 2 4d2 1 0

4 5 x

0 5 x 2 4

0 5 sx 2 4d2

20x

y

20

16

)(4, 0

161284

8

4

44

12

f sxd 5 sx 2 4d2

47.

x-intercepts

vertex

s32, 94d 5 21sx 2

32d2

194

y 5 21sx2 2 3x 194d 1

94

x 5 30 5 x

0 5 2xsx 2 3d

0 5 2x2 1 3x

)0

4x

4,9

23

y

3

2

(3),0 0

1 2

1

1

( ,

f sxd 5 2x2 1 3x45.

x-intercepts

vertex

s32, 29

4df sxd 5 sx 2

32d2

294

f sxd 5 sx2 2 3x 194d 2

94

x 5 30 5 x

0 5 xsx 2 3d

0 5 x2 2 3x0

4x

),0

y

0

1 2

94,

32

1

1

3

2

(),( 3

f sxd 5 x2 2 3x

53.

x-intercepts

vertex

x1

y

5

3

4)43( ,

2

10, )

0),( 123

(

6 4

5

y 5 2sx 1 3d2 1 4

y 5 2sx2 1 6x 1 9d 2 5 1 9

x 5 2125 5 x

0 5 sx 1 5dsx 1 1d

0 5 x2 1 6x 1 5

f sxd 5 2sx2 1 6x 1 5d51.

x-intercepts

vertex

0),(5x

6

y

5

3

4

)0,(3

)1,(4

532 4

2

1

1

2

1

5 sx 2 4d2 2 1

y 5 sx2 2 8x 1 16d 1 15 2 16

x 5 35 5 x

0 5 sx 2 5dsx 2 3d

0 5 x2 2 8x 1 15

f sxd 5 x2 2 8x 1 15

Section 7.3 Graphs of Quadratic Functions 373

57.

vertex

x-intercepts

1x

y

3

2

11

),2 0

3

3

), 22

(

, 0)4(

( 3

1

x 5 2224 5 x

0 5 sx 1 4dsx 1 2d

0 5 x2 1 6x 1 8

y 5 2sx 1 3d2 2 2

y 5 2sx2 1 6x 1 9d 1 16 2 18

f sxd 5 2sx2 1 6x 1 8d 59.

vertex

x-intercepts

4x

y

3

2

0),(3

321

),(1 2

, 01 )(

2

3

1

x 5 213 5 x

0 5 sx 2 3dsx 1 1d

0 5 x2 2 2x 2 3

y 512 sx 2 1d2 2 2

y 512 sx2 2 2x 1 1d 2

32 2

12

f sxd 512sx2 2 2x 2 3d

55.

x-intercepts

x 5 23 ± !2

x 526 ± 2!2

2

x 526 ± !8

22

x 526 ± !36 2 28

22

x 526 ± !62 2 4s21ds27d

2s21d

0 5 2x2 1 6x 2 7

g sxd 5 2x2 1 6x 2 7 vertex

q sxd 5 2sx 2 3d2 1 2

q sxd 5 2sx2 2 6x 1 9d 1 9 2 7

q sxd 5 2sx2 2 6x 1 9 2 9d 2 7

q sxd 5 2sx2 2 6xd 2 7

q sxd 5 2x2 1 6x 2 7 0),2

5x

323(),(

y

3

2

, 0)243

3(21

11

3

2

1


73.

Keystrokes:

1 6 2 8 11

vertex

−2

−8

10

10

5 s2, 0.5d

y 516s2x2 2 8x 1 11d

65.

Vertical shift 2 units up.

x

y

−2 −1−3 21 3

1

3

5

4

−1

h sxd 5 x2 1 2 67.

Horizontal shift 2 units left.

x

y

−2−4−5 −1−3 1

1

5

4

−1

h sxd 5 sx 1 2d2 69.

Horizontal shift 1 unit right.

Vertical shift 3 units up.

x

y

−2 −1 2 41 3

1

3

2

6

4

h sxd 5 sx 2 1d2 1 3

71.

Horizontal shift 3 units left.

Vertical shift 1 unit up.

x

y

−2−3 −1−4−5−6

4

2

1

5

3

−1

h sxd 5 sx 1 3d2 1 1

x xx x


61.

< 5.83, 2.17

x 524 ± !120

65

12 ± !303

x 524 ± !576 2 456

6

0 5 3x2 2 24x 1 38

y 535

sx 2 4d2 2 2

y 535

sx2 2 8x 1 16d 1385

2485

y 515

s3x2 2 24x 1 38d 63.

x < 3.87, 23.87

x 5 ±!15

x2 5 15

13

x2 5 5

0 5 213

x2 1 5

f sxd 5 213

sx 2 0d2 1 5

f sxd 5 213

x2 1 5

f sxd 5 5 213

x2

, 0

0,30

3

x

033

124

)4, 2

8

y

66

12

2

4

2

(2

2

5),(0

x,

6

051 )

2

(

66

4

2

,

26

)15 0(

4

6

2

y

75.

Keystrokes:

.7 2.7 2.3

vertex5 s21.9, 4.9d

−6

−10

6

8

y 5 20.7x2 2 2.7x 1 2.3

1x 2X,T, X,T,5Y GRAPHx2c x2cu u

Section 7.3 Graphs of Quadratic Functions 375

77. vertex point

21 5 a

24 5 4a

0 5 4a 1 4

0 5 as22 2 0d2 1 4

y 5 asx 2 0d2 1 4

5 s22, 0d5 s0, 4d 79. vertex point

1 5 a

4 5 4a

2 5 4a 2 2

2 5 as0 1 2d2 2 2

y 5 asx 1 2d2 2 2

y 5 asx 2 s22dd2 1 s22d

5 s0, 2d5 s22, 2d

y 5 2x2 1 4

y 5 21sx 2 0d2 1 4

y 5 x2 1 4x 1 2

y 5 x2 1 4x 1 4 2 2

y 5 sx 1 2d2 2 2

y 5 1sx 1 2d2 2 2

81. vertex point

224 5 a

22 5 as4d

4 5 as4d 1 6

4 5 as0 2 2d2 1 6

y 5 asx 2 2d2 1 6

5 s0, 4d5 s2, 6d 83. vertex

y 5 1sx 2 2d2 1 1 5 x2 2 4x 1 5

5 s2, 1d a 5 1

y 5 212 x2 1 2x 1 4

y 5 212 x2 1 2x 2 2 1 6

y 5 212sx2 2 4x 1 4d 1 6

y 5 212 sx 2 2d2 1 6

85. vertex point

y 5 1sx 2 2d2 2 4 5 x2 2 4x

1 5 a

4 5 as4d

0 5 as0 2 2d2 2 4

5 s0, 0d5 s2, 24d 87. vertex point

y 512 sx 2 3d2 1 2 5

12 x2 2 3x 1

132

12 5 a

2 5 as4d

4 5 as1 2 3d2 1 2

5 s1, 4d5 s3, 2d

89. vertex point

24 5 a

1 5 as1d 1 5

1 5 as0 2 s21dd2 1 5

5 s0, 1d5 s21, 5d

91. Horizontal shift 3 units right

y 5 24x2 2 8x 1 1

y 5 24x2 2 8x 2 4 1 5

y 5 24sx2 1 2x 1 1d 1 5

y 5 24sx 1 1d2 1 5

93. Horizontal shift 2 units right

Vertical shift 3 units down

95.

(a)

(b)

Maximum height feet5 16

y 5 2112

sx 2 12d2 1 16

y 5 21

12sx2 2 24x 1 144d 1 4 1 12

y 5 21

12x2 1 2x 1 4

y 5 4 feet

y 5 21

12s0d2 1 2s0d 1 4

y 5 21

12x2 1 2x 1 4

(c)

feet < 25.86

x 524 ± !576 1 192

2

0 5 x2 2 24x 2 48

0 5 21

12x2 1 2x 1 4


97.

The maximum height of the diver is 14 ft.

y 5 249 sx 2 3d2 1 14

y 5 249 sx2 2 6x 1 9d 1 4 1 10

y 5 249 sx2 2 6x 1 9 2 9d 1 10

y 5 249 sx2 2 6xd 1 10

y 5 249 x2 1

249 x 1 10 99. (a)

(b) vertex 1993, 110,800 reserves5 s3.65, 110, 810d

00

6

125

101. (a)

P 5 3000 1 15x 2320 x2

P 5 3000 1 15x 2 0.15x2

P 5 3000 2 15x 1 30x 5 0.15x2

P 5 s100 1 xds30 2 0.15xd

P 5 s100 1 xdf90 2 xs0.15d 2 60g

P 5 s100 1 xdf90 2 xs0.15dg 2 s100 1 xd60

103.

Keystrokes:

2 100

when A is maximum

0

0

2000

100

x < 50

A 52p

s100x 2 x2d 105.

y 51

2500x2

y 51

2500sx 2 0d2 1 0

1

25005 a

100250,000

5 a

100 5 as250,000d

100 5 as500 2 0d2 1 0

(b)

vertex

order size for maximum profit

radiosP 5 100 1 50 5 150

5 s50, 3375d

P 5 2320 sx 2 50d2 1 3375

P 5 2320 sx2 2 100x 1 2500d 1 3000 1 375

P 5 23

20 x2 1 15x 1 3000

(c) Recommend pricing scheme if price reductions are restricted to orders between 100 and 150 orders.

107. The graph of the quadratic function is a parabola.f sxd 5 ax2 1 bx 1 c

109. To find any x-intercepts, set and solve the resulting equation for x.

To find the y-intercept, set and solve the resulting equation for y.x 5 0

y 5 0

111. The discriminant of a quadratic function tells how many x-intercepts the parabola has. If positive, there are 2 x-intercepts; ifzero, 1 x-intercept; and if negative, no x-intercepts.

113. Find the y-coordinate of the vertex. This is the maximum (or minimum) value of a quadratic function.


1. A 5 kr 2 2. z 5kxy2 3. Distance:

Distance varies jointly proportionalto rate and time.

d 5 rt

x xx x

2 x 2X,T, X,T,5Y GRAPH4 p u u


4. Volume:

The volume of a cube varies directly as the cube of the length of the sides.

V 5 s3

5. if

23

5 k

2436

5 k

24 5 ks6d2

then 6 5ks6d2

4

z 52x2

3yz 5 6, x 5 6, y 5 4z 5

kx2

y

6.

(a)

is a solution.

(c)


16 ≤ 4

4 1 12 ≤ 4

2s2d 2 3s24d ≤? 4

s5, 2d

4 ≤ 4

10 2 6 ≤ 4

2s5d 2 3s2d ≤? 4

2x 2 3y ≤ 4

(b)

is a solution.

(d)


6 ≤ 4

6 2 0 ≤ 4

2s3d 2 3s0d ≤? 4

s22, 4d

216 ≤ 4

24 2 12 ≤ 4

2s22d 2 3s4d ≤? 4

7.

Shaded region:x 1 2y ≤ 11

x 1 2y 5 11 line

2y 2 6 5 2x 1 5

y 2 3 5 212

x 152

y 2 3 5 212

sx 2 5d

m 53 2 55 2 1

5 224

5 212

8.

Shaded region:x 2 3y > 25

x 2 3y 5 25 line

3y 2 9 5 x 2 4

y 2 3 513

x 243

y 2 3 513

sx 2 4d

m 53 2 1

4 2 s22d 526

513

9.

y

−3

1

−1

2

3

x2−4 1−3

−2

−1

x > 22 10.

x2 4−1 1

y

1

2

5

3

−1

4

3

2x 1 3y ≤ 9 11.

x4−2 2 3−1

y

1

−2

−4

−1

−3

1

2x 2 y ≤ 4


12.

1 5 a

4 5 as4d

3 5 as4d 2 1

y 5 1sx 2 3d2 2 13 5 as5 2 3d2 2 1 13.

214 5 a

21 5 as4d

3 5 as4d 1 4

y 5 214sx 2 5d2 1 43 5 as3 2 5d2 1 4

14. vertex

y 5 214

sx 1 3d2 1 2

y 5 214

sx2 1 6x 1 9d 214

194

y 5 214

sx2 1 6x d 214

5 s23, 2d x-intercepts

x < 20.17 and 25.83

x 526 ± !32

25

26 ± 4!22

523 ± 2!2

2

x 526 ± !36 2 4

2

x 526 ± !62 2 4s1ds1d

2s1d

0 5 x2 1 6x 1 1

0 5 214

sx2 1 6x 1 1d

−8 −4 −2

−2

2

−6

−4

4

2

y

x

( 3, 2)−

( 3 2 2, 0)− − ( 3 + 2 2, 0)−

15. vertex

y 5 2sx 2 1d2 2 9

y 5 2sx2 2 2x 1 1d 2 7 2 2

y 5 2sx2 2 2x d 2 7

5 s1, 29d x-intercepts

x < 21.12 and 3.12

x 52 ± 3!2

25 1 ±

3!22

x 54 ± 6!2

4

x 54 ± !72

4

x 54 ± !16 1 56

4

x 52s24d ± !s24d2 2 4s2ds27d

2s2d

0 5 2x2 2 4x 2 7( )

−4 −2 2 4 6

−10

−8

y

x

(1, 9)−

1 , 0− 3 22( ) 1 + , 03 2

2

16.

1

6005 k

2

12005 k

.0212

5 k

.02 5 ks12d

g 5 kt

minutes 30 5 t

s.05ds600d 5 t

.05 51

600t

17.

9x 1 14y ≤ 200

900x 1 1400y ≤ 20,000

Section 7.4 Conic Sections 379

18.

maximum height feet5 55

y 5 20.005sx 2 100d2 1 55

y 5 20.005sx2 2 200x 1 10,000d 1 5 1 50

y 5 20.005sx2 2 200x d 1 5

y 5 20.005x2 1 x 1 5

Section 7.4 Conic Sections

1. (c)x2 1 y2 5 9 3. (e)x2

41

y2

95 1 5. (a)x2 2 y2 5 4

7. center: radius: 5

x2 1 y2 5 25

x2 1 y2 5 52

x2 1 y2 5 r2

s0, 0d, 9. center: radius:

or 9x2 1 9y2 5 4 x2 1 y2 549

x2 1 y2 5 s23d2

x2 1 y2 5 r2

23s0, 0d,

11. center: point:

x2 1 y2 5 64

x2 1 y2 5 82

x2 1 y2 5 r2

r 5 8

r 5 !64

r 5 !s0 2 0d2 1 s8 2 0d2

s0, 8ds0, 0d, 13. center: point:

x2 1 y2 5 29

x2 1 y2 5 s!29d2

x2 1 y2 5 r2

r 5 !29

r 5 !25 1 4

r 5 !s5 2 0d2 1 s2 2 0d2

s5, 2ds0, 0d,

15. center: radius: 10

sx 2 4d2 1 sy 2 3d2 5 100

sx 2 4d2 1 sy 2 3d2 5 102

sx 2 hd2 1 sy 2 kd2 5 r2

s4, 3d, 17. center: radius: 9

sx 2 5d2 1 sy 1 3d2 5 81

sx 2 5d2 1 fy 2 s23dg2 5 92

sx 2 hd2 1 sy 2 kd2 5 r2

s5, 23d,

19. center: point:

r 5 2

r 5 !4 1 0

r 5 !f0 2 s22dg2 1 s1 2 1d2

s0, 1ds22, 1d,

sx 1 2d2 1 sy 2 1d2 5 4

fx 2 s22dg2 1 sy 2 1d2 5 22

sx 2 hd2 1 sy 2 kd2 5 r2

21. center: point:

5 !17

r 5 !1 1 16

r 5 !s4 2 3d2 1 s6 2 2d2

s4, 6ds3, 2d,

sx 2 3d2 1 sy 2 2d2 5 17

sx 2 3d2 1 sy 2 2d2 5 s!17d2

sx 2 hd2 1 sy 2 kd2 5 r2


25.

center

radius

−8 −4 −2 2 84

−8

−4

8

4

2

y

x

5 6

5 s0, 0d

x2 1 y2 5 36 27. center

radius

x1

1

y

1

1

34

34

14

14

14

34

34

r 512

512 x2 1 y2 5

14

5 s0, 0d 4x2 1 4y2 5 1

29.

center

radius

–3 –2 1 2 3 4 5 6

−2

−1

1

2

3

4

5

6

7

x

y

5 2

5 s2, 3d

sx 2 2d2 1 sy 2 3d2 5 4 31.

center

radius

y

x−2−6−8

−2

−4

−6

−8

2

2

5 3

5 s252, 23d

sx 152d2

1 sy 1 3d2 5 9

33.

center

radius

–1 2 4 5

−2

−1

1

2

3

4

x

y

5 2

5 s2, 1d

sx 2 2d2 1 sy 2 1d2 5 4

sx2 2 4x 1 4d 1 sy2 2 2y 1 1d 5 21 1 4 1 1

x2 2 4x 1 y2 2 2y 5 21

x2 1 y2 2 4x 2 2y 1 1 5 0 35.

center

radius

y

x−6 −5 −4 −3 −2 −1 1 2

1

−2

−3

−4

−6

−7

5 2

5 s21, 23d

sx 1 1d2 1 sy 1 3d2 5 4

sx2 1 2x 1 1d 1 sy2 1 6y 1 9d 5 26 1 1 1 9

x2 1 2x 1 y2 1 6y 5 26

x2 1 y2 1 2x 1 6y 1 6 5 0

23.

radius center5 s0, 0d5 4

x

y

1

3

5

2

3 521−1−2

−2

−3

−3

−5

−5

x2 1 y2 5 16


37.

Keystrokes:

30

30y2

y1

y 5 ±!30 2 x2

y2 5 30 2 x2

x2 1 y2 5 30

Y5 ! x 2 X,T,u x2 d ENTER

! x 2 X,T,u x2 d GRAPH

−10

−15

10

15

39.

Keystrokes:

10 2

10 2y2

y1

y 5 ±!10 2 sx 2 2d2

y2 5 10 2 sx 2 2d2

sx 2 2d2 1 y2 5 10


−4

−4

4

8

x 2 d

! x 2 X,T,u x2 dx 2 d GRAPHx2c

x2c

41. center:

vertices:

co-vertices:

major axis is x-axis so

minor axis is y-axis so

x2

161

y2

95 1

x2

42 1y2

32 5 1

b 5 3

a 5 4

x2

a2 1y2

b2 5 1

s0, 23d, s0, 3d

s24, 0d, s4, 0d

s0, 0d 43. center:

vertices:

co-vertices:

major axis is x-axis so

minor axis is y-axis so

x2

41

y2

15 1

x2

22 1y2

12 5 1

b 5 1

a 5 2

x2

a2 1y2

b2 5 1

s0, 21d, s0, 1d

s22, 0d, s2, 0d

s0, 0d

45. center:

vertices:

co-vertices:

major axis is y-axis so

minor axis is x-axis so

x2

91

y2

165 1

x2

32 1y2

42 5 1

b 5 3

a 5 4

x2

b2 1y2

a2 5 1

s23, 0d, s3, 0d

s0, 24d, s0, 4d

s0, 0d 47. center:

vertices:

co-vertices:

major axis is y-axis so

minor axis is x-axis so

x2

11

y2

45 1

x2

12 1y2

22 5 1

b 5 1

a 5 2

x2

b2 1y2

a2 5 1

s21, 0d, s1, 0d

s0, 22d, s0, 2d

s0, 0d


49. center:

major axis (vertical) 10 units

minor axis 6 units

x2

91

y2

255 1

x2

32 1y2

52 5 1

b 5 3 a 5 5

x2

b2 1y2

a2 5 1

s0, 0d 51. center:

major axis (horizontal) 20 units

minor axis 12 units

x2

1001

y2

365 1

x2

102 1y2

62 5 1

a 5 10 b 5 6

x2

a2 1y2

b2 5 1

s0, 0d

53. Vertices:

Co-Vertices:

x1

1

43

5

−1−2

−3

−5

y

2 3 5

−5−4

s0, 2d, s0, 22d

s24, 0d, s4, 0d 55. Vertices:

Co-Vertices:

x531−1

−2

−5

−3−4

y

21

3

5

−5 4

s2, 0d, s22, 0d

s0, 4d, s0, 24d

57. Vertices:

Co-Vertices:

x2

y

1

2

2

1

2 1 1

10, 432, 10, 2

432

1253

, 02, 153

, 02 59.

Vertices:

Co-Vertices:

x2

2

1

y

1

2

2 1 1

s1, 0d, s21, 0d

s0, 2d, s0, 22d

x2

11

y2

45 1

4x2 1 y2 2 4 5 0

61.

Vertices:

Co-Vertices: s0, ±!10ds±4, 0d

x2

161

y2

105 1

10x2

1601

16y2

1605

160160

y

x−2

−2

−4

2

4

2

10x2 1 16y2 2 160 5 0


63.

Keystrokes:

4 2

4 2

Vertices: s±2, 0d

y2

−3

−2

2

3

y1

y 5 ±!4 2 x2

2

y2 54 2 x2

2

2y2 5 4 2 x2

x2 1 2y2 5 4

Y5 ! x x x22 X,T,u d d4 ENTER

! x x x22 X,T,u d d4x2c GRAPH

65.

Keystrokes:

12 3

12 3

Vertices: s0, ±2!3d

y2

−6

−4

4

6

y1

y 5 ±!12 2 3x2

y2 5 12 2 3x2

3x2 1 y2 2 12 5 0


! x 2 X,T,u x2 dx2c GRAPH

67. Vertices:

Asymptotes:

Equation:

x2

92

y2

95 1

x2 2 y2 5 9

y 5 2x y 5 x

y 5 233

x y 533

x

x642

66

2

4

2

y

46

6

2

4

s3, 0d, s23, 0d

69. Vertices:

Asymptotes:

Equation: y2 2 x2 5 1

y 5 ±x

y

x−2 −1

−2

1 2

2

s0, ±1d


71. Vertices:

Asymptotes:

Equation:x2

92

y2

255 1

y 5 253

x

y 553

x

x64

y

2

4

66

6 4

4

6

s3, 0d, s23, 0d

73. Vertices:

Asymptotes:

Equation:y2

42

x2

95 1

y 5 223

x

y 523

x

y

x−4 4

−4

4

s0, ±2d

75. Vertices:

Asymptotes:

Equation:x2

12

y2

94

5 1

y 5 ±32

x

y 5 ±

321

x

−3 −2 2 3

−3

−2

2

3

x

ys±1, 0d

77. Vertices:

Asymptotes:

Equation:

x2

162

y2

45 1

2y2

41

x2

165 1

4y2

2162

x2

2165

216216

4y2 2 x2 1 16 5 0

y 5 224

x 5 212

x

y 524

x 512

x

x6

66

y

4

2

6

6

4

2

s4, 0d, s24, 0d


79.x2

162

y2

645 1 81.

y2

162

x2

645 1 83.

x2

812

y2

365 1 85.

y2

12

x2

14

5 1

87.

Keystrokes:: 16 4

: 16 4y2

y1

±!x2 2 164

5 y

x2 2 16

45 y2

x2 2 16 5 4y2

x2 2 4y2 5 16

x2

162

y2

45 1

Y5 ! x x X,T,u x2 2 d 4 d ENTER

! x x X,T,u x2 2 d 4 dx2c GRAPH

5

8

−5

−8

89.

Keystrokes:5 10 2

5 10 2y2:

y1:

±!5x2 1 102

5 y

5x2 1 10

25 y2

5x2 1 10 5 2y2

5x2 2 2y2 1 10 5 0

! x x X,T,u x2 d 4 d

x2c GRAPH

Y5 1 ENTER

! x x X,T,u x2 d 4 d1

5

8

−5

−8

91. Parabola 93. Ellipse 95. Hyperbola

97. Circle 99. Line 101.

x2 1 y2 5 20,250,000

x2 1 y2 5 45002

103. (a) (equation of circle)

of the rectangle is also the point on the circle,so y-coordinate equals:

area 5 4x!625 2 x2

area 5 2x ? 2s!625 2 x2d width 5 2s!625 2 x2d

y 5 !625 2 x2

y2 5 625 2 x2

x2 1 y2 5 625

sx, yd

x2 1 y2 5 625 (b)

x < 17.68

1500

30

−50

−5


105. Equation of ellipse

or

y 5 17.435596 < 17 feet

y2 5 304

y2

16005 0.19

452

25001

y2

16005 1

x2

25001

y2

16005 1

5x2

502 1y2

402 5 1 107.

x2

1441

y2

645 1

b 5 12 b 5 8

a 5 8 a 5 12

0 5 sa 2 12dsa 2 8d

0 5 a2 2 20a 1 96

0 5 2a2 1 20a 2 96

96 5 as20 2 ad

96 < ab

301.59

p5 ab

301.59 5 pab

A 5 pab

b 5 20 2 a

a 1 b 5 20

109. The four types of conics are circles, parabolas, ellipses, and hyperbolas.

111. An ellipse is the set of all points such that the sum of the distances between and two distinct fixed pointsis a constant.

x2

a2 1y2

b2 5 1 or x2

b2 1y2

a2 5 1

sx, ydsx, yd

113. An ellipse is a circle if the coefficients of the second degree terms are equal.

115. The central rectangle of a hyperbola can be used to sketch its asymptotes because the asymptotes are the extended diagonalsof the central rectangle.

117. is the top half of the hyperbola x2

42

y2

95 1.y 5

32!x2 2 4

Section 7.5 Graphs of Rational Functions

1. (a) (b) (c) Domain:

s2`, 1d < s1, `d

x Þ 1

x 2 1 Þ 0

x

y

−3 −2 −1 2 3 54

4

3

2

1

−4

x 0 0.5 0.9 0.99 0.999

y 2400024002402824

x 2 1.5 1.1 1.01 1.001

y 4 8 40 400 4000

x 2 5 10 100 1000

y 4 1 0.44444 0.0404 0.004

Section 7.5 Graphs of Rational Functions 387

5. (a) (b)

(c) Domain:

s2`, 23d < s23, 3d < s3, `d

x Þ 3 x Þ 23

sx 2 3dsx 1 3d Þ 0

x2 2 9 Þ 0

x

y

−2 4 6

6

4

2

−2

−4

−6

x 2 2.5 2.9 2.99 2.999

y 215002149.7214.7522.72721.2

x 4 3.5 3.1 3.01 3.001

y 1.714 3.231 15.246 150.25 1500.2

x 4 5 10 100 1000

y 1.714 0.938 0.330 0.030 0.003

7.

Domain:

Vertical asymptote:


y 5 0

x 5 0

s2`, 0d < s0, `d

x Þ 0

x2 Þ 0

f sxd 55x2

9.

Domain:

Vertical asymptote:

Horizontal asymptote: since the degree of thenumerator is equal to the degree of the denominator andthe leading coefficients are 1.

y 5 1

x 5 28

s2`, 28d < s28, `d

x Þ 28

x 1 8 Þ 0

f sxd 5x

x 1 8

3. (a) (b) (c) Domain:

s2`, 3d < s3, `d

x Þ 3

x 2 3 Þ 0

x

y

−1 2 6 71 54

4

3

1

5

6

−2

x 2 2.5 2.9 2.99 2.999

y 1 0 299829828

x 4 3.5 3.1 3.01 3.001

y 3 4 12 102 1002

x 4 5 10 100 1000

y 3 2.5 2.143 2.010 2.001

11.

Domain:

Vertical asymptote:

Horizontal asymptote: since the degree of thenumerator is equal to the degree of the denominator andthe leading coefficient of the numerator is 2 and the lead-ing coefficient of the denominator is 3.

y 523

t 5 3

s2`, 3d < s3, `d

t Þ 3

3t 2 9 Þ 0

g std 52t 2 53t 2 9

13.

Domain:

Vertical asymptote:

Horizontal asymptote: since the degree of thenumerator is equal to the degree of the denominator andthe leading coefficient of the numerator is and theleading coefficient of the denominator is 23.

25

y 553

x 513

s2`, 13d < s13, `d

13 Þ x

1 Þ 3x

1 2 3x Þ 0

y 53 2 5x1 2 3x


19.

Domain:

Vertical asymptotes:


y 5 1

x 5 1, x 5 21

s2`, 21d < s21, 1d < s1, `d

x Þ 1 x Þ 21

sx 2 1dsx 1 1d Þ 0

x2 2 1 Þ 0

y 5x2 2 4x2 2 1

21.

Domain:

Vertical asymptote:

Horizontal asymptote: since the degree of thenumerator is equal to the degree of the denominator andthe leading coefficients are 1.

y 5 1

z 5 0

s2`, 0d < s0, `dz Þ 0

gszd 5zz

?11

22z

5z 2 2

z

g szd 5 1 22z

23.

Domain:

Vertical asymptote:

Horizontal asymptote: none since the degree of thenumerator is greater than the degree of the denominator.

x 5 0

s2`, 0d < s0, `d

x Þ 0

gsxd 5 2x 14x

5xx

?2x1

14x

52x2 1 4

x25. matches with graph (d).

Vertical asymptote:

Horizontal asymptote:y 5 0

x 5 21

x 1 1 5 0

f sxd 52

x 1 1

27. matches with graph (b).

Vertical asymptote:


x 5 1

x 2 1 5 0

f sxd 5x 2 2x 2 1

29. (d) 31. (a)

15.



y 5 0

t 5 0, t 5 1

s2`, 0d < s0, 1d < s1, `d

t Þ 1

t Þ 0 t 2 1 Þ 0

tst 2 1d Þ 0

g std 53

tst 2 1d 17.

Domain:

no real solution

Vertical asymptote: none


y 5 2

s2`, `d

x2 1 1 Þ 0

y 52x2

x2 1 1


39.

intercept:

intercept: none, numerator is never zero

Vertical asymptote:


–2 1 6

−4

−3

−2

−1

1

2

3

4

x

y

y 5 0

x 5 2

2 2 x 5 0

x-

gs0d 51

2 2 05

12

y-

g sxd 51

2 2 x41.

intercept:

intercept:

Vertical asymptote:


x

y

−6 −2

4

2

−4

−2

2 4

y 5 0

x 5 24

xsx 1 4d 5 0

x2 1 4x 5 0

0 53

x 1 4; none

0 53x

x2 1 4x5

3xxsx 1 4dx-

y 53s0d

02 1 4s0d 5 undefined, noney-

y 53x

x2 1 4x

33.

intercept:

intercept: none, numerator is never zero.

Vertical asymptote:


y

x2 4 6 8

8

6

4

2

−2

y 5 0

x 5 0

x-

gs0d 550

5 undefined, noney-

g sxd 55x

35.

intercept:


Vertical asymptote:


y

x6 8 10 12

8

6

4

2

−22

−4

−6

−8

y 5 0

x 5 4

x 2 4 5 0

x-

gs0d 55

0 2 45 2

54

y-

g sxd 55

x 2 4

37.

intercept:


Vertical asymptote:


y 5 0

x 5 2

x 2 2 5 0

x-

f s0d 51

0 2 25 2

12

y-

f sxd 51

x 2 2

–2 3 4 5 6

−4

−3

−2

−1

1

2

3

4

x

y


45.

intercept: undefined, none.

intercept:

Vertical asymptote:

Horizontal asymptote: since the degree of thenumerator is equal to the degree of the denominatorand the leading coefficient of the numerator is 2 andthe leading coefficient of the denominator is 1.

y

x

4

2−8 4−6 6−4 8

6

8

y 5 2

x 5 0

x 5 22

2x 1 4 5 0x-

y 52s0d 1 4

05y-

y 52x 1 4

x 47.

intercept:

intercept:

Vertical asymptote: none, has no real solutions.

Horizontal asymptote: since the degree of thenumerator is equal to the degree of the denominatorand the leading coefficient of the numerator is 2 andthe leading coefficient of the denominator is 1.

–4 –3 –2 –1 1 2 3 4

–3

–2

3

4

5

x

y

y 5 2

x2 1 1 5 0

x 5 0x-

y 52s0d2

02 1 15 0y-

y 52x2

x2 1 1

49.

intercept:


Vertical asymptote: none,

no real solution


y 5 0

x2 1 1 Þ 0

x-

y 54

02 1 15 4y-

y 54

x2 1 1

y

x1 2 3 4−1−2−3−4

1

−2

−3

2

5

43.

intercept:

intercept:

none, since is undefined.

Vertical asymptote:

Horizontal asymptote: since the degrees are equaland the leading coefficient of the numerator is 3 and theleading coefficient of the denominator is 1.

y 5 3

u 5 3

usu 2 3d 5 0

u2 2 3u 5 0

hs0d 0 5 u,

0 5 3u

0 53u

u 2 3

0 53u2

u2 2 3u5

3u2

usu 2 3dx-

hs0d 53s0d2

02 2 3s0d 5 undefined, noney-

h sud 53u2

u2 2 3u

u

y

−4 −2 2 4 6 8 10

10

8

6

4

2

−2

−4


51.

intercept:

intercept:

Vertical asymptote:


t 5 0

23

5 t

2 5 3t

0 5 3t 2 2

0 5 3 22t

x-

gs0d 5 3 220

5 undefined, noney-

g std 5 3 22t

y

t1 2 3 4−1−2−3−4

2

−1

1

4

5

53.

intercept:

intercept:

Vertical asymptote:

Horizontal asymptote: since the degree of the numerator is less than the degree of the denominator.y 5 0

x 5 2 x 5 22

sx 2 2dsx 1 2d 5 0

x2 2 4 5 0

x 5 2, x 5 22

0 5 x

0 5 2x

0 5 2x

x2 2 4x-

y 520

02 2 45 0y-

y

x1−3−4

−3

−4

1

4

3

2

−1

−2

y 5 2x

x2 2 4

55.

intercept:

intercept:



Horizontal asymptote: since the degree of the numerator is equal to the degree of the denominator and the leading coefficient of the numerator is 3 and the leading coefficient of the denominator is 1.

y 5 3

x 5 2 x 5 2

x 2 2 5 0 x 2 2 5 0

sx 2 2dsx 1 1d 5 0

x2 2 x 2 2 5 0

x 5 0

x2 5 0

3x2 5 0x-

y

x−6 −4 4 6

2

4

6

8

10

12

8 10

y 53s0d2

02 2 0 2 25

022

5 0y-

f sxd 53x2

x2 2 x 2 2


63.

Domain:


Horizontal asymptote:

Keystrokes:

6 1

y 5 0

s2`, `d

−6

−1

6

7 t 2 1 1 Þ 0

f std 56

t2 1 1

Y5 x x

GRAPH4 1X,T,u x2

59.

Domain:

Vertical asymptote:


Keystrokes:

3 2y1

y 5 0

x 5 22

s2`, 22d < s22, `d

x Þ 22

−8

−6

4

6 x 1 2 Þ 0

f sxd 53

x 1 2

Y5 4 x 1

x

GRAPHX,T,

61.

Domain:

Vertical asymptote:


Keystrokes:

3 1

y 5 1

x 5 1

s2`, 1d < s1, `d x Þ 1

−7

−4

8

6 x 2 1 Þ 0

h sxd 5x 2 3x 2 1

x 2

x

4 x 2 GRAPHX,T, X,T,Y5 u u

u

57.

intercept:

intercept:


Horizontal asymptote: since the degrees are equal and the leading coefficients are 1.

f sxd 5sx 2 2dsx 1 2dsx 2 5dsx 1 2d 6 gives a hole in graph at x 5 22

y 5 1

x 5 5 x 5 22 → hole in graph

sx 2 5dsx 1 2d 5 0

x2 2 3x 2 10 5 0

x 5 2 undefined at x 5 22

0 5 sx 2 2dsx 1 2d 0 5 x2 2 4

0 5x2 2 4

x2 2 3x 2 10x-

x

y

−2 42 6 8 10

6

4

2

−2

−4

−6

f s0d 502 2 4

02 2 3s0d 2 105

410

525

y-

f sxd 5x2 2 4

x2 2 3x 2 10


65.

Domain:

Vertical asymptote:


Keystrokes:

2 1

y 5 2

x 5 0

s2`, 0d < s0, `d x Þ 0

−9

−2

9

10 x2 Þ 0

y 52sx2 1 1d

x2

Y5 x x

GRAPH41x x

73. (a) is the horizontal asymptote, since the degree of the numerator is less than the degree of the denominator. The meaning in the context of the problem is that the chemical is eliminated from the body.

(b) Keystrokes: 2 4 25

Maximum occurs when t < 2.5.0

05

0.5C 5 0

Y5 4 x 1

x

GRAPH

X,T,u x2 x2X,T,u

X,T,u X,T,u x2

67.

Domain:


Horizontal asymptote: since the degree of the numerator is less than the degree of the denominator.

Keystrokes: 3 1 2 or

2 d6d 4 ss4

y 5 0

x 5 0, x 5 2x2 2 2x 5 0, xsx 2 2d 5 0,

s2`, 0d < s0, 2d < s2, `d x Þ 2

−7

−5

8

5 x Þ 0 x 2 2 Þ 0

y 53x

11

x 2 25

4x 2 6x2 2 2x

y 53x

11

x 2 2

69. Reduce to lowest terms.

Keystrokes: 4 2 2

There is no vertical asymptote because the fraction is not reduced to lowest terms.

gsxd 54 2 2xx 2 2

52s2 2 xd

x 2 25 22 −3

−3

3

1gsxd

71. (a)

(b)

C 52500 1 0.50s10,000d

10,0005 $0.75

C 52500 1 0.50s1000d

10005 $3

C 52500 1 0.50x

x, 0 < x

Average cost 5Cost

Number of units(c) Keystrokes:

2500 .5

Horizontal asymptote

since the degree of the numerator is equal tothe degree of the denominator and the leading coefficientof the numerator is 0.50 and the leading coefficient of thedenominator is 1. As the number of units producedincreases, the average cost is approximately $0.50.

00

50

2500

C 5 $0.50

Y5 x x

GRAPH42 x 2

xX,T, X,T,

Y5

Y 5

x x

GRAPH

GRAPH

4 1 4 2

22

X,T, X,T,

X,T,X,T,X,T, x2

u u

uuu

u u

Y5 x 1

x

4 GRAPHX,T, X,T,u u


81. (c)

x-intercept:

x 548.44.79

5 10.10

4.79x 5 48.4

0 5 48.4 2 4.79x

0 548.4 2 4.79x

1 2 0.13x

x Þ 7.69

x Þ21

20.13

20.13x Þ 21

Domain: 1 2 0.13x Þ 0

y 548.4 2 4.79x

1 2 0.13x

since the degrees are equal.

(the excluded value of the domain)

Vertical asymptote: x < 7.69

y < 36.85

Horizontal asymptote: y 524.7920.13

(d) Keystrokes:

48.4 4.79 1 .13

Plot

in

1 then enter 1, 2, 3, 4, 5, 6, in and enter 50.1, 51.9, 54.8, 59.3, 73.6, 78.7, in

1

The model appears to be accurate for the restricted domain.

—CONTINUED—

85

6.545

0.5

L2.L1

s1, 50.1d, s2, 51.9d, s3, 54.8d, s4, 59.3d, s5, 73.6d, s6, 78.7d

x 2 X,T,u d 4 x 2 X,T,u d GRAPH

STAT

STAT PLOT ON GRAPH

Y5

75. (a) answers will vary.

(b)

P 5 21x 1400

x 2 400

x5 y

P 5 2sl 1 wd 400 5 x ? y

P 5 2l 1 2w A 5 x ? y

x

y

(c) Domain:

(d) Minimum perimeter:

Keystrokes:

2 400

00

15

1500

20 units 3 20 units

x > 0 or s0, `d

Y5 x 1 4

x

GRAPHX,T,u X,T,u

77. y 52sx 1 1d

x 2 379. y 5

x 2 6sx 2 4dsx 1 2d



1. P varies directly as the cube of t. P 5 kt 3 3. z varies inversely as the square of s. z 5ks2

5.

y 5 6 3!x

6 5123!8

5 k

12 5 k 3!8

y 5 k 3!x 7.

T 51

18rs2

118

5 k

5000

90,0005 k

5000 5 ks0.09ds1000d2

T 5 krs2

9.

x4−4 2

y

−2

2

−2

6

y > 4 11.

x41

y

3

2

1

2

1

1 3

x ≥ 2

x 2 2 ≥ 0

83. An asymptote of a graph is a line to which the graphbecomes arbitrarily close as or increases withoutbound.

|y||x|85. No, not when the domain is all reals. For example,

has no vertical asymptote.f sxd 51

x2 1 1

81. —CONTINUED—

(e) The models are not accurate for the years before 1991 and after 1996. Use the quadratic model to estimate the value ofthe shipment in 1998, because the rational function evaluated at is negative.x 5 8


13.

x21

3

y

1

1

12

2x 1 y < 1 or y < 22x 1 1 15.

y

4

2

x−2 2 3−1 1

−2

3

−1

2x 1 3

4≤ y

2x 1 3 ≤ 4y

2x 1 1 ≤ 4y 2 2

2sx 2 1d ≤ 4y 2 2

17.

Keystrokes:

12 3 2

7 10 1 1

y ≤ 12 232x

DRAW

X,T,u

, ENTER

2 x 4 d

d

20

10

−4

−4

19.

Keystrokes:

7 1 1 10

y ≥ 2x

x 1 y ≥ 0

DRAW

4

4

−4

−4

21.

vertex5 s4, 213d

5 sx 2 4d2 2 13

5 sx2 2 8x 1 16d 1 3 2 16

f sxd 5 x2 2 8x 1 3 23.

vertex5 s14, 23

8 d 5 2su 2

14d2

1238

5 2su2 212u 1

116d 1 3 2

18

5 2su2 212ud 1 3

h sud 5 2u2 2 u 1 3

25.

x-intercepts vertex

s24, 216d x 5 0 x 5 28

y 5 sx 1 4d2 2 16 0 5 xsx 1 8d

y 5 x2 1 8x 1 16 2 16 0 5 x2 1 8x

y 5 x2 1 8x

−2−4−6−10 2

−16

4

8

x

y

( 8, 0)− (0, 0)

( 4, 16)− −

Y5

x Y-VARSx2c

Y-VARSx , d ENTER

Y5 x2c X,T,u


27.

x-intercepts vertex

x

y

−2 2 4 6

−2

−4

2

4

(1, 0) (5, 0)

(3, 4)−

s3, 24d x 5 5 x 5 1

y 5 sx 2 3d2 2 4 0 5 sx 2 5dsx 2 1d

y 5 sx2 2 6x 1 9d 1 5 2 9 0 5 x2 2 6x 1 5

y 5 x2 2 6x 1 5

29.

Vertical shift 3 units up

x

y

−2−3 −1 21 3

1

5

2

6

4

h sxd 5 x2 1 3 31.

Horizontal shift 2 units left


x

y

−2−3 −1−4−5 1

2

1

−2

−3

−4

h sxd 5 sx 1 2d2 2 3 33. vertex:

y 5 22sx 2 3d2 1 5

y 5 asx 2 hd2 1 k

a 5 22

s3, 5d

35. vertex: y-intercept:

2 5 a

8 5 as4d

3 5 as4d 2 5

3 5 as0 2 2d2 2 5

y 5 asx 2 2d2 2 5

y 5 2sx 2 2d2 2 5 or y 5 2x2 2 8x 1 3 y 5 asx 2 hd2 1 k

s0, 3ds2, 25d:

37. Parabola: vertex: passes through the point

y 5116sx 2 5d2 1 0 or y 5

116x2 2

58x 1

2516

116 5 a

1 5 as16d

1 5 as1 2 5d2 1 0

y 5 asx 2 hd2 1 k

s1, 1ds5, 0d;

39. (c) matches 4x2 1 4y2 5 81 41. (a) matches y2

42 x2 5 1


43. (b) matches y 5 2x2 1 6x 2 5 45.

parabola

x2

25 y

x2 5 2y

x2 2 2y 5 0

x32

5

y

2

3

4

1

1

1123

47.

circle

y

x−12 −4

−4

4

4

−12

12

12

x2 1 y2 5 64 49.

parabola

y

x2 4 6 8

2

4

6

8

y 5 sx 2 6d2 1 1

51.

ellipse

x6

66

y

4

6 2

4

6

2

x2

251

y2

45 1 53.

circle

x221

2

y

1

12

1

2

x2

94

1y2

94

5 1

4x2

91

4y2

95 1

4x2 1 4y2 2 9 5 0

55. Parabola: vertex: passes through the point

y 5116sx 2 5d2 1 0 or y 5

116x2 2

58x 1

2516

116 5 a

1 5 as16d

1 5 as1 2 5d2 1 0

y 5 asx 2 hd2 1 k

s1, 1ds5, 0d;


59. Circle: center: ; radius:

x2 1 y2 5 400

20s0, 0d

57. Ellipse: vertices: co-vertices:

x2

41

y2

255 1

s22, 0d, s2, 0ds0, 25d, s0, 5d;

61. Hyperbola: vertices: asymptotes:

a 5 3 b 532

x2

92

y2

94

5 1x2

32 2y2

1322

2 5 1x2

a2 2y2

b2 5 1

y 5 212

x, y 512

xs23, 0d, s3, 0d;

63.

(b)

x

y

4

2

−2

−4

−6

4

6

8 10

f sxd 55

x 2 665.

(a)

x

y

4

8

12

16

20

−4−4 4 8 12 16−8

f sxd 56x

x 2 5

67.

y-intercept: undefined; none

x-intercept:

none

vertical asymptote:

horizontal asymptote: since the degree of thenumerator is less than the degree of the denominator.

y 5 0

x 5 0

x2 5 0

y

x−1−2

−1

−2

−3

−4

−5

1 2

0 5 25

0 525x2

f s0d 52502 5

f sxd 525x2 69.

y-intercept:

x-intercept:

vertical asymptote:

horizontal asymptote: since the degrees are equaland the leading coefficient of the numerator is 3 and theleading coefficient of the denominator is 1.

y 5 3

x 5 2

y

x−2−4−6

−4

2

4

6

8

10

4 6 8

22 5 x

0 5 3x 1 6

0 53x 1 6x 2 2

P s0d 53s0d 1 6

0 2 25 23

P sxd 53x 1 6x 2 2


71.

y-intercept:

x-intercept:

vertical asymptote:

horizontal asymptote: since the degrees are equaland the leading coefficient of the number is 1 and theleading coefficient of the denominator is 21.

y 5 21

x 5 1

y

x2 3 4−1−2

−2

−3

−4

2

22 5 x

0 5 2 1 x

0 52 1 x1 2 x

g s0d 52 1 01 2 0

5 2

g sxd 52 1 x1 2 x

73.

y-intercept:

x-intercept:

vertical asymptote: none


y 5 0

x2 1 1 Þ 0

y

x−1−2−3 1 2 3

−0.75

0.25

0.50

0.75

0 5 x

0 5x

x2 1 1

f s0d 50

02 1 15 0

f sxd 5x

x2 1 1

75.

y-intercept:

x-intercept:

none

vertical asymptote:


y 5 0

x 5 1

y

x−2 −1 2 3 4

1

3

4

5

6

0 5 4

0 54

sx 2 1d2

h s0d 54

s0 2 1d2 5 4

h sxd 54

sx 2 1d277.

y-intercept:

x-intercept:

vertical asymptote:


y 5 0

x 5 1 x 5 21

x2 2 1 5 0

y

x2 3

2

3

−1

−2

−3

0 5 x

0 5x

x2 2 1

y 50

02 2 15 0

y 5x

x2 2 1


79.

y-intercept:

x-intercept:

vertical asymptote:


y 5 2

x 5 2 x 5 22

x2 2 4 5 0

y

x−3−6−9 3 6 9

3

2

1

0 5 x

0 5 2x2

0 52x2

x2 2 4

y 52s0d2

02 2 45 0

y 52x2

x2 2 481.

y-intercept:

x-intercept:

none

vertical asymptote:

horizontal asymptote: since the degree of thenumber is less than the degree of the denominator.

y 5 0

x 5 21

y

x−8 42−4

2

−6

4

6

0 5 1

0 51

x 1 1

y 50 2 4

02 2 3s0d 2 45

2424

5 1

y 5x 2 4

x2 2 3x 2 45

x 2 4sx 2 4dsx 1 1d 5

1x 1 1

y 5x 2 4

x2 2 3x 2 4

83. Vertical asymptote:


Zero of the function:

y 53x

x 2 4

x 5 0

y 5 3

x 5 485.

150 pounds 5 F

6 51

25F

125

5 k

4

1005 k

4 5 ks100d

d 5 kF

87.

x < 945 units

x 5 944.91118

x 55000!28

5000 5 k

1000 5k

!25

x 5k!p

89.

Ordered pair solutions:


s12, 11d, s8, 15d

s0, 20d, s25, 0d, s10, 12d,

y ≥ 2.8x 1 20

10y ≥ 28x 1 200

8x 1 10y ≥ 200

8x 1 10y ≥ 200

5 10 15 20 30x

20

25

y

25

5

10

15

30


91. (a) Keystrokes:

10 3 6

(b)

(c)

(d)

The ball is 31.9 feet from the child when it hits the ground.

x 5 25s23 ± !11.4d 5 15 ± 5!11.4 5 31.9

x 523 ± !11.4

215

x 523 ± !9 1 2.4

215

x 5

23 ± !32 2 4121

102s6d

2121

102

0 52110

x2 1 3x 1 6

5 15

523

215

5 23

2121

102

x 5 2b

2a

y 5 6 feet

y 5 0 1 0 1 6

y 5 21

10s0d2 1 3s0d 1 6

Y5 x2c X,T,u x2 4 1 X,T,u 1 GRAPH

5 28.5 feet

5 222.5 1 45 1 6

5 21

10s225d 1 45 1 6

y 51

10s15d2 1 3s15d 1 6

93.

x2 1 y2 5 50002

x2 1 y2 5 r2

0

0

32

32

95. (a)

Ns25d 520s4 1 3 ? 25d1 1 0.05s25d 5

15802.25

< 702.2 thousand

Ns10d 520s4 1 3 ? 10d1 1 0.05s10d 5

6801.5

< 453.3 thousand

Ns5d 520s4 1 3 ? 5d1 1 0.05s5d 5

3801.25

5 304 thousand (b) The population is limited by the horizontal asymptotethousand fish.N 5 1200



1. S 5kx2

y2.

v 514!u

14 5 k

32 5 k!36

v 5 k!u 3.

x4−4 2

y

−2

2

−2

6

y < 4

4.

y <32

x 2 3

y <2322

x 16

22

22y > 23x 1 6

y

4

2

x4−2 2 6

−2

3x 2 2y > 65.

10x 1 7y ≤ 35

7y 5 210x 1 35

y 5210

7x 1 5

y 2 5 5210

7x

y 2 5 5210

7sx 2 0d

m 55 2 0

0 272

?22

51027

6.

x & y-intercept

x-interceptss0, 0d, s4, 0d; 0, 4 5 x

2 ± 2 5 x

±2 5 x 2 2

4 5 sx 2 2d

28 5 22sx 2 2d2

0 5 22sx 2 2d2 1 8

s0, 0d; y 5 0

y 5 22s4d 1 8

y 5 22s0 2 2d2 1 8

vertex 5 s2, 8dy

8

6

x−2 2 6

2

4

(0, 0)

(2, 8)

(4, 0)

y 5 22sx 2 2d2 1 8


7.

Circle

4x

2

y

4

2

24

2

4

x2 1 y2 5 9 8.

Ellipse

4x

2

y

3

2

1

12 14

3

2

x2

91

y2

165 1

9.

Hyperbola

x2 4 6

y

66

2

2

4

46

4

6

x2

92

y2

165 1 10.

Parabola

y

x1 3 4 6

1

2

3

4

2 5

y 5 sx 2 3d2

11. Circle with center at and radius 5

x2 1 y2 5 25

x2 1 y2 5 52

x2 1 y2 5 r2

s0, 0d 12. Parabola with vertex and passing through

8 5 a

64 5 as8d

s9 2 1d2 5 af6 2 s22dg

sy 2 1d2 5 8sx 1 2d sy 2 kd2 5 asx 2 hd

s6, 9ds22, 1d

13.x2

91

y2

1005 1 14.

or

x2 2 4y2 5 9

x2

92

4y2

95 1

x2

92

y2

94

5 1


15.

x-intercept: none, numerator is never zero

y-intercept:

Vertical asymptote:


y 5 0

x 5 3

x 2 3 5 0

f s0d 53

0 2 35 21

x86

y

2

4

4

6

−2

−4

−6

2

f sxd 53

x 2 3

16.

x-intercept:

y-intercept:



y 5 0

x 5 5 x 5 23

x 2 5 5 0 x 1 3 5 0

y 5 0

x 5 0

x86

y

2

4

4

6

−2

−4

−6

−2

f sxd 53x

x2 2 2x 2 155

3xsx 2 5dsx 1 3d

17.

V 5 240 cubic meters 180 5 K

V 5180.75

.75 5180V

1 5K

180

.75V 5 180 P 5180V

P 5KV

18.


2x 1 3y ≤ 2400

250 500 1000 1250x

1000

y

250

500

750

1250

750

20x 1 30y ≤ 24,000

19.

passengers will produce a maximum revenue n 5 120

R 5 21

20sn 2 120d2 1 720

R 5 21

20sn2 2 240n 1 14,400d 1 720

R 5 21

20sn2 2 240nd, 80 ≤ n ≤ 160


CHAPTER 7 Linear Models and Graphs of Nonlinear Models

SECTION 7.1 Variation

1. For some there correspondsmore than one value of y.

x

x

y

−1 1 3 542

4

2

1

3

−1

−2

2. For each there correspondsexactly one value of y.

x

x

y

−1−2 1 3 42

4

2

1

3

−1

−2

3.

Domain: s2`, `df sxd 5 x2 2 4x 1 9 4.

Domain:

s2`, 0d < s0, `d x Þ 0

x2 Þ 0 x2 1 1 Þ 0

x2sx2 1 1d Þ 0

hsxd 5x 2 1

x2sx2 1 1d

5.

Yes, graphs are the same.

−4

−20

5

50

5 s2x 2 3dsx 1 3dsx 2 3d

f sxd 5 2x3 2 3x2 2 18x 1 27 6.

5 2x3 2 3x2 2 18x 1 27

5 2x3 2 18x 2 3x2 1 27

s2x 2 3dsx 1 3dsx 2 3d 5 s2x 2 3dsx2 2 9d

7.

x2 2 9 5 sx 2 3dsx 1 3d

0

218x 1 27

218x 1 27

2x3 2 3x2

2x 2 3 ) 2x3 2 3x2 2 18x 1 27

x2 2 9

2x3 2 3x2 2 18x 1 272x 2 3

8.

0

23x2 1 27

23x2 1 27

2x3 2 18x

2x3 2 3x2 2 18x 1 27

x2 2 95 x2 2 9 ) 2x3 2 3x2 2 18x 1 27

2x 2 3


1. index radicand5 6x5 4,4!6x 2. in radical form is n!a.a1n

3.

x <32

22x22

<2322

22x > 23

7 2 2x 2 7 > 4 2 7

7 2 2x > 4

7 2 3x 1 x > 4 2 x 1 x

7 2 3x > 4 2 x 4.

x < 5

2x2

<102

2x < 10

2x 2 8 1 8 < 2 1 8

2x 2 8 < 2

2x 1 12 2 20 < 2

2sx 1 6d 2 20 < 2 5.

x <125

5x5

<125

5x < 12

2x 1 3x < 12

121x6

1x42 < s1d12

x6

1x4

< 1

6.

x ≤ 211

2x21

≤1121

2x ≥ 11

5 2 5 2 x ≥ 16 2 5

5 2 x ≥ 16

215 2 x2 2 ≥ s8d2

5 2 x

2≥ 8 7.

1 < x < 5

22 1 3 < x 2 3 1 3 < 2 1 3

22 < x 2 3 < 2

|x 2 3| < 2

9.

5 4 1 h

5hs4 1 hd

h

54h 1 h2

h

54 1 4h 1 h2 2 3 2 4 1 3

h

f s2 1 hd 2 f s2d

h5

s2 1 hd2 2 3 2 s22 2 3dh

f sxd 5 x2 2 3 10.

523

7s7 1 hd

523h

7hs7 1 hd

521 2 21 2 3h

7hs7 1 hd

521 2 3s7 1 hd

7hs7 1 hd

5

37 1 h

237

h?

7s7 1 hd7s7 1 hd

f s2 1 hd 2 f s2d

h5

3s2 1 hd 1 5

23

2 1 5h

f sxd 53

x 1 5

SECTION 7.2 Graphs of Linear Inequalities


11.

Keystrokes:

5

Reflection in the x-axis

2

3

−2

−3

g sxd 5 2x5

Y5 X,T,u > GRAPHx2c

12.

Keystrokes:

5

Reflection in the y-axis

2

3

−2

−3

g sxd 5 s2xd5

Y5 X,T,u GRAPHx d >x2c

8.

or

or x > 8 x < 2

x 2 5 1 5 > 3 1 5 x 2 5 1 5 < 23 1 5

x 2 5 > 3 x 2 5 < 23

|x 2 5| > 3

9.

Keystrokes:

5 2


2

5

−4

−4

g sxd 5 x5 2 2

Y5 X,T,u > 2 GRAPH

10.

Keystrokes:

2 5

Horizontal shift 2 units to the right

2

5

−2

−1

g sxd 5 sx 2 2d5

Y5 X,T,u GRAPHx 2 d >

SECTION 7.3 Graphs of Quadratic Functions

1.

(Recall then multiply byFOIL.)

sx 1 bd2 5 sx 1 bdsx 1 bd

sx 1 bd2 5 x2 1 2bx 1 b2 2.

To complete the square, take one-half of and square it.

s12 bd2

b

x2 1 5x 1254

3.

5 211x

s4x 1 3yd 2 3s5x 1 yd 5 4x 1 3y 2 15x 2 3y 4.

5 241v

s215u 1 4vd 1 5s3u 2 9vd 5 215u 1 4v 1 15u 2 45v

5.

5 6x2 1 9

2x2 1 s2x 2 3d2 1 12x 5 2x2 1 4x2 2 12x 1 9 1 12x 6.

5 24

5 y2 2 y2 2 4y 2 4 1 4y

y2 2 sy 1 2d2 1 4y 5 y2 2 sy2 1 4y 1 4d 1 4y


1. illustrates the Additive Inverse Property.s3t 1 1d 2 s3t 1 1d 5 0

2. illustrates the Distributive Property.3xsx 2 2d 5 3x2 2 6x

3. illustrates the Associative Property of Multiplication.2s3yd 5 s2 ? 3dy

4. illustrates the Commutative Property of Addition.23 1 x 5 x 2 3

5. sx2 ? x3d4 5 sx213d4 5 sx5d4 5 x20 6. 422 ? x2 5x2

42 5x2

16

7.15y23

10y2 55 ? 3y2322

5 ? 25

32

y25 53

2y5 8. 13x2

2y 222

5 1 2y3x22

2

54y2

9x4

9.3x2y3

18x21y2 53x22s21dy322

6 ? 35

1x3y1

65

x3y6

10. sx2 1 1d0 5 1

7.

5 2|x|y!6y

!24x2y3 5 !4 ? 6 ? x2 ? y2 ? y 8.

5 3 3!5

5 3!33 ? 5

3!9 ? 3!15 5 3!9 ? 15

9. s12a24b6d1y2 5!12b6

a 4 5!4 ? 3 ? b6

a 4 52b3

a2!3 10. s161y3d3y4 5 161y3?3y4 5 161y4 5 4!16 5 2

11.

Reject

t 5 !5 < 2.24 seconds

2!5

t 5 ±!5

t 2 5 5

16t 2 5 80

0 5 216t 2 1 80

s0 5 80 h 5 0

h 5 216t 2 1 s0 12.

Reject

t 5!25 ? 6

45

5!64

< 3.06 seconds

2!15016

t 5 ±!15016

t 2 515016

16t 2 5 150

0 5 216t 2 1 150

s0 5 150 h 5 0

h 5 216t 2 1 s0

SECTION 7.4 Conic Sections


12. Verbal model: 1500

Labels: Original number of persons

Number of persons now

Equation:

discard x 1 3 5 18

x 5 218 x 5 15

0 5 sx 2 15dsx 1 18d

0 5 x2 1 3x 2 270

0 5 1500x2 1 4500x 2 405,000

135,000x 1 405,000 2 135,000x 5 1500x2 1 4500x

135,000sx 1 3d 2 135,000x 5 1500xsx 1 3d

135,000

x2

135,000x 1 3

5 1500

5 x 1 3

5 x

52

Total costOriginalnumberof persons

Total costOriginalnumber ofpersons now

11. Verbal model: 8

Labels: Original number of persons

Number of persons now

Equation:

people

discard people x 1 3 5 12

x 5 9x 5 212

0 5 sx 1 12dsx 2 9d

0 5 x2 1 3x 2 108

0 5 8x2 1 24x 2 864

288x 1 864 2 288x 5 8x2 1 24x

288sx 1 3d 2 288x 5 8xsx 1 3d

288

x2

288x 1 3

5 8

5 x 1 3

5 x

52

Total costOriginalnumberof persons

Total costOriginalnumber ofpersons now

1. Leading coefficient in is 7. It is the coefficient of the term.ax2-

7x2 1 3x 2 4 2. Degree is 5.

sx 4 1 3dsx 2 4d 5 x5 2 4x4 1 3x 2 12

SECTION 7.5 Graphs of Rational Functions


11.

Labels:

Equation:

Height 5 8 meters

Base 5 20 meters

x 2 12 5 8 meters

x 5 20 meters x 5 28

0 5 sx 2 20dsx 1 8d

0 5 x2 2 12x 2 160

80 512

x2 2 6x

A 512

? x ? sx 2 12d

Height 5 x 2 12

Base 5 x

Area 5 A 5 80

3. Many answers

For some there corresponds more than one value of y.x

x

y

−1−2 2 31

3

1

−1

−3

4. Many answers

For each there corresponds exactly one value of y.x

x

y

−1 1 3 42

4

1

3

−1

−2

5. 22x5s5x3d 5 210x8 6. 3xs5 2 2xd 5 15x 2 6x2

7.

5 4x2 2 60x 1 225

s2x 2 15d2 5 s2x 2 15ds2x 2 15d 8.

5 21x2 2 16x 2 20

s3x 1 2ds7x 2 10d 5 21x2 1 14x 2 30x 2 20

9.

5 x2 2 y2 1 2x 1 1

5 x2 1 2x 1 1 2 y2

fsx 1 1d 2 yg fsx 1 1d 1 yg 5 sx 1 1d2 2 y2 10.

5 x3 1 27

sx 1 3dsx2 2 3x 1 9d 5 x3 2 3x2 1 9x 1 3x2 2 9x 1 27

AreaVerbal Model:

512

? Base ? Height 12.

Labels:

Equation:


x 5 255 x 5 15 inches

x 1 55 5 0 x 2 15 5 0

0 5 sx 1 55dsx 2 15d

0 5 x2 1 40x 2 825

825 5 x2 1 40x

825 5 x ? x 1 4s10 ? xd

Area of one side 5 10 ? x

Area of bottom 5 x ? x

Surface area 5 825

Surfacearea

VerbalModel:

1 4 ?5Area ofbottom

Area of one side

C H A P T E R 8Systems of Equations

Section 8.1 Systems of Equations . . . . . . . . . . . . . . . . . . . .407

Section 8.2 Linear Systems in Two Variables . . . . . . . . . . . . .416

Section 8.3 Linear Systems in Three Variables . . . . . . . . . . . . .424

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .433

Section 8.4 Matrices and Linear Systems . . . . . . . . . . . . . . . .437

Section 8.5 Determinants and Linear Systems . . . . . . . . . . . . .446


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .469

407

C H A P T E R 8Systems of Equations

Section 8.1 Systems of EquationsSolutions to Odd-Numbered Exercises

1. (a)

Solution

10 5 10

22 1 12 5?

10

22s1d 1 3s4d 5?

10

9 5 9

1 1 2s4d 5?

9

s1, 4d (b)

Not a solution

1 Þ 9

3 2 2 5?

9

3 1 2s21d 5?

9

s3, 21d 3. (a)

Not a solution

20 Þ 46

6 1 14 5?

46

22s23d 1 7s2d 5?

46

s23, 2d (b)

Solution

0 5 0

26 1 6 5?

0

3s22d 1 6 5?

0

46 5 46

4 1 42 5?

46

22s22d 1 7s6d 5?

46

s22, 6d

5. (a)

Not a solution

32 Þ 22.5

3s8d 1 2s4d 5?

22.5

12 5 12

4s8d 2 5s4d 5?

12

s8, 4d (b)

Solution

22.5 5 22.5

3s12d 1 2s22d 5

?22.5

12 5 12

4s12d 2 5s22d 5

?12

s12, 22d 7. (a)

Solution

169 5 169

17s5d 2 7s212d 5?

169

169 5 169

52 1 s212d2 5?

169

s5, 212d (b)

Not a solution

149 Þ 169

s27d2 1 s10d2 5?

169

s27, 10d

9. Solve each equation for y.

Slopes are equal; therefore the system is inconsistent.

y 5 212 x 1 3

2y 5 2x 1 6

x 1 2y 5 6

y 5 212 x 1

32

2y 5 2x 1 3

x 1 2y 5 3


Slopes are equal; therefore the system is inconsistent.

y 523 x 1 4

23y 5 22x 2 12

2x 2 3y 5 212

y 523 x 2 1

12y 5 8x 2 12

28x 1 12y 5 212


Lines are the same; therefore the system is consistent anddependent.

y 514 x 1

74

4y 5 x 1 7

2x 1 4y 5 7

y 514 x 1

74

212y 5 23x 2 21

3x 2 12y 5 221


Slopes are not equal; therefore the system is consistent.

y 553 x 2

13

23y 5 25x 1 1

5x 2 3y 5 1

y 532 x 1

34

24y 5 26x 2 3

6x 2 4y 5 23

408 Chapter 8 Systems of Equations

21. No solution

Solve first equation for y.

Slopes are the same.

y 5 2x 1 4

x 1 y 5 4

y 5 2x 2 1

x 1 y 5 21

23. One solution

Solve first equation for y.

Substitute into second equation.

11, 132

y 513

y 53 2 2s1d

3

x 5 1

7x 5 7

7x 2 3 5 4

5x 2 3 1 2x 5 4

5x 2 313 2 2x3 2 5 4

y 53 2 2x

3

3y 5 3 2 2x

25. Infinite number of solutions

Solve each equation for y.

Slopes are the same; lines are the same.

y 512 x 1 2

22y 5 2x 2 4

x 2 2y 5 24

y 5 0.5x 1 2

20.5x 1 y 5 2

27. No solution

Solve second equation for y.

Substitute into first equation.

no real solution

0 5 2x2 2 x 1 4

x 2 2x2 5 4

x2 5 y

x2 2 y 5 0


Keystrokes:

2 3 2

2 3 2

Inconsistent

−9

−6

9

6

y2

y1

y 523 x 2 2

212 y 5 2

13 x 1 1

13 x 2

12 y 5 1

y 523 x 1 2

3y 5 2x 1 6

22x 1 3y 5 6


Keystrokes:

2 3 2

1

One solution

−4

−2

14

10

y2

y1

y 523 x 1 2

3y 5 2x 1 6

22x 1 3y 5 6

y 5 x 1 1

2y 5 2x 2 1

x 2 y 5 21

Y5

x

x 4

4

d

d

X,T,u

X,T,u 1

2 ENTER

GRAPH X,T,u

X,T,u ENTERY5 4

1 GRAPH

dx 1

Section 8.1 Systems of Equations409


The lines representing the two equations are the same.System is dependent and has infinitely many solutions.

4 + 5 = 20x y

y

2

1

−1

3

4

x−1 2 531 4

5

x + y = 445

y 5 245 x 1 4

y 5 245 x 1 4 5y 5 24x 1 20

45 x 1 y 5 44x 1 5y 5 20

29.

The two lines intersect in a point and the coordinates are s1, 2d.

x

y

−2 2 3 4

2

1

−1

−2

−3

(1, 2)

y x= + 1

y x= + 3−

1

33. Solve first equation for y.


x

y

−1−2 2 4

2

1

−1

−2

−3

(3, 1)

x = 3

3 4 = 5x y−

3

y 534 x 2

54

24y 5 23x 1 5

3x 2 4y 5 5


The two lines intersect in a point and the coordinates ares2, 0d.

x

y

−1−2 2 3 4

2

1

−1

−2

−3

(2, 0)

y x= 2−

y x= + 2−3

1

y 5 x 2 2

y 5 2x 1 2 2y 5 2x 1 2

x 1 y 5 2x 2 y 5 2



x

y

−2 4 10

2

−2

−6

(10, 0)

4 5 = 40x y−

2 12 14

−12

2 5 = 20x y−

y 525 x 2 4

25y 5 22x 1 20

2x 2 5y 5 20

y 545 x 2 8

25y 5 24x 1 40

4x 2 5y 5 40


The lines representing the two equations are the same.System is dependent and has infinitely many solutions.

x y+ = 2

y

2

1

−1

3

4

x−1 2 531 4

5

3 + 3 = 6x y

y 5 2x 1 2

x 1 y 5 2

y 5 2x 1 2

3y 5 23x 1 6

3x 1 3y 5 6



The two lines intersect in a point and the coordinates ares3, 21d.

y

1

−1

x−1 531

−2

−4

75y x += 4

5−

y x= 3−23

y 523 x 2 3 y 5 2

45x 1

75

23y 5 22x 1 9 5y 5 24x 1 7

2x 2 3y 5 9 4x 1 5y 5 7

43. Keystrokes:

4

Points of intersection are and s2, 4d.s0, 0d

−5

−3

7

6

(2, 4)

(0, 0)

y2

y1

45. Keystrokes:

3

3 3 3

Points of intersection are and s1, 1d.s0, 0d

−6

−8

10

8

(1, 1)

(0, 0)

y2

y1

47. Solve for x in first equation.


(2, 1)

5 2

x 5 2s1d

y 5 1

8y 5 8

6y 1 2y 5 8

3s2yd 1 2y 5 8

x 5 2y

49.


s4, 3d

y 5 3

22y 5 26

4 2 2y 5 22

x 5 4 51. Solve for y.


s1, 2d

y 5 2

y 5 3 2 1

x 5 1

3x 5 3

2x 2 3 1 x 5 0

2x 2 s3 2 xd 5 0

y 5 3 2 x

Y5 X,T,u x2 ENTER

X,T,u X,T,u x2 GRAPH2

Y5 ENTERX,T,u

X,T,u X,T,u X,T,ux2 1

>

> 2 GRAPH


59. Solve for y.


s22, 21d

y 5 21

y 5 2138 1

58

y 51316 s22d 1

1016

x 5 22

18x 5 236

5x 1 13x 1 10 5 226

5x 1 16s1316x 1

1016d 5 226

y 51316 x 1

1016

16y 5 13x 1 10

61. Solve for x.


132

, 322

5215 1 21

45

32

x 5

215 1 141322

4

y 532

204y 5 306

2270 1 252y 2 48y 5 36

18s215 1 14yd 2 48y 5 36

181215 1 14y4 2 2 12y 5 9

x 5215 1 14y

4

4x 5 215 1 14y

63. Solve for y.


1203

, 403 2

y 5403

y 5 2203

1603

y 5 2203

1 20

x 5203

23x 5 220

2x 2 5x 5 220

15

x 212

x 5 22

15

x 212

x 1 10 5 8

15

x 112

s2x 1 20d 5 8

y 5 2x 1 20

57. Solve for y.


s10, 4d

y 5 4

y 5 285s10d 1 20

x 5 10

25x 5 250

9x 1 16x 2 200 5 50

9x 2 10s285 x 1 20d 5 50

y 5 285 x 1 20

5y 5 28x 1 100

53. Solve for x.


s4, 22d

x 5 4

x 5 2 2 s22d

y 5 22

25y 5 10

2 2 y 2 4y 5 12

x 5 2 2 y

55. Solve for x.


s7, 2d

x 5 27 1 7s2d 5 7

y 5 2

13y 5 26

27 1 7y 1 6y 5 19

x 5 27 1 7y


73. Solve for y.


and s1, 21ds24, 14d

y 5 21y 5 14

y 5 23s1d 1 2y 5 23s24d 1 2

x 5 1x 5 24

sx 1 4dsx 2 1d 5 0

x2 1 3x 2 4 5 0

x2 1 3x 2 2 5 2

x2 2 s23x 1 2d 5 2

y 5 23x 1 2

71. Solve for y.


and s26, 8ds8, 26d

y 5 8y 5 26

y 5 2s26d 1 2y 5 28 1 2

x 5 26x 5 8

sx 2 8dsx 1 6d 5 0

x2 2 2x 2 48 5 0

2x2 2 4x 2 96 5 0

x2 1 x2 2 4x 1 4 5 100

x2 1 s2x 1 2d2 5 100

y 5 2x 1 2

67.


and s2, 12ds252, 75

4 dy 5 12 5 75

4

y 5 3s4dy 5 3s254 d

y 5 3s2d2y 5 3s252d2

x 5 2x 5 252

s2x 1 5dsx 2 2d 5 0

2x2 1 x 2 10 5 0

6x2 1 3x 2 30 5 0

3x 1 6x2 5 30

3x 1 2s3x2d 5 30

y 5 3x2

69. Solve for x.


and s2, 5ds23, 0d

x 5 2x 5 23

x 5 23 1 5x 5 23 1 0

y 5 5y 5 0

ysy 2 5d 5 0

y2 2 5y 5 0

9 2 6y 1 y2 1 y 2 9 5 0

4s23 1 yd2 1 y 5 9

x 5 23 1 y

75. Solve for y.


s24, 23ds0, 5d

y 5 23y 5 5

y 5 5 1 2s24dy 5 5 1 2s0d

x 5 24x 5 0

5xsx 1 4d 5 0

5x2 1 20x 5 0

x2 1 25 1 20x 1 4x2 2 25 5 0

x2 1 s5 1 2xd2 5 25

y 5 5 1 2x

2y 5 25 2 2x

65. Substitute into second equation.

and s2, 8ds23, 18d

5 8 5 18

5 2s4d 5 2s9d

y 5 2s2d2y 5 2s23d2

x 5 2x 5 23

sx 1 3dsx 2 2d 5 0

x2 1 x 2 6 5 0

2x2 1 2x 2 12 5 0

2x2 5 22x 1 12

y 5 22x 1 12


83. Verbal Model:

Labels: Total cost

Cost per unit

Number of units

Initial cost

Total revenue

Price per unit

System:

Break-even point occurs when so

10,000 items

10,000 5 x

8000 5 0.80x

1.20x 1 8000 5 2.00x

R 5 C

R 5 2.00x

C 5 1.20x 1 8000

5 2.00

5 R

5 8000

5 x

5 1.20

5 C

Numberof units?

Price perunit

5Total Revenue

Initialcost

1Numberof units?

Cost perunit

5Total cost

79. Answers will vary. Write equations so that satisfieseach equation.

x 1 y 5 9

2x 2 3y 5 27

s4, 5d 81. Answers will vary. Write equations so that satisfies each equation.

2x 1 3y 5 25

7x 1 y 5 29

s21, 22d

or2x 1 3y 5 23

x 2 y 5 21or

x 2 y 5 1

x 1 y 5 23

77.

or

Solve each equation for y.

y 53

275x 2 0.72

y 523

2275x 1

1982275

y 5 0.01x 1 2

2275y 5 23x 1 198 y 51

100x 1 2

3x 2 275y 5 198x 2 100y 5 2200

s2992, 79825 ds2.992, 31.92d

x 5 2200 1 100s31.92d 5 2200 1 3192 5 2992

y 5 31.92

y 579825

25y 5 798

2600 1 300y 2 275y 5 198

3s2200 1 100yd 2 275y 5 198

x 5 2200 1 100y

by substitution

Keystrokes:

.01 2

3 275 .72

3

3

−1

−3

y2

y1 Y5 ENTER

GRAPHX,T,ud

1X,T,u

4x 2


85. Verbal Model:

Labels: Total cost

Cost per unit

Number of units

Initial cost

Total revenue

Price per unit

System:

Break-even point occurs when so

6250 units

6250 5 x

10,000 5 1.60x

1.65x 1 10,000 5 3.25x

R 5 C

R 5 3.25x

C 5 1.65x 1 10,000

5 3.25

5 R

5 10,000

5 x

5 1.65

5 C

Numberof units?

Price perunit

5Total Revenue

Initialcost

1Numberof units?

Cost perunit

5Total cost

87. Verbal Model:

Labels: Amount at 8%

Amount at 9.5%

System:

Solve for x.


at 9.5%

at 8% x 5 20,000 2 5000 5 $15,000

y 5 $5000

0.015y 5 75

1600 2 0.08y 1 0.095y 5 1675

0.08s20,000 2 yd 1 0.095y 5 1675

x 5 20,000 2 y

0.08x 1 0.095y 5 1675

x 1 y 5 20,000

5 y

5 x

16755Amountat 9.5%?9.5%1

Amountat 8%?8%

20,0005Amountat 9.5%

1Amountat 8%


93. Verbal Model:

Labels: Larger number

Smaller number

System:


s120, 40d

x 5 120

x 5 3s40d y 5 40

4y 5 160

3y 1 y 5 160

x 5 3y

x 1 y 5 160

5 y

5 x

Smallernumber?35

Largernumber

1605Smallernumber

1Largernumber

91. Verbal Model:

Labels: Larger number

Smaller number

System:

Solve for x.


s25, 18d

x 5 25

x 5 18 1 7

y 5 18

3y 5 54

3y 1 7 5 61

y 1 7 1 2y 5 61

x 5 y 1 7

x 2 7 5 7

x 1 2y 5 61

5 y

5 x

75Smallernumber

2Largernumber

615Smallernumber?21

Largernumber

89. Verbal Model:

Labels: Amount in 8% fund

Amount in 9.5% fund

System:

Solve for x.


at 8.5%

at 8% x 5 25,000 2 12,000 5 $13,000

y 5 $12,000

0.005y 5 60

2000 2 0.08y 1 0.085y 5 2060

0.08s25,000 2 yd 1 0.085y 5 2060

x 5 25,000 2 y

0.08x 1 0.085y 5 2060

x 1 y 5 25,000

5 y

5 x

20605Amount in8.5% fund?8.5%1

Amount in8% fund?8%

25,0005Amount in8.5% fund

1Amount in8% fund


Section 8.2 Linear Systems in Two Variables

1.

s2, 0d

y 5 0

2y 5 0

2 2 y 5 0

x 5 2

3x 5 6

x 2 y 5 2

x

y

−1 1 3 4 5

2

1

3

−1

−2

−3

x y− = 2

2 + = 4x y(2, 0)

2x 1 y 5 4 3.

s5, 3d

x 5 5

x 2 3 5 2

y 5 3

x 2 y 5 2

)3,(5

1y2

543

y 2

x

4

y

3

2

1

x

11

x2

1 2

2x 1 2y 5 1

5.

s2, 23d

y 5 23

6 1 y 5 3

3s2d 1 y 5 3

x 5 2

5x 5 10

2x 2 y 5 7

y

x

(2, −3)

−2 42 5 6

1

−1

−2

−3

−4

−5

2

3 2x − y = 7

3x + y = 3

3x 1 y 5 3 7.

No solution

y

1

x

2x

1

y

1

2

21

3y3x

122

8

0 Þ 11

23x 1 3y 5 8 ⇒ 23x 1 3y 5 8

x 2 y 5 1 ⇒ 3x 2 3y 5 3

9.

All solutions to

5

x5

10

y3x

y

2

1

6y

21

2x

11

4

3

x 2 3y 5 5

0 5 0

22x 1 6y 5 210 ⇒ 22x 1 6y 5 210

x 2 3y 5 5 ⇒ 2x 2 6y 5 10

Section 8.2 Linear Systems in Two Variables417

13.

s3, 2d

y 5 2

2y 5 4

3 1 2y 5 7

x 5 3

4x 5 12

x 1 2y 5 5

3x 2 2y 5 5 15.

s22, 5d

x 5 22

4x 5 28

4x 1 5 5 23

y 5 5

4y 5 20

24x 1 3y 5 23

4x 1 y 5 23 17.

s2, 1d

y 5 1

25y 5 25

3s2d 2 5y 5 1

x 5 2

5x 5 10

2x 1 5y 5 9

3x 2 5y 5 1

19.

s3, 24d

y 5 24

2y 5 4

3s3d 2 y 5 13

x 5 3

11x 5 33

6x 2 2y 5 26 3x 2 y 5 13 ⇒

5x 1 2y 5 7 5x 1 2y 5 7 ⇒ 21.

s21, 21d

x 5 21

x 2 3s21d 5 2

y 5 21

2y 5 22

3x 2 7y 5 4 3x 2 7y 5 4 ⇒ 23x 1 9y 5 26 x 2 3y 5 2 ⇒

23.

s5, 21d

y 5 21

10 1 y 5 9

2s5d 1 y 5 9

x 5 5

5x 5 25

3x 2 y 5 16

2x 1 y 5 9 25.

s7, 22d

u 5 7

2u 5 14

2u 1 3s22d 5 8

v 5 22

2v 5 2

6u 1 8v 5 26 3u 1 4v 5 13 ⇒ 26u 2 9v 5 24 2u 1 3v 5 8 ⇒

11.

112

, 322

y 532

y 521228

28y 5 212

21122 2 8y 5 211

x 5 12

x 5 2346

46x 5 23

40x 1 24y 5 56 5x 1 3y 5 7 ⇒11

3x

y

3

x

,1

2 8

y

2

21

73x y

1

1

5

21 2

6x 2 24y 5 233 2x 2 8y 5 211 ⇒


27.

Inconsistent

0 Þ 10

224x 1 10y 5 6 ⇒ 224x 1 10y 5 6

12x 2 5y 5 2 ⇒ 24x 2 10y 5 4 29.

s32, 1d

23s3

2d 2 s2s

s

5

5

5

021

1

r 5 32

r 5 5738

38r 5 57

30r 1 12s 5 57 10r 1 4s 5 19 ⇒ 10r 1 4s 5 19 ⇒ 8r 2 12s 5 0 23r 2 s 5 0 ⇒ 2r 2 3s 5 0 ⇒

31.

s6, 3d

x 1 y6 1 y

y

5 95 95 3

x 5 6

8x 5 48

3x 1 3y 5 27 x 1 y 5 9 ⇒ 5x 2 3y 5 21 0.05x 2 0.03y 5 0.21 ⇒

33.

s22, 21d

7u7uu

2 10s1d 5

5

5

2421422

26v 5

v 5

22621

0.7u 2

0.3u 2

v0.8v

5

5

20.40.2

⇒⇒

7u 2

3u 2

10v8v

5

5

242 ⇒⇒

21u 2

221u 1

30v 5 21256v 5 214

39.

s12.5, 4.948d

4x4x4xx

2

2

10s4.948d49.48

5

5

5

5

0.520.52

5012.5

y 5 4.948

210y 5 249.48

4x 2 10y 5 0.52 4x 2 10y 5 0.52 ⇒ 24x 5 250 2x 5 25 ⇒ 41.

s23, 7d

5 7

5 26 1 13

y 5 2s23d 1 13

x 5 23

7x 5 221

3x 1 4x 1 26 5 5

3x 1 2s2x 1 13d 5 5

y 5 2x 1 13

3x 1 2y 5 5

35.

All solutions of the form x 1 1.4y 5 5

0 5 0

25x 2 7y 5 225 x 1 1.4y 5 5 ⇒ 5x 1 7y 5 25 5x 1 7y 5 25 ⇒ 37.

Inconsistent

0 Þ 5

23x 1 2y 5 23 2x 1 23y 5 21 ⇒ 3x 2 2y 5 8 32x 2 y 5 4 ⇒


53.

No solution inconsistent ⇒

2x 2 3y 5 224 ⇒ 23y 5 22x 2 24 ⇒ y 523 x 1 8

210x 1 5y 5 25 ⇒ 15y 5 10x 1 25 ⇒ y 523 x 1

53

55.

Inconsistent; no solution

0 Þ 230

22x 1 4y 5 30 ⇒ 210x 1 20y 5 150

5x 2 10y 5 40 ⇒ 10x 2 20y 5 80

so 2k

5 12

⇒ k 5 4

22x 1 ky 5 30 ⇒ y 52k

x 13016

5x 2 10y 5 40 ⇒ y 512

x 2 4 57. Answers will vary. Write equations so that satisfies each equation.

0 5 0

3 1 212322 5

?0

x 2 4y 5 9

x 1 2y 5 0

s3, 232d

43.

s2, 7d

y 5

y 5

y 5

5s2d 2 310 2 3

7

2 5 x

14 5 7x

0 5 7x 2 14

2y 5 2x 2 11 y 5 22x 1 11 ⇒ y 5 5x 2 3 y 5 5x 2 3 ⇒ 45.

s15, 10d

y 5 10

2 15 1 y 5 25

x 5 15

2x 1 y 5 25

2x 2 y 5 20

47.

s4, 3d

5 3 5 4 2 1

y 5 4 214s4d

x 5 4

32x 1 8 212x 5 12

32x 1 2s4 214xd 5 12

y 5 4 214x

14x 1 y 5 4

32x 1 2y 5 12 49.

Many solutions consistent ⇒

28x 1 10y 5 26 ⇒ 10y 5 8x 2 6 ⇒ y 545 x 2

35

4x 2 5y 5 3 ⇒ 25y 5 24x 1 3 ⇒ y 545 x 2

35

51.

One solution consistent ⇒

5x 1 2y 5 8 ⇒ 2y 5 25x 1 8 ⇒ y 5 252 x 1 4

22x 1 5y 5 3 ⇒ 5y 5 2x 1 3 ⇒ y 5 25 x 135

9 5 9

3 2 412322 5

?9


59. Verbal Model:

Labels: Total cost

Cost per unit

Number of weeks

Initial cost

Total revenue

Price per unit

System:

Break-even point occurs when

122 weeks

121.4285 < x

85,000 5 700x

7400x 1 85,000 5 8100x

R 5 C

R 5 8100x

C 5 7400x 1 85,000

5 8100

5 R

5 85,000

5 x

5 7400

5 C

Numberof units?

Price perunit

5Total revenue

Initialcost

1Numberof units?

Cost perunit

5Total cost

61. Verbal Model:

Labels: Amount in 8% bond

Amount in 9.5% bond

System:

$15,000 at 8%

$5,000 at 9.5% x 5 15,000

x 1 5000 5 20,000

y 5 5000

0.015y 5 75

0.08x 1 0.095y 5 1675 ⇒ 0.08x 1 0.095y 5 1675

x 1 y 5 20,000 ⇒ 20.08x 2 0.08y 5 21600

5 y

5 x

Totalinterest

5Interest in9.5% bond

1Interest in8% bond

Totalinvestment

5Amount in9.5% bond

1Amount in8% bond


65. Verbal Model:

Labels: Plane speed

Speed of air

System:

mph y 5 50

2y 5 250

550 2 y 5 500

5

5

1100550 mph

2xx

x 1 y 51800

3 ⇒ x 1 y 5 600

x 2 y 518003.6 ⇒ x 2 y 5 500

5 y

5 x

Speed intohead wind

5Speedof air

1Plane speed(still air)

Speed intohead wind

5Speedof air

2Plane speed(still air)

67. Verbal Model:

Labels: Number of adult tickets

Number of children tickets

System:

y 5 500 2 375 5 125 children tickets

x 5 375 adult tickets

3.5x 5 1312.50

7.50x 1 2000 2 4.00x 5 3312.50

7.50x 1 4.00s500 2 xd 5 3312.50

y 5 500 2 x

7.50x 1 4.00y 5 3312.50

x 1 y 5 500

5 y

5 x

3312.505Value of children tickets

1Value ofadult tickets

5005Number of children tickets

1Number ofadult tickets

63. Verbal Model:

Labels: Time at 55 mph

distance at 40 mph for 2 hours at 55 mph for x hours

distance at 50 mph for hours

System: Since

hours x 5 4

5x 5 20

80 1 55x 5 100 1 50x

40s2d 1 55sxd 5 50s2 1 xd

D1 5 D2

D2 5 50s2 1 xd

D1 5 40s2d 1 55sxd

2 1 xD2 5

1D1 5

5 x

Time?Rate5Distance


69. Verbal Model: 12 8

Labels: Cost of regular gasoline

Cost of premium gasoline

System:

y 5 0.11 1 1.11 5 $1.22 premium

x 5 $1.11 regular

20x 5 22.20

12x 1 0.88 1 8x 5 23.08

12x 1 8s0.11 1 xd 5 23.08

y 5 0.11 1 x

12x 1 8y 5 23.08

5 y

5 x

Cost of regulargasoline

1$0.115Cost of premiumgasoline

$23.0852Cost of premiumgasoline112Cost of regular

gasoline1

71. Verbal Model:

Labels: Number liters Solution 1

Number liters Solution 2

System:

x 5 20 2 8 5 12 liters at 40% alcohol solution

y 5 8 liters at 65% alcohol solution

25y 5 200

800 2 40y 1 65y 5 100

40s20 2 yd 1 65y 5 20s50d

x 5 20 2 y

0.40x 1 0.65y 5 20s0.50d

x 1 y 5 20

5 y

5 x

20(0.50)5Value ofSolution 2

1Value ofSolution 1

205Number of litersSolution 2

1Number of litersSolution 1


75. (a) (b)

y 5 232x 1

236 5 21.5x 1 35

6

b 5236

6b 5 23

6b 2 9 5 14

3b 1 3s232d 5 7

m 5 232

22m 5 3

3b 1 5m 5 4

y

x

3

2

1

2 31−1

3b 1 3m 5 7

73. Verbal Model:

Labels: Amount of $5.65 variety

Amount of $8.95 variety

System:

lbs of $5.65 variety

lbs of $8.95 varietyy 5 10 2 x 5 10 2 6.1 5 3.9

x < 6.1

23.3x 5 220

5.65x 1 89.5 2 8.95x 5 69.5

5.65x 1 8.95s10 2 xd 5 69.5

y 5 10 2 x

5.65x 1 8.95y 5 6.95s10d x 1 y 5 10

5 y

5 x

Total cost5Cost for $8.95 variety

1Cost for$5.65 variety

105Amount of$8.95 variety

1Amount of$5.65 variety

77. (a) Solve by substitution.

The memorial is 10 feet deep.

s0, 210d x 5 0

y 5 210 225 x 5 2561 x

y 5225s0d 2 102

25 x 2 10 5 2561 x 2 10

y 5 2561 x 2 10

y 52

25 x 2 10 (b)

122 feet and 125 feet

125 5 x

10 52

25 x

0 5225 x 2 10

122 5 x

10 55

61 x

0 55

61 x 2 10

79. When solving a system by elimination, you can recognize that it has infinitely many solutions when adding a nonzero multiple of one equation to another equation to eliminate a variable, you get for the second equation.0 5 0

83. Substitution may be better than elimination when it is easy to solve for one of the variables in one of the equations of the system.

81. (a) Obtain coefficients for x or y that differ only in sign by multiplying all terms of one or both equations by suitable chosen constants.

(b) Add the equations to eliminate one variable, and solve the resulting equation.

(c) Back-substitute the value obtained in Step (b) into either of the original equations and solve for the other variable.

(d) Check your solution in both of the original equations.


Section 8.3 Linear Systems in Three Variables

83. Substitution may be better than elimination when it is easy to solve for one of the variables in one of the equations of the system.

1. (a)

not a solution

(c)

solution

1 5 1

1 2 6 1 6 5 1

1 1 3s22d 1 2s3d 5?

1

s1, 22, 3d

9 2 4 Þ 1

0 1 3s3d 1 2s22d 5?

1

s0, 3, 22d (b)

solution

(d)

not a solution

7 Þ 1

22 1 15 2 6 5 1

22 1 3s5d 1 2s23d 5?

1

s22, 5, 23d

1 5 1

12 1 15 2 26 5 1

12 1 3s5d 1 2s213d 5?

1

s12, 5, 213d

81. (a) Obtain coefficients for x or y that differ only in sign by multiplying all terms of one or both equations by suitable chosen constants.

(b) Add the equations to eliminate one variable, and solve the resulting equation.

(c) Back-substitute the value obtained in Step (b) into either of the original equations and solve for the other variable.

(d) Check your solution in both of the original equations.

3.

s22, 21, 25d

x 5 22

x 2 18 5 4

x 1 2 2 20 5 4

x 2 2s21d 1 4s25d 5 4

y 5 21

3y 5 23

3y 2 s25d 5 2 5.

s14, 3, 21d

x 5 14

x 2 10 5 4

x 2 6 2 4 5 4

x 2 2s3d 1 4s21d 5 4

z 5 21

3 1 z 5 2

7. The two systems are not equivalent because when the first equation was multiplied by and added to the second equation the constant term should have been 211.

22

9.

This operation eliminated the x-term from the secondequation.

y 5 14

2x 1 3y 5 6

x 2 2y 5 8 11.

This operation eliminated the x-term in Equation 2.

2y 1 8z 5 9

2x 1 y 1 5z 5 4

x 2 2y 1 3z 5 5

Section 8.3 Linear Systems in Three Variables425

17.

s2, 23, 22d

x 5 2

x 2 5 5 23

x 1 s23d 1 s22d 5 23

y 5 23

y 5 293

y 2143 5

2233

y 173s22d 5

2233

z 5 22

y 173 z 5 2

233

x 1 y 1 z 5 23

273 z 5

143

y 173 z 5 2

233

x 1 y 1 z 5 23

y 5 23

y 173 z 5 2

233

x 1 y 1 z 5 23

25y 5 15

23y 2 7z 5 23

x 1 y 1 z 5 23

13.

s1, 2, 3d

z 5 3

y 5 1

x 5 1

z 5 3

y 5 2

x 1 z 5 4

23z 5 29

y 5 2

x 1 z 5 4

4x 1 z 5 7

y 5 2

x 1 z 5 4

15.

s1, 2, 3d

x 5 1

x 1 2 1 3 5 6

y 5 2

y 1 13s3d 5 3

z 5 3

y 1 13 z 5 3

x 1 y 1 z 5 6

23x 5 29

y 1 13 z 5 3

x 1 y 1 z 5 6

23y 2 4z 5 218

y 1 13 z 5 3

x 1 y 1 z 5 6

23y 2 4z 5 218

23y 2 z 5 29

x 1 y 1 z 5 6

3x 2z 5 0

2x 2 y 1 z 5 3

x 1 y 1 z 5 6


21.

s24, 8, 5d

x 5 24

x 1 5 5 1

y 5 8

y 253s5d 5 2

13

z 5 5

y 253 z 5 2

13

x 1 z 5 1

3y 2 4z 5 4

y 2 53 z 5 213

x 1 z 5 1

3y 2 4z 5 4

3y 2 5z 5 21

x 1 z 5 1

3y 2 4z 5 4

5x 1 3y 5 4

x 1 z 5 1

3y 2 4z 5 4

5x 1 3y 5 4

2x 1 2z 5 2

23.

No solution

Inconsistent

y 1 5z 5 35

y 1 5z 5 2

x 1 y 1 8z 5 3

2y 2 5z 5 23

2y 2 5z 5 22

x 1 y 1 8z 5 3

x 1 3z 5 0

2x 1 y 1 11z 5 4

x 1 y 1 8z 5 3

19.

No solution

Inconsistent

0 5 12

3y 1 4z 5 8

x 1 2y 1 6z 5 5

26y 2 8z 5 24

3y 1 4z 5 8

x 1 2y 1 6z 5 5

x 2 4y 2 2z 5 1

3y 1 4z 5 8

x 1 2y 1 6z 5 5

x 2 4y 2 2z 5 1

2x 1 y 2 2z 5 3

x 1 2y 1 6z 5 5

25.

s 310, 25, 0d

x 55

10 22

10 53

10

x 115 5

12

x 112s2

5d 132s0d 5

12

y 525

y 1 0 525

z 5 0

y 1 z 525

x 112 y 1

32 z 5

12

4z 5 0

5y 1 5z 5 2

2x 1 y 1 3z 5 1

5y 1 9z 5 2

5y 1 5z 5 2

2x 1 y 1 3z 5 1

6x 1 8y 1 18z 5 5

2x 1 6y 1 8z 5 3

2x 1 y 1 3z 5 1


29.

s21, 5, 5d

x 5 21

x 1 5 5 4

x 1 15 2 10 5 4

x 1 3s5d 2 2s5d 5 4

y 5 5

y 2 5 5 0

z 5 5

y 2 z 5 0

x 1 3y 2 2z 5 4

22x 2 6y 1 5z 5 23

3x 1 10y 2 7z 5 12

x 1 3y 2 2z 5 4

22x 2 6y 1 5z 5 23

3x 1 10y 2 7z 5 12

2x 1 6y 2 4z 5 8

31.

let s12 2

12a, 35a 1

25, ada 5 z

x 512 2

12 z

x 112 z 5

12

y 535 z 1

25

0 5 0

y 2 35 z 5 25

x 1 12 z 5 12

20y 2 12z 5 8

y 2 35 z 5 25

x 1 12 z 512

20y 2 12z 5 8

5y 2 3z 5 2

x 1 12z 5 12

6x 1 20y 2 9z 5 11

5y 2 3z 5 2

x 1 12z 5 12

6x 1 20y 2 9z 5 11

5y 2 3z 5 2

2x 1 z 5 3 33.

let s14 2

12a, 12a 1

54, ada 5 z 0 5 0

514 2

12z

x 112 z 5

14 y 2

12 z 5 54

x 116 z 1

512 1

13 z 5

23 x 1 13 y 1

13 z 5 23

x 113s1

2 z 154d 1

13 z 5

23 28y 1 4z 5 210

y 512 z 1

54 y 2

12 z 5 54

x 1 13 y 113 z 5 23

283 y 1

43 z 5 2

103

243 y 1 23 z 5 25

3

x 1 13 y 113 z 5 23

5x 2 y 1 3z 5 0

4x 1 2z 5 1

x 1 13 y 113 z 5 23

5x 2 y 1 3z 5 0

4x 1 2z 5 1

3x 1 y 1 z 5 2

27.

s24, 2, 3d

x 5 24

2x 5 28

2x 1 12 5 4

2x 1 4s3d 5 4

y 5 2

y 1 3 5 5

z 5 3

2x 1 4z 5 4

y 1 z 5 5

2z 5 23

2x 1 4z 5 4

y 1 z 5 5

23y 2 4z 5 218

2x 1 4z 5 4

y 1 z 5 5

2x 2 3y 5 214

2x 1 4z 5 4

y 1 z 5 5


35.

s1, 21, 2d z 5 2

x 5 1 y 5 21

x 1 3s2d 5 7 x 1 3z 5 7

2z 5 4

y 5 21

x 1 3z 5 7

10y 1 2z 5 26

y 5 21

1x 1 3z 5 7

10y 1 2z 5 26

13y 5 213

1x 1 3z 5 7

2x 1 10y 1 8z 5 8

1x 1 3z 5 7

2x 1 13y 1 6z 5 1

2x 1 10y 1 8z 5 8

0.1x 1 0.3z 5 0.7

0.2x 1 1.3y 1 0.6y 5 0.1 37.

Let

s 613 a 1

1013, 5

13 a 14

13, ad x 56

13 z 11013

z 5 a x 26

13 z 51013

x 12013 x 1

1613 2

2613 z 5

2613

x 1 4s 513 x 1

413d 2 2z 5 2

y 5513 z 1

413

0 5 0

y 25

13 z 5 413

x 1 4y 2 2z 5 2

213y 1 5z 5 24

y 25

13 z 5 413

x 1 4y 2 2z 5 2

213y 1 5z 5 24

13y 2 5z 5 4

x 1 4y 2 2z 5 2

5x 1 7y 2 5z 5 6

23x 1 y 1 z 5 22

x 1 4y 2 2z 5 2

39.

y 2

s212, 2, 10d

5

2 1 y 1 2 5 6

24s212d 1 y 1 0.2s10d 5 6

x 5 212

26x 5 3

26x 1 1.1s10d 5 14

z 5 10

0.2z 5 2

26x 1 1.1z 5 14

24x 1 y 1 0.2z 5 6

28x 1 2y 1 0.6z 5 14

6x 2 3y 1 0.5z 5 24

24x 1 y 1 0.2z 5 6 41.

or

Many correct answers. Write equations so that satisfies each equation.

s4, 23, 2d

x 2 2z 5 0

2x 1 y 1 2z 5 9

x 1 y 1 z 5 3

3x 1 y 1 3z 5 15

y 1 2z 5 1

x 1 2y 2 z 5 24


45.

a

s 5 216t2 1 48t

232 5

s0

0

0

v0 1

2s48d 1

a 1

0 5

48 5

64 5

2s0

32 s0

2s0

2v0 1

v0 1

1

a 1

64 5

48 5

0 5

2s0

32 s0

16s0

2v0 1

v0 1

12v0 2

a 1

2

64 5

48 5

2576 5

2s0

3s0

16s0

2v0 1

2v0 2

12v0 2

a 1

2

2

64 5

296 5

2576 5

2s0

s0

s0

2v0 1

2v0 1

6v0 1

a 1

2a 1

9a 1

64 5

32 5

0 5

s0

s0

s0

32 5

32 5

0 5

12 as1d2 1 v0s1d 1

12 as2d2 1 v0s2d 1

12 as3d2 1 v0s3d 1

43.

s 5 216t 2 1 144

232 5 a

256 5 a 1 0 1 288

0 5 v0

216 5 v0 132 s144d

144 5 1 s0

216 5 v0 1 32 s0

256 5 a 1 2v0 1 2s0

1152 5 2 6v0 2 8s0

216 5 v0 1 32 s0

256 5 a 1 2v0 1 2s0

21152 5 2 6v0 2 8s0

2432 5 2 2v0 2 3s0

256 5 a 1 2v0 1 2s0

0 5 92 a 1 3v0 1 s0

80 5 2a 1 2v0 1 s0

256 5 a 1 2v0 1 2s0

0 5 92 a 1 3v0 1 s0

80 5 2a 1 2v0 1 s0

128 5 12 a 1 v0 1 s0

0 512 as3d2 1 v0s3d 1 s0

80 512 as2d2 1 v0s2d 1 s0

128 512 as1d2 1 v0s1d 1 s0

47.

6 5 22b 2 3c

1 5 a 1 b 1 c

24 5 c

10 5 4a 1 2b 1 c

1 5 a 1 b 1 c

24 5 c

10 5 as2d2 1 bs2d 1 c

1 5 as1d2 1 bs1d 1 c

24 5 as0d2 1 bs0d 1 c

y 5 2x2 1 3x 2 4

2 5 a

1 5 a 2 1

1 5 a 1 3 1 s24d

3 5 b

26 5 22b

6 5 22b 1 12

6 5 22b 2 3s24d

c 5 24


49.

y 5 x2 2 4x 1 3

1

24

3

5

5

c 5

b

a

212

12

3

212 c 5

32 c 5

c 5

b 1

a

212

12

232

212 c 5

32 c 5

212 c 5

b 1

a 1

0

12

0

c 5

32c 5

4c 5

b 1

b 1

3b 1

a 1

0

21

0

c 5

3c 5

8c 5

b 1

22b 2

26b 2

a 1

0 5 as3d2 1 bs3d 1 c ⇒ 0 5 9a 1 3b 1 c

21 5 as2d2 1 bs2d 1 c ⇒ 21 5 4a 1 2b 1 c

0 5 as1d2 1 bs1d 1 c ⇒ 0 5 a 1 b 1 c 51.

y 5 21x 2 1 2x 1 0

c

0

2 1

2

a

a

0 5

23 5

21 5

3c

23c

6s2d

b

2

2 5

12 5

0 5

c

3c

1

2

b

12b

16b

a 2

23 5

4 5

12 5

c

c

c

b 1

b 1

2b 1

a 2

a 1

4 1

23 5

1 5

0 5

c

c

c

bs21d 1

bs1d 1

bs2d 1

as21d2 1

as1d2 1

as2d2 1

23 5

1 5

0 5

53.

yesy 512 s6d2 2

12 s6d 5

12 s36d 2

12 s6d 5 18 2 3 5 15

y 512 x 2 2

12 x 5

12 xsx 2 1d

3 5

3 5

92 5

12 5

9a 1 3s212d 1 0

9a 232

9a

a

23 5

23 5

212 5

243b 2

79s0d

243b

b

0 53

18c

0 5 c

3 5 9a 1 3b 1 c

23 5 2

43b 2

79c

0 53

18c

⇒

3 5 9a 1 3b 1 c

23 5 2

43b 2

79c

53 5 2

103 b 2

169 c

⇒

3 5 9a 1 3b 1 c

6 5 16a 1 4b 1 c

10 5 25a 1 5b 1 c

10 5 as5d2 1 bs5d 1 c

6 5 as4d2 1 bs4d 1 c

3 5 as3d2 1 bs3d 1 c


55.

x 2 1 y 2 2 4x 5 0

E 5 0

22E 5 0

2s24d 2 2E 1 0 5 28

D 5 24

4D 5 216

4D 1 0 5 216

4D 1 F 5 216

2D 2 2E 1 F 5 28

F 5 0

Es0d 1 F 5 0

Es22d 1 F 5 0

Es0d 1 F 5 0

Ds0d 1

Ds2d 1

Ds4d 1

02 1

s22d2 1

02 1

02 1

22 1

42 1

57.

x 2 1 y 2 2 6x 2 8y 5 0

F 5 0

F 1 3s26d 2 s28d 5 210

D 5 26

D 1 3s28d 5 230

E 5 28

D 1 3 5 230

F 1 3D 2 E 5 210

120E 5 2160

D 1 3E 5 230

F 1 3D 2 E 5 210

25D 1 5E 5 210

D 1 3E 5 230

F 1 3D 2 E 5 210

3D 1 9E 5 290

25D 1 5E 5 210

F 1 3D 2 E 5 210

F 1 6D 1 8E 5 2100

F 2 2D 1 4E 5 220

F 1 3D 2 E 5 210

6D 1 8E 1 F 5 2100

22D 1 4E 1 F 5 220

3D 2 E 1 F 5 210

62 1 82 1 Ds6d 1 Es8d 1 F 5 0

s22d2 1 42 1 Ds22d 1 Es4d 1 F 5 0

32 1 s21d2 1 Ds3d 1 Es21d 1 F 5 0

59.

—CONTINUED—

15 F 5 210

15 F 5 212

85 F 5 264

E 1

2E 1

8E 1

D 1

15F 5 210

F 5 252

F 5 234

E 1

6E 1

5E 1

D 1

4D 1

23D 1

F 5 234

F 5 252

F 5 250

23D 1 5E 1

4D 1 6E 1

5D 1 5E 1

s23d2 1 52 1

42 1 62 1

52 1 52 1

Ds23d 1 Es5d 1 F 5 0

Ds4d 1 Es6d 1 F 5 0

Ds5d 1 Es5d 1 F 5 0


59. —CONTINUED—

x 2 1 y 2 2 2x 2 4y 2 20 5 0

22

24

220

5

5

F 5

E

D

24

26

220

110 F 5

110 F 5

F 5

1

E 1

D

24

26

216

110 F 5

110 F 5

45 F 5

1

E 1

D

210

26

264

15 F 5

110 F 5

85 F 5

E 1

E 1

8E 1

D 1

61.

Spray X: 20 gal

Spray Y: 18 gal

Spray Z: 16 gal

x

y

z

5 20

5 18

5 16

⇒

x 1

y 2

2.5z 5

z 5

z 5

60

2

16

⇒

x 1

1y 2

2.5z 5

2.5z 5

1z 5

60

28

2

⇒

x

.4x

.4x

1 1y

1

1

2.5z

.5z

5 60

5 16

5 26

.20x

.40x

.40x

1 1y

1 .50z 5 12

1 .50z 5 16

5 26

63.

String: 50

Wind: 20

Percussion: 8

x

y

z

5

5

5

50

20

8

⇒

x

y

1 1.25z 5

5

z 5

60

20

8

⇒

x

y

1

1

1.25z 5

5

.125z 5

60

20

1

⇒

x 1 .75y

y

.075y

1

1

1.25z 5

5

.125z 5

75

20

2.5

⇒

x 1 .75y

.1y

0.75y

1

1

1.25z 5

5

.125z 5

75

2

2.5

⇒

x 1

.20x 1

.10x 1

.75y 1

.25y 1

.15y 1

1.25z 5 75

.25z 5 17

.25z 5 10

.40x 1 .30y 1 .50z 5 30

.20x 1 .25y 1 .25z 5 17

.10x 1 .15y 1 .25z 5 10


65. (d)

(e) Students: 140; Nonstudents: 25; Major contributors: 35

(f)

(not possible)

3195

3195

465

2465

7

5

5

5

5

15s182 2 xd

2730 2 15x

27x

x

8x 1

8x 1

182

3195

182 2 x

5

5

5

y

15y

y

x 1

8x 1

200

4995

18

z 5

100z 5

z 5

y 1

15y 1

x 1

8x 1

200

140

35 5

x 5

25 1x 1

3520025

z 5

5s35d 5

y 5

y 1

200200

1995

z 5

5z 5

57z 5

y 1

y 1

x 1

2003395

2200

z 5

92z 5

5z 5

y 1

7y 1

2y 2

x 1

2004995

0

z100z

4z

5

5

5

y15y

1

1

2

x8xx

1

1

67. Substitute into the first equation to obtainor x 5 2 2 6 5 24.x 1 2s3d 5 2

y 5 3 69. Answers will vary.


1.

This is not the solution.

This is a solution.

26 5 26

20 1 6 5 26

2s10d 1 1.5s4d 5?

26

2 5 2

50 2 48 5 2

5s10d 2 12s4d 5?

2s10, 4d

5 1 24 Þ 2

5s1d 2 12s22d 5?

2s1, 22d 2.

No solution

y

x−4 −1 1 3 4

−1

−3

−4

−5

2

3−6x + 9y = 9

2x − 3y = 6


9.

s5, 2d

x 5 5

2x 5 10

2x 2 6 5 4

2x 2 3s2d 5 4

y 5 2

2x 2 3y 5 4 10.

s23, 24d, s1, 4d

y 5 4y 5 24

x 5 1x 5 23

0 5 sx 1 3dsx 2 1d

0 5 x2 1 2x 2 3

5 2 x2 5 2x 1 2

5 2 x2 5 2sx 1 1d

y 5 2sx 1 1d

y 5 5 2 x2

3.

One solution

y

x42 3 5 6

1

−2

−1

5

3

6

4

3 2 4x y− =

x y− =−2 4

(4, 4)(4, 4)

−2

4.

Two solutions

y

x−3 −2 2 3 4 5

−3

−4

−5

2

1

3

y = 1 + 2x − x2

y = x − 1

(2, 1)

(−1, −2)

5.

s4, 2d

y

x1−1 52 6 7

−2

−1

−3

−4

3

2

1

4

(4, 2)

x = 4

2x − y = 6

6.

s2, 21d

y

x−2 5 6

−4

−3

−2

−1

2

3

y = (5x − 13)

y = (1 − 2x)1

1

3

3

(2, −1)

7.

s8, 0d

y

x−2 642

−6

−8

−4

6

4

8

(8, 0)

2x + 7y = 16

3x + 2y = 248.

s5, 12d, s212, 5d

y

x−16 −4−8 4 8 16

−8

−16

16

x2 + y2 = 169 7x − 17y = −169

(−12, 5)

(5, 12)


11.

19013

, 34132

53413

545013

241613

y 5 5190132 2 32

x 52270239

59013

239x 5 2270

6x 2 45x 1 288 5 18

6x 2 9s5x 2 32d 5 18

y 5 5x 2 326x 2 9y 5 18

2y 5 25x 1 32⇒ 5x 2 y 5 32 12.

s5, 10d

5 5

5 20 2 15

x 5 2s10d 2 15

y 5 10

1.1y 5 11

0.4y 2 3 1 0.7y 5 8

x 5 2y 2 150.2s2y 2 15d 1 0.7y 5 8

2x 5 22y 1 15⇒ 2x 1 2y 5 15

0.2x 1 0.7y 5 8

13.

s8, 1d

y 5 1

x 5 8

y 5 1

x 1 10y 5 18

248y 5 248

x 1 10y 5 18

5x 1 2y 5 42

x 1 10y 5 18 14.

s22, 4d

y 5 4

x 5 22

y 5 4

x 1 113 y 5 38

3

2923 y 5 2

3683

x 1 113 y 5 38

3

7x 2 5y 5 234

x 1 113 y 5 38

3

7x 2 5y 5 234

3x 1 11y 5 38


17.

2x 2 y 5 32

x 1 y 5 22 18.

22x 1 y 1 z 5 26

x 1 2y 2 z 5 14

x 1 y 2 z 5 11

19. Verbal model:

Labels: Amount Solution

Amount Solution

System of equations:

20x 1 50y 5 600

x 1 y 5 20

0.20x 1 0.50y 5 0.30s20d

x 1 y 5 20

2 5 y

1 5 x

5 0.30 ? 20AmountSolution 2

1 0.50AmountSolution 1

0.20

AmountMixture

5AmountSolution 2

1AmountSolution 1

By substitution

gallons at 20% solution

galllons at 50% solution 20 2 x 5 623

x 5 1313

230x 5 2400

20x 1 1000 2 50x 5 600

20x 1 50s20 2 xd 5 600

y 5 20 2 x

15.

s12, 21

2, 1d c 5 1

b 5 212

a 5 12

c 5 1

b 1 32 c 5 1

a 212 c 5 0

26b 2 8c 5 25

b 1 32 c 5 1

a 1 b 1 c 5 1

26b 2 8c 5 25

22b 2 3c 5 22

a 1 b 1 c 5 1

9a 1 3b 1 c 5 4

4a 1 2b 1 c 5 2

a 1 b 1 c 5 1 16.

s5, 21, 3d

z 5 3

y 5 21

x 5 5

z 5 3

y 1112 z 5 31

2

x 1 4z 5 17

532 z 5 159

2

y 1112 z 5 31

2

x 1 4z 5 17

25y 2 z 5 2

2y 1 11z 5 31

x 1 4z 5 17

x 2 5y 1 3z 5 19

23x 1 2y 2 z 5 220

x 1 4z 5 17

Section 8.4 Matrices and Linear Systems437

Section 8.4 Matrices and Linear Systems

1. 4 3 2 3. 2 3 3 5. 4 3 1

7. 3 421

258

::

22104 9. 3

152

1023

4

2340

:::

2064 11. 35

012

234

::

7124

13.

x 2 2y 5 3

4x 1 3y 5 8 15.

4x 1 2y 5 3

3y 2 z 5 5

x 1 2z 5 210 17.

x 1 6y 2 7z 5 23

22x 1 15y 1 5z 1 w 5 9

5x 1 8y 1 2z 5 21

19.

310

42

321422R1 1 R2

312

410

354 21.

312

228

23

15419 R1

392

2188

6154

23.

3100

113

42

25

20

2165

4415 R2

3100

153

42220

21644

23R1 1 R2

2R1 1 R3

313

22

181

41012

21364

20.

y 5 x2 1 3x 2 2

c 5 22

b 5 3

a 5 1

c 5 22

b 5 3

a 1 c 5 21

23c 5 6

b 5 3

a 1 c 5 21

22b 2 3c 5 0

b 5 3

a 1 b 1 c 5 2

22b 2 3c 5 0

22b 5 26

a 1 b 1 c 5 2

8 5 as2d2 1 bs2d 1 c ⇒ 4a 1 2b 1 c 5 8

24 5 as21d2 1 bs21d 1 c ⇒ a 2 b 1 c 5 24

2 5 as1d2 1 bs1d 1 c ⇒ a 1 b 1 c 5 2


25.

310

21

3242

15 R2

310

225

3210422R1 1 R2

312

221

3244 27.

31

0

0

1

275

111042

32R3 1 R2

31

0

32

1

14

111041

5 R2

31

0

32

5

14

11242R1 1 R2

3 122

32

2

14

5414 R1

3 422

62

154 29.

3100

110

021

50

21423R2 1 R3

3100

113

027

50

2142R1 1 R2

23R1 1 R3

31

223

121

6

027

5210

144

31.

3100

2110

2161

134541

2R215R3

3100

2120

21125

1644R3

R2

3100

2102

215

12

146424R1 1 R2

6R1 1 R3

314

26

2124

8

211

18

1804

33.

3100

110

2124

8

31

2242R2

3100

121

0

2148

321224R2 1 R3

3100

121

1

2144

32121422R1 1 R2

23R1 1 R3

3123

112

2121

3584

35.

s22, 23d

x 5 22

x 1 6 5 4 y 5 23

x 2 2s23d 5 4x 2 2y 5 4

37.

s13, 22d

x 5 13

x 2 10 5 3

x 1 5s22d 5 3

y 5 22

x 1 5y 5 3 39.

s8, 0, 22d

x 5 8

x 2 4 5 4

x 2 0 1 2s22d 5 4

z 5 22

y 2 z 5 2

x 2 y 1 2z 5 4

41.

s95, 13

5 d31

0

0

1

::

95

135422R2 1 R1

310

21

::

713542

15 R2

310

225

::

7213423R1 1 R2

313

21

::

784 43.

s1, 1d

310

01

::

114

23 R2 1 R1

310

223

1::

13

14316 R2

31

0

223

163

::

13

163425R1 1 R2

315

223

2::

13

7416 R1

365

242

::

274 45.


310

220

::

21.56422R1 1 R2

312

2224

::

21.534

2R1

3212

224

::

1.534

y 5 0

y 1 2 5 2

y 2 s22d 5 2


51.

s1, 2, 21d

x 5 1 y 5 2

x 1 4 5 5 y 542

x 1 2s2d 5 5y 232 s21d 5

72z 5 21

3100

210

02

32

1

:::

572

214225 R3

31

0

0

2

1

0

0

232

252

:::

572

22524

27R2 1 R3

3100

217

02

32

2

:::

572

124212 R2

3100

222

7

032

:::

5271243R1 1 R3

310

23

222

1

032

:::

527234

12 R1

320

23

422

1

032

:::

10272342R1 1 R2

322

23

421

032

:::

103

234 53.

s1, 21, 2d

3100

010

001

:::

121

242

12R3 1 R112R3 1 R2

31

0

0

0

1

0

12

212

1

:::

2

22

242R3

3100

010

12

21212

:::

2

22

143R2 1 R1

23R2 1 R3

3100

2313

22

12

21

:::

8222541

2 R2

2R1 1 R3

3101

2320

221

1

:::

824

34

47.

s2, 23, 2d

x 5 2

x 1 6 2 2 5 6 y 5 23

x 2 2s23d 2 s2d 5 6y 1 4s2d 5 5z 5 2

3100

2210

2141

:::

65241

233 R3

3100

2210

214

233

:::

65

2664210R2 1 R3

3100

221

10

2147

:::

65

216424R1 1 R3

3104

2212

2143

:::

6584 49.

let (a is any real number)

s1 1 2a, 2 1 3a, ad

a 5 z

x 5 1 1 2z y 5 2 1 3z

x 1 s2 1 3zd 2 5z 5 3y 2 3z 5 2

3100

110

2523

0

:::

32043R2 1 R3

3100

11

23

2523

9

:::

32

2642R2

3100

12123

2539

:::

3222642R1 1 R2

22R1 1 R3

3112

10

21

252221

:::

3104


59.

let

then

s212a 2 1, 1 1 4a, ad 5 21 2 12a

5 2 2 3 2 12a

x 5 2 2 3s1 1 4ady 5 1 1 4a

z 5 a

3100

310

024

0

:::

21042R3

R2

3100

30

21

004

:::

20

21422R1 1 R2

22R1 1 R3

3122

365

004

:::

2434

55.

s34, 24, 24d

x 5 34

x 2 16 5 18 y 5 24

x 2 8 2 8 5 18 y 1 22 5 18

x 1 2s24d 1 2s24d 5 18y 2112 s24d 5 18z 5 24

3100

210

22

112

1

:::

1818

24412 R2

22R3

3100

220

2211

212

:::

1836243

2 R2 1 R3

3100

22

23

2211

16

:::

1836

25242R1 1 R2

23R1 1 R3

31

223

222

3

2215

22

:::

18024

R2

R1

322

13

2223

2152

22

:::

01824 57.


3100

110

310

:::

02

12

1422R2 1 R3

3100

112

312

:::

02

12

04212 R2

3100

122

2

322

2

:::

010422R1 1 R2

2R1 1 R3

3121

103

345

:::

0104

R2

R1

3211

013

435

:::

1004

61.

s2, 5, 52d x 5 2 y 5 5

x 152 2

52 5 2 y 2 5 5 0

x 112s5d 2 s5

2d 5 2y 2 2s52d 5 0z 5

52

3100

12

10

2122

1

:::

205241

8 R3

3100

12

10

2122

8

:::

20

20423R2 1 R3

3100

12

1

3

21

22

2

:::

2

0

20412 R1

227 R2

3200

12

72

3

2272

:::

40

204232 R1 1 R2

2R1 1 R3

323

24

122

1

2246

:::

46

124


65. Verbal model:

Labels: certificates of deposit

municipal bonds

blue-chip stocks

growth stocks


—CONTINUED—

3100

101

01

26

01

2132

:::

375,000125,000

2625,0004212 R3

3100

10

22

01

12

01

13

:::

375,000125,000

1,250,0004210R1 1 R3

310

10

108

01

12

01

13

:::

375,000125,000

5,000,0004R1

R2

31001

801

1210

1310

:::

5,000,000

125,000375,0004

0.10x

x

1 0

1

.08y

y

1 0.12z 1

z 1

0.13w 5

w 5

5

50,000125,000375,000

w 5

z 5

y 5

x 5

5 375,00Bonds1CDs

5 125,000G stocks1BC stocks

5 50,000G stocks1 0.13BC stocks1 0.12Bonds1 0.08CDs0.10 ?

63. Verbal model:

Labels: Money 1

Money 2

Money 3

$800,000 at 8%, $500,000 at 90%, $200,000 at 12%

5z

5y

5x

Money5 4 ?Money 1

113,00050.12 Money 310.09 Money 210.08 Money 1

1,500,0005Money 31Money 21Money 1

? ??

System ofequations:

x 5 800,000 y 5 500,000

x 1 500,000 1 200,000 5 1,500,000y 1 4s200,000d 5 1,300,000z 5 200,000

3100

110

141

:::

1,500,0001,300,000

200,00042R3

3100

110

14

21

:::

1,500,0001,300,0002200,0004R2 1 R3

3100

11

21

14

25

:::

1,500,0001,300,000

21,500,000428R1 1 R2

2R1 1 R3

3181

190

112

24

:::

1,500,000 13,300,000

04 x 5 4z

0.08x 1 0.09y 1 0.12y 5 133,000 x 1 y 1 z 5 1,500,000


65. —CONTINUED—

so let

then

If

CD

M Bonds

BC Stocks

G Stocks5 $100,000

5 $25,000

5 $175,000

5 $200,000

s 5 $100,000

y 5 .5s 1 125,000

y 2 .5w 5 125,000

z 5 2s 1 125,000

z 1 w 5 125,000

x 5 2.5s 1 250,000

x 1 .5w 5 250,000

w 5 s

3100

001

010

.51

2.5

:::

250,000125,000125,000426R2 1 R1

6R2 1 R3

3100

001

61

26

132

12

132

:::

1,000,000

125,0002625,00042R3 1 R1

Certificates of deposit:

Municipal bonds:

Blue-chip stocks:

Growth stocks:s

125,000 2 s

125,000 1 .5s

250,000 2 .5s

67. Verbal model:

Labels: Pounds Nut 1

Pounds Nut 2

Pounds Nut 3


15 pounds at $3.50, 10 pounds at $4.50, 25 pounds at $6.00

x 5 15 y 5 10

x 1 10 1 25 5 50y 1 2.5s25d 5 72.5z 5 25

3100

110

12.5

1

:::

50

72.52541

100 R2

2R3

3100

1100

0

125021

:::

50

725022542350R1 1 R2

2R1 1 R3

31

3501

1450

1

1600

0

:::

50

24,750254

x 1 y 5 25

3.50x 1 4.50y 1 6.00z 5 50s4.95d x 1 y 1 z 5 50

5 z

5 y

5 x

25 pounds5PoundsNut 21

PoundsNut 1

50(4.95)56.00 (Nut 3)14.50 (Nut 2)13.50 (Nut 1)

50 pounds5PoundsNut 3

1PoundsNut 2

1PoundsNut 1


71.

—CONTINUED—

3100

010

21232

212

:::

218

2242R2 1 R1

23R2 1 R3

3100

113

132

4

:::

7

822

4212 R2

212 R3

3100

12226

12328

:::

7216244424R1 1 R2

29R1 1 R3

3149

123

111

:::

712194

19 5 as3d2 1 bs3d 1 c ⇒ 19 5 9a 1 3b 1 c

12 5 as2d2 1 bs2d 1 c ⇒ 12 5 4a 1 2b 1 c

7 5 as1d2 1 bs1d 1 c ⇒ 7 5 a 1 b 1 c

69. Verbal model:

Labels: Number

Number 2

Number 3


s5, 8, 20d

x 5 5 y 5 8

x 1 8 1 20 5 33y 112 s20d 5 18z 5 20

3100

110

112

1

:::

33182041

3 R3

3100

110

112

3

:::

331860424R2 1 R3

3100

114

112

5

:::

3318

132412 R2

3100

124

115

:::

3336

1324R1 1 R2

4R1 1 R3

31

2124

110

101

:::

33304

z 5 4x

y 5 3 1 x

x 1 y 1 z 5 33

5 z

5 y

1 5 x

Number 15 4 ?Number 3

Number 15 3 1Number 2

335Number 31Number 21Number 1


71. —CONTINUED —

y 5 x2 1 2x 1 4

a 5 1, b 5 2, c 5 4

31

0

0

0

1

0

0

0

1

:::

1

2

4412 R3 1 R1

232 R3 1 R2

31

0

0

0

1

0

21232

1

:::

21

8

4422R3

73.

y 5 210.5x2 1 25.5x 2 7

x 5 210.5 y 5 25.5

x 1 20.5s27d 5 27y 1 1.5s27d 5 15z 5 27

3100

010

20.51.5

1

:::

2715

27422R3

3100

010

20.51.5

20.5

:::

27153.54

2R2 1 R1

23R2 1 R3

3100

113

11.5

4

:::

815

48.54212 R2

212 R3

3100

12226

123

281

:::

8230297424R1 1 R2

29R1 1 R3

3149

123

111

:::

82

2254225 5 as3d2 1 bs3d 1 c ⇒ 225 5 9a 1 3b 1 c

2 5 as2d2 1 bs2d 1 c ⇒ 2 5 4a 1 2b 1 c

8 5 as1d2 1 bs1d 1 c ⇒ 8 5 a 1 b 1 c

75.

—CONTINUED —

3100

10

22

12223

:::

22212212423R1 1 R2

24R1 1 R3

3134

132

111

:::

222182204

42 1 22 1 Ds4d 1 Es2d 1 F 5 0 ⇒ 4D 1 2E 1 F 5 220

32 1 32 1 Ds3d 1 Es3d 1 F 5 0 ⇒ 3D 1 3E 1 F 5 218

12 1 12 1 Ds1d 1 Es1d 1 F 5 0 ⇒ D 1 E 1 F 5 22


77. (a)

so

(b) Keystrokes: (c) Maximum height feet

.004 .6 6 Point at which the ball struck the ground feet

00

180

30

5 159.4

5 28.5

y 5 20.004x2 1 0.6x 1 6

c 5 6

b 5 0.6

a 5 20.004

3100

010

001

:::

20.004

0.664

.0008R3 1 R1

2.06R3 1 R2

3100

010

20.00080.06

1

:::

20.0088

0.966 4

2.04R2 1 R1

3100

0.0410

0.00160.06

1

:::

0.0296

0.96642

150 R2

3100

0.04250

0

0.001623

1

:::

0.0296

2486422500R1 1 R2

31

25000

0.04500

0.001611

:::

0.0296

2664

1625 R1

3625

25000

25500

111

:::

18.52664

R1

R2

R3

30

6252500

02550

111

:::

618.5

264 26 5 as50d2 1 bs50d 1 c ⇒ 26 5 2500a 1 50b 1 c

18.5 5 as25d2 1 bs25d 1 c ⇒ 18.5 5 625a 1 25b 1 c

6 5 as0d2 1 bs0d 1 c ⇒ 6 5 c

75. —CONTINUED —

x2 1 y2 2 5x 2 3y 1 6 5 0

D 5 25 E 5 23

D 1 3 5 22 E 1 9 5 6

D 1 s23d 1 6 5 22E 132s6d 5 6F 5 6

3100

110

132

1

:::

226642

12 R2

212 R3

3100

122

0

12322

:::

222122124R2

R3

Y5 x2c GRAPHx2X,T,u X,T,u1 1


Section 8.5 Determinants and Linear Systems

1. detsAd 5 |23 14| 5 2s4d 2 3s1d 5 8 2 3 5 5 3. detsAd 5 | 5

2523| 5 5s3d 2 s26ds2d 5 15 1 12 5 27

5.

5 40 2 40 5 0

detsAd 5 | 5210

248| 5 5s8d 2 s210ds24d 7. 5 6 2 0 5 6detsAd 5 |20 6

3| 5 2s3d 2 0s6d

9.

5 221 2 3 5 224

detsAd 5 |2712

36| 5 s27ds3d 2 s1

2ds6d 11.

5 .09 2 .25 5 20.16

detsAd 5 |0.30.5

0.50.3| 5 s0.3ds0.3d 2 s0.5ds0.5d

13.

(second row)

5 224

5 s26ds4d

5 2s6d|31 211| 1 0 1 0

detsAd 5 |264 301

2101|

79.

2x2 2 9xsx 2 2d3 5

2x 2 2

21

sx 2 2d2 210

sx 2 2d3

2R2 1 R33100

010

001

:::

221

2104

4R1 1 R2

24R1 1 R23100

01

22

001

:::

221284

R1

R33

124

4

01

22

001

229

04

34

241

2210

100

:::

029

24

81. (a) Interchange two rows.

(b) Multiply a row by a nonzeroconstant.

(c) Add a multiple of a row toanother row.

83. The one matrix can be obtainedfrom the other by using the ele-mentary row operations.

85. There will be a row in the matrixwith all zero entries except in thelast column.

Section 8.5 Determinants and Linear Systems447

17.

(first column)

5 s2ds215d 5 230

5 s2d|30 125| 2 0 1 0

detsAd 5 |200 430

61

25| 19.

(second row)

5 25 1 8 5 3

5 s21ds5d 1 s21ds28d

5 2s1d|21 34| 1 s21d|22

034| 2 0

detsAd 5 |2210

221

1

304|

21.

(first row)

5 30 2 0 2 30 5 0

5 s1ds30d 2 s4ds0d 1 s22ds15d

5 s1d|61 264| 2 s4d| 3

2226

4| 1 s22d| 322

61|

detsAd 5 | 13

22

461

2226

4|

23.

(third column)

5 222 2 30 2 23 5 275


5 s1d|42 523| 2 s6d|23

22

23| 1 s1d|234

25|

detsAd 5 |2342

253

161|

25.

(second row)

5 258

5 260 1 2

5 s23ds20d 1 s2ds1d

5 2s3d|44 223| 1 s2d| 1

2122

3| 2 0

detsAd 5 | 13

21

424

2203| 27.

(third column)

5 102

5 s3ds34d

5 0 2 0 1 3|24 257|

detsAd 5 | 24

27

257

25

003|

15.

(third column)

5 4 2 6 5 22

5 s2ds2d 2 0 1 s3ds22d

5 s2d| 322

10| 2 s0d| 1

2210| 1 s3d|13 1

1|detsAd 5 | 1

322

110

203|


35. Keystrokes:

3 3 3 1

2 1 1 2 2 3 10

Solution is 232.

29.

(third row)

5 20.22

5 20.1 2 0.44 1 0.32

5 s5ds20.02d 2 s4ds0.11d 1 s4ds0.08d

5 s5d|0.20.2

0.30.2| 2 s4d| 0.1

20.30.30.2| 1 s4d| 0.1

20.30.20.2|

detsAd 5 | 0.120.3

5

0.20.2

4

0.30.2

4|

31.

(third row)

5 x 2 5y 1 2

5 22y 1 2 1 x 2 3y

5 s22dsy 2 1d 1 s1dsx 2 3yd

5 s22d|y1

11| 2 0 1 s1d|x

3y1|

detsAd 5 | x3

22

y10

111|

33. Keystrokes:

3 3 5 3 2 7 5 7

0 6 1

Solution is 248.

MATRX 1MATH 1

EDIT 1 ENTER

QUIT

ENTERENTER

ENTER ENTER ENTER

ENTER

ENTER ENTER ENTER ENTERx2c

x2c

x2c

ENTER ENTER

ENTER ENTER ENTER

ENTER

ENTER ENTER ENTER ENTER

ENTER ENTERx2c

x2cx2c

MATRX ENTER

MATRX

QUIT

37. Keystrokes:

3 3 .2 .8 .3

.1 .8 .6 10 5

1

Solution is 26.37

QUIT

MATRX ENTER ENTER

ENTER

ENTER

ENTER ENTER

ENTER

ENTER ENTER

ENTER ENTERx2c

x2c x2c

MATRX 1MATH 1MATRX

EDIT 1MATRX

ENTEREDIT 1

MATRX 1MATH 1MATRX


47.

123

, 122

y 5Dy

D5

|36 511|

65

33 2 306

536

512

x 5Dx

D5

| 511

614|

65

70 2 666

546

523

D 5 |36 614| 5 42 2 36 5 6

336

614

::

5114

39.

s1, 2d

y 5Dy

D5

| 121

51|

35

1 2 s25d3

563

5 2

x 5Dx

D5

|51 21|

35

5 2 23

533

5 1

D 5 | 121

21| 5 1 2 s22d 5 3

3 121

21

::

514 41.

s2, 22d

y 5Dy

D5

|35 224|

2115

12 2 s210d211

522

2115 22

x 5Dx

D5

|224

43|

2115

26 2 16211

5222211

5 2

D 5 |35 43| 5 9 2 20 5 211

335

43

::

2244

43.

134

, 2122

y 5Dy

D5

|2012

1121|

25765

420 2 1322576

5288

25765 2

12

x 5Dx

D5

|1121

8224|

25765

2264 2 1682576

524322576

534

D 5 |2012

8224| 5 2480 2 96 5 2576

32012

8224

::

11214 45.

Cannot be solved by Cramer’s Rule because

Solve by elimination.


0 Þ 26

4x 2 8y 5 10 2x 2 4y 5 5 ⇒

24x 1 8y 5 16 24x 1 8y 5 16 ⇒

D 5 0.

D 5 |20.42

0.824| 5 1.6 2 1.6 5 0

320.42

0.824

::

1.654

49.

—CONTINUED—

5 214 1 9 1 60 5 55

5 s1ds214d 1 s23ds23d 1 s6ds10d

D 5 |425 212

22

136| 5 s1d|25 2

22| 2 s3d|45 2122| 1 s6d|42 21

2|3425

212

22

136

:::

251014


49. —CONTINUED—

s21, 3, 2d

50 1 100 1 10

555

11055

5 2

5s5ds0d 1 s2ds50d 1 s1ds10d

55

z 5 |425 212

22

25101|

555

s5d|212

2510| 2 s22d|42 25

10| 1 s1d|42 212|

55

5248 2 87 1 300

555

16555

5 3


55

y 5 |425 25101

136|

555

s1d|25 101| 2 s3d|45 25

1| 1 s6d|42 2510|

55

5222 2 33

555

25555

5 21


55

x 5 |25101

212

22

136|

555

s1d|101

222| 2 s3d|25

12122| 1 s6d|25

1021

2|55

51.

—CONTINUED—

52216 2 468

2525

252252

5 1

5s4ds254d 2 s6ds278d

252

x 5 |11113

42426

460|

2525

s4d|113

2426| 2 s6d|11

34

26| 1 0

252

5 252

5 s4ds0d 2 s6ds242d 1 0

D 5 |346 42426

430| 5 s4d|46 24

26| 2 s6d|36 426| 1 0

3346

42426

460

:::

111134


53.

—CONTINUED—

526 2 6 1 8

25

242

5 22


2

b 5 |335 124

49

17|2

5

s3d|24 917| 2 s2d|35 9

17| 1 s4d|35 24|

2

54 2 30 1 28

25

22

5 1


2

a 5 |124 359

49

17|2

5

s1d|59 917| 2 s2d|39 4

17| 1 s4d|35 49|

2

5 2

5 12 2 45 1 35


D 5 |335 359

49

17| 5 s3d|59 917| 2 s3d|39 4

17| 1 s5d|35 49|

3335

359

49

14

:::

1244

51. —CONTINUED—

11, 12

, 322

5162 2 312 1 528

2525

378252

532


252

z 5 |346 42426

11113|

2525

s3d|2426

113| 2 s4d| 4

26113| 1 s6d| 4

241111|

252

52216 1 342

2525

126252

512

5s4ds254d 2 s6ds257d

252

y 5 |346 11113

460|

2525

s4d|46 113| 2 s6d|36 11

3| 1 0

252


55.

Cannot be solved by Cramer’s Rule because D 5 0.

5 0

5 225 1 20 1 5


D 5 |521 2327

223

8| 5 s5d| 227

238| 2 s2d|23

2728| 1 s1d|23

22

23|3521

232

27

223

8

:::

23

244

57.

y 5Dy

D5

|239

220|

2815

2198281

5229

x 5Dx

D5

|220

1023|

2815

266281

52227

D 5 |239

1023| 5 281

3239

1023

::

2204

53. —CONTINUED—

s1, 22, 1d

52 2 24 1 24

25

22

5 1


2

c 5 |335 359

124|

25

s1d|35 59| 2 s2d|35 3

9| 1 s4d|33 35|

2

Keystrokes:

det D

2 2

Enter each number in matrix followed by

2 2

Enter each number in matrix followed by

2 2

Enter each number in matrix followed by ENTER

ENTERENTERMATRX

det Dy

ENTER

ENTERENTERMATRX

det Dx

ENTER

ENTERENTERMATRX

QUIT

QUIT

QUIT

ENTER

ENTER

ENTER

MATRX 1MATH 1MATRX

EDIT 1

EDIT 2

EDIT 3

MATRX 2MATH 1MATRX

MATRX 3MATH 1MATRX


67.

Area or 15125 1

12 s31d 5

312

5 31

5 19 1 13 2 1


|x1

x2

x3

y1

y2

y3

111| 5 |22

31

121

6

111| 5 s1d|31 21

6| 2 s1d|221

16| 1 s1d|22

31

21|sx1, y1d 5 s22, 1d, sx2, y2d 5 s3, 21d, sx3, y3d 5 s1, 6d

59.

Keystrokes:

3 3

Enter each number in matrix followed by .

3 3


3 3


3 3


ENTERENTERMATRXdet Dz

ENTERENTERMATRXdet Dy

ENTERENTERMATRXdet Dx

ENTERENTERMATRXdet D

15116

, 27

16, 2

13162

z 5Dz

D5 |311 22

32

82325|

485

23948

5 21316

y 5Dy

D5 |311 8

2325

369|

485

22148

52716

x 5Dx

D5 | 8

2325

2232

369|

485

15348

55116

D 5 |311 2232

369| 5 48

61.

x 5 1x 5 6

sx 2 6dsx 2 1d 5 0

x2 2 7x 1 6 5 0

10 2 7x 1 x2 2 4 5 0

s5 2 xds2 2 xd 2 4 5 0

63.

Area5 112

s32d 5 16

|x1

x2

x3

y1

y2

y3

111| 5 |048 3

05

111| 5 32

sx1, y1d 5 s0, 3d, sx2, y2d 5 s4, 0d, sx3, y3d 5 s8, 5d

65.

Area5 112

s14d 5 7

5 s1ds14d 5 14

|x1

x2

x3

y1

y2

y3

111| 5 |031 0

15

111| 5 s1d|31 1


ENTER

ENTER

ENTER

ENTER

ENTER

ENTER

ENTER

ENTER

QUIT

QUIT

QUIT

QUIT

MATRX 1MATH 1MATRX

EDIT 4

EDIT 1

MATRX 2MATH 1MATRX

MATRX 3MATH 1MATRX

EDIT 3

EDIT 2

MATRX 4MATH 1MATRX


69.

Area512s33

4 d 5338

5 334

5 34 1

304

5 34 1

152

5 212s23

2d 1 1s152 d

5 0 212s5

2 2 4d 1 1s152 2 0d

|x1

x2

x3

y1

y2

y3

111| 5 |0524 1

2

03

111| 5 0|03 1

1| 212| 5

2

411| 1 1| 5

2

403|

sx3, y3d 5 s4, 3dsx2, y2d 5 s52, 0dsx1, y1d 5 s0, 12d

71. Verbal model:

Equation:

Let

Area

Let

Area512 s9d 5 4.5

5 24s1d 2 1s213d 5 24 1 13 5 9

|x1

x2

x3

y1

y2

y3

111| 5 |345 5

04

111| 5 24|54 1

1| 1 0 2 1|35 54|


512 s23d 5 11.5

5 24s23d 2 1s211d 5 12 1 11 5 23

|x1

x2

x3

y1

y2

y3

111| 5 |21

43

205

111| 5 24|25 1

1| 1 0 2 1|213

25|


5 16

A 5 11.5 1 4.5

Area of Triangle 2

1Area of

Triangle 15

Area of Shaded Region

73. Verbal Model:

Equation:

Let

Area512 s19d 5 9.5

|x1

x2

x3

y1

y2

y3

111| 5 |23

21

2122

2

111| 5 19


5 26.5

5 36 2 9.5

A 5 s9ds4d 2 9.5

Area of Triangle

2Area of

Rectangle5

Area of Shaded Region


77. Let

The three points are collinear.

5 0

5 26 1 6

5 s21ds6d 1 s2ds3d

|x1

x2

x3

y1

y2

y3

111| 5 |21

02

1182

111| 5 s21d|82 1

1| 1 0 1 s2d|118

11|


79.

The three points are collinear.

5 0

5 9 2 15 1 6


5 s1d|14 215| 2 s1d|21

4

255| 1 s1d|21

12521||x1

x2

x3

y1

y2

y3

111| 5 |21

14

2521

5

111|


81. Let

The three points are not collinear.

5 25815

5 21815 2

4015

5 2135 1

75 2

83

5 s1ds2135 d 2 s1ds27

5d 1 s1ds283d

|x1

x2

x3

y1

y2

y3

111| 5 |22

23

13

115

111| 5 s1d|23 1

15| 2 s1d|22

3

1315| 1 s1d|22

2

13


75.

From diagram the coordinates of A, B, C are determined to be and

Area5 212 s2500d 5 250 mi2

|x1

x2

x3

y1

y2

y3

111| 5 | 0

1028

2025

0

111| 5 2500

Cs28, 0d.As0, 20d, Bs10, 25d

A

C

B


83.

3x 2 5y 5 0

s1ds3x 2 5yd 5 0

s1d|x5

y3 | 5 0

|x05

y03

111| 5 0

sx1, y1d 5 s0, 0d, sx2, y2d 5 s5, 3d

87.

9x 1 10y 1 3 5 0

92 x 1 5y 132 5 0

x| 32

2311| 2 y|22

311| 1 1|22

3

32

23| 5 0

| x22

3

y32

23

111| 5 0

sx2, y2d 5 s3, 23dsx1, y1d 5 s22, 32d, 89.

32x 2 30y 1 44 5 0

23.2x 1 3y 2 4.4 5 0

26.4x 1 6y 2 8.8 5 0

xs3.6 2 10d 2 ys2 2 8d 1 1s20 2 28.8d 5 0

x|3.610

11| 2 y|28 1

1| 1 1|28 3.610| 5 0

|x28

y3.610

111| 5 0

sx2, y2d 5 s8, 10dsx1, y1d 5 s2, 3.6d

91.

y 5 2x2 2 6x 1 1

c 5 |014 01

22

12321|

265

s1d|14 122|

265

s1ds26d26

5 1

5s21ds3d 1 s1ds33d

265

3626

5 26

b 5 |014 12321

111|

265

2s1d|14 11| 1 s1d|14 23

21|26

5s1ds3d 1 s1ds215d

265

21226

5 2

a 5 | 12321

01

22

111|

265

s1d| 122

11| 2 0 1 s1d|23

211

22|26

D 5 |014 01

22

111| 5 s1d|14 1

22| 5 s1ds26d 5 26

3014

01

22

111

:::

123214

21 5 as22d2 1 bs22d 1 c ⇒ 21 5 4a 1 2b 1 c

23 5 as1d2 1 bs1d 1 c ⇒ 23 5 a 1 b 1 c

1 5 as0d2 1 bs0d 1 c ⇒ 1 5 1 c

85.

7x 2 6y 2 28 5 0

14x 2 12y 2 56 5 0

256 1 7x 2 2y 1 7x 2 10y 5 0

s1ds256d 2 s27x 1 2yd 1 s1ds7x 2 10yd 5 0

s1d| 1022

727| 2 s1d| x

22y

27| 1 s1d| x10

y7| 5 0

| x10

22

y7

27

111| 5 0

sx1, y1d 5 s10, 7d, sx2, y2d 5 s22, 27d

−2.3 1.3

−7.08

25.08


93.

y 512

x2 2 2x

c 5Dc

D5 | 4

416

2224

622

0|248

50

2485 0

b 5Db

D5 | 4

416

622

0

111|

2485

96248

5 22

a 5Da

D5 | 6

220

2224

111|

2485

224248

512

D 5 | 44

16

2224

111| 5 2483

44

16

2224

111

:::

622

04 0 5 a s4d2 1 b s4d 1 c ⇒ 6 5 16a 1 4b 1 c

22 5 a s2d2 1 b s2d 1 c ⇒ 22 5 4a 1 2b 1 c

−3

−3

7

7 6 5 as22d2 1 bs22d 1 c ⇒ 6 5 4a 2 2b 1 c

95.

or

—CONTINUED—

56 1 21 2 9

265

1826

5 23


26

a 5 |2125

1

121

2

114|

265

s21d|212

14| 2 s1d|25

114| 1 s1d|25

121

2|26

5 26

5 26 2 2 1 2


D 5 |111 121

2

114| 5 s1d|21

214| 2 s1d|12 1

4| 1 s1d| 121

11|

3111

121

2

114

:::

2125

141 5 a 1 2b 1 4c 14 5 as1

2d2 1 bs12d 1 c ⇒ 14 5 14 a 1

12 b 1 c

25 5 as21d2 1 bs21d 1 c ⇒ 25 5 a 2 b 1 c

1.1

1.2

−6

−1.2

21 5 as1d2 1 bs1d 1 c ⇒ 21 5 a 1 b 1 c


95. —CONTINUED—

y 5 23x2 1 2x


265

9 2 3 2 626

5 0

c 5 |111 121

2

2525

1|26

5

s1d|212

251| 2 s1d|12 21

1| 1 s1d| 121

2125|

26

5221 1 5 1 4

265

21226

5 2


26

b 5 |111 2125

1

114|

265

s1d|251

14| 2 s1d|21

114| 1 s1d|21

2511|

26

97. (a)

(b)

—CONTINUED—

a2 5 |743.4791.4870.7

567

111|

225

231.322

5 15.65detsAd 5 |253649

567

111| 5 22

as7d2 1 bs7d 1 c 5 870.7 ⇒ 49a2 1 7b 1 c 5 870.7

as6d2 1 bs6d 1 c 5 791.4 ⇒ 36a2 1 6b 1 c 5 791.4

as5d2 1 bs5d 1 c 5 743.4 ⇒ 25a2 1 5b 1 c 5 743.4

s5, 743.4d s6, 791.4d s7, 870.7d

y1 5 12.15t2 2 93.55t 1 748.7

c1 5 |253649

567

584.7624.8689.2|

225

21497.422

5 748.7

5 293.55 5 12.15

b1 5 |253649

584.7624.8689.2

111|

225

187.122

a1 5 |584.7624.8689.2

567

111|

225

224.322

detsAd 5 |253649

567

111| 5 22

as7d2 1 bs7d 1 c 5 689.2 ⇒ 49a 1 7b 1 c 5 689.2

as6d2 1 bs6d 1 c 5 624.8 ⇒ 36a 1 6b 1 c 5 624.8

as5d2 1 bs5d 1 c 5 584.7 ⇒ 25a 1 5b 1 c 5 584.7

s5, 584.7d s6, 624.8d s7, 689.2d



1. (a) (b)

Not a solution Solution

9 5 9 2 5 2 37 Þ 2

5s3d 1 6s21d 5?

93s3d 1 7s21d 5?

23s3d 1 7s4d 5?

2

s3, 21ds3, 4d

99. (a)

(b)

k 512

2k 5 1

2k 2 1 5 0

14k 2 32k 2 1

, 4k 2 12k 2 12

y 5Dy

D5

| k1 2 k

13|

2k 2 15

3k 2 1s1 2 kd2k 2 1

53k 2 1 1 k

2k 2 15

4k 2 12k 2 1

x 5Dx

D5

|13 1 2 k

k |2k 2 1

5k 2 3s1 2 kd

2k 2 15

k 2 3 1 3k2k 2 1

54k 2 32k 2 1

D 5 | k1 2 k

1 2 k

k | 5 k2 2 s1 2 kd2 5 k2 2 s1 2 2k 1 k2d 5 k2 2 1 1 2k 2 k2 5 2k 2 1

3 k1 2 k

1 2 k

k :

:

134

101. A determinant is a real number associated with a squarematrix.

103. The minor of an entry of a square matrix is the determi-nant of the matrix that remains after deleting the row andcolumn in which the entry occurs.

97. —CONTINUED—

(c) (d)

(e) The trade deficit is increasing.010

−2800

4

5 23.5t2 1 30.6t 2 224.2

s15.65t2 2 124.15t 1 972.9dy1 2 y2 5 s12.15t2 2 93.55t 1 748.7d 212,500

100

4

y1

y2

y2 5 15.65t2 2 124.15t 1 972.9

5 972.9 5 2124.15

c2 5 |253649

567

743.4791.4870.7|

225

21945.822

b2 5 |253649

743.4791.4870.7

111|

225

248.322



Point of intersection is s1, 1d.

x3

0y

)11,

2

x

1

(

y

3

1

2

1

yx1 2

y 5 x

2y 5 2x y 5 2x 1 2

x 2 y 5 0x 1 y 5 2


No solution

1

x

y

3

3

y

x

x

1

y

1

2

1

3

1

2

23

4

2

y 5 x 2 3

y 5 x 1 1 2y 5 2x 1 3

2x 1 y 5 1x 2 y 5 3



x

4

8

0

6

)8

yx

x2

4

(4

y

22

2

4

66

8

10

,

y

y 5 2x

y 5 x 1 4 2y 5 22x

2x 1 y 5 42x 2 y 5 0


Infinite number of solutions.

x

y

−1 1 3 4 5

2

4

1

3

−1

2 + = 4x y

− − −4 2 = 8x y

y 5 22x 1 4

22y 5 4x 2 8 y 5 22x 1 4

24x 2 2y 5 282x 1 y 5 4

3. (a)

Solution

30 5 30

80 2 50 5?

30

20s4d 1 10s25d 5?

30

41 5 41

42 1 s25d2 5?

41

s4, 25d (b)

Not a solution

193 Þ 41

72 1 122 5?

41

s7, 12d



x

y

−2−3 1 2 3

2

1

3

4

−2

(0, 1)

3 2 = 2x y− −−5 + 2 = 2x y

y 552 x 1 1 y 5

32 x 1 1

2y 5 5x 1 2 22y 5 23x 2 2

25x 1 2y 5 23x 2 2y 5 22



Keystrokes:

5 3 1

7

Solution is s3, 4d

y2

y1

12

14

−6

−10

y 5 21x 1 7 y 553 x 2 1

2y 5 22x 1 14 23y 5 25x 1 3

2x 1 2y 5 145x 2 3y 5 3

Y5 cs 4 2 ENTER

x2c 1 GRAPH


Keystrokes:

4

2 3 11 3

Solutions are and s213, 235

9 ds1, 23d

y2

y1

y 523 x 2

113

23y 5 22x 1 11

2x 2 3y 5 11y 5 x2 2 4−3

−5

5

2

Y5 2 ENTER

cs 4 2 4 GRAPH

19.

s2, 21d

x 5 2

x 5 22 2 4s21d

y 5 21

25y 5 5

24 2 8y 1 3y 5 1

2s22 2 4yd 1 3y 5 1

x 5 22 2 4y 21.

No solution

28 Þ 7

10x 2 10x 2 8 5 7

10x 2 2s5x 1 4d 5 7

10x 2 415x 1 42 2 5 7

y 55x 1 4

2

2y 5 5x 1 4

X,T,u

X,T,u

X,T,u x2

X,T,u


27.

s0, 21d, s21, 0d

y 5 0y 5 21

x 5 21x 5 0

2xsx 1 1d 5 0

2x2 1 2x 5 0

x2 1 1 1 2x 1 x2 5 1

x2 1 s21 2 xd2 5 1

y 5 21 2 x

x 1 y 5 21

x2 1 y2 5 1

23.

s210, 25d

x 5 210

x 57s25d 1 5

3

y 5 25

8y 5 240

35y 1 25 2 27y 5 215

5s7y 1 5d 2 27y 5 215

517y 1 53 2 2 9y 5 25

x 57y 1 5

3

3x 5 7y 1 5 25.

s22, 20d, s21, 5d

5 5 5 20

y 5 5s21d2y 5 5s22d2

x 5 21x 5 22

sx 1 2dsx 1 1d 5 0

x2 1 3x 1 2 5 0

5x2 1 15x 1 10 5 0

5x2 5 215x 2 10

y 5 215x 2 10

y 5 5x2

29.

s0, 0d

y 5 0

0 1 y 5 0

5 05 0

2xx

2x 1 y 5 0 ⇒ 22x 2 y 5 0

x 1 y 5 0 ⇒ x 1 y 5 0

31.

s52, 3d

y 5 3

2y 5 23

5 2 y 5 2

2s52d 2 y 5 2

5 555

5522 5

52

22xx

6x 1 8y 5 39 ⇒ 6x 1 8y 5 39

2x 2 y 5 2 ⇒ 16x 2 8y 5 16


33.

s20.5, 0.8d

x 5 212 5 20.5

2x 5 21

2x 1 2.4 5 1.4

2x 1 3s0.8d 5 1.4

y 5 0.8

2y 5 20.8

0.2x 1 0.3y 5 0.14 ⇒0.4x 1 0.5y 5 0.20 ⇒

2x 1 3y 5

4x 1 5y 5

1.42

⇒ ⇒

24x4x

2 6y 5

1 5y 5

22.8 2

35.

s2, 23, 3d

x 5 2

x 1 3 2 6 5 21

x 2 s2 3d 2 2s3d 5 21

5 23y

3

0

z 5

3 5y 1

21

0

2z

z

5

5

y 2

y 1

x 2

21

0

9

2z

z

3z

5

5

5

y 2

y 1

x 2

21

0

9

2z

z

12z

5

5

5

y 2

y 1

9y 1

x 2

21

0

9

2z

5z

12z

5

5

5

y 2

5y 1

9y 1

x 2

21

22

4

2z

z

2z

5

5

5

y 2

3y 1

4y 1

x 2

2x 1

5x 1

37.

s0, 1, 22d

x 5 0

x 2 1 2 s22d 5 1

y 5 1

2y 5 21

2y 1 s22d 5 23

z 5 22

9z 5 218

2y 1 z 5 23

x 2 y 2 z 5 1

7y 1 2z 5 3

2y 1 z 5 23

x 2 y 2 z 5 1

3x 1 4y 2 z 5 6

22x 1 y 1 3z 5 25

x 2 y 2 z 5 1 39.

s1, 0, 24d

y 5 0

2y 5 0

5s1d 2 y 1 2s24d 5 23

24

17

1

5

5

5

z

24s24d

x

x

17

2332

23

5

5

5

4z

83z

2z

2

1

y

2

x

5x

17

226

23

5

5

5

4z

11z

2z

2

1

1

y

2

x

18x

5x

17

214

23

5

5

5

4z

3z

2z

2

1

1

4y

y

1

2

x

22x

5x


41.

x 5 10

s10, 212dx 2 s212d 5 22y 5 212

310

211

::

2221241

9 R2

310

219

::

222108425R1 1 R2

315

214

::

2224

2R1

R1

R2 321

514

::

22224

3 521

41

::

22224 43.

x 5 .6

s0.6, 0.5dx 212 s.5d 5 .35y 5 .5

310

212

1::

.35542

13 R2

310

212

23::

.3521.5424R1 1 R2

314

212

25::

.352.14

12 R1

324

2125

::

.72.14

10R1

10R2

3.2.4

2.12.5

::

.072.014

45.

s245 , 22

5 , 285d

x 5245 y 5

225

x 1 2s225 d 1 6s28

5d 5 4y 1178 s28

5d 5 1z 5 285

3100

210

6178

1

:::

41

2854

215 R3

3100

210

6178

25

:::

41848R2 1 R3

3100

21

28

6178

222

:::

41041

8 R2

3100

28

28

617

222

:::

48043R1 1 R2

24R1 1 R3

31

234

220

621

2

:::

424164 47.

s12, 21

3 , 1d x1 5

12

x1 1 1 532

x1 212 1

32 5

32

x1 132 s21

3 d 132 s1d 5

32

x2 5 213

x2 2 1 5 243

x3 5 1

3100

32

10

32

21

0

:::

34

245

142128 R3

31

0

0

32

1

0

32

21

228

:::

32

243

22849R2 1 R3

3100

32

129

32

21219

:::

32

243

2164213 R2

3100

32

2329

32

3219

:::

32

4216426R1 1 R2

212R1 1 R3

316

12

32

69

32

1221

:::

32

1324

326

12

369

312

21

:::

31324

12 R1

49. detsAd 5 | 710

1015| 5 s7ds15d 2 s10ds10d 5 105 2 100 5 5


51.

5 251

5 227 2 24

5 s3ds29d 2 0 1 s2ds212d

(thirdrow)

5 s3d|63 30| 2 0|86 3

0| 1 2|86 63|

detsAd 5 |863 630

302| 53.

(third row)

5 1

5 s6ds16d 1 s5ds219d

5 6| 322

24| 2 0 1 5|81 3

22|detsAd 5 |816 3

220

245|

57.

Cannot be solved by Cramer’s Rule because . Solveby elimination.


0 Þ 269

12x 2 8y 5 25

212x 1 8y 5 264

D 5 0

D 5 | 312

2228| 5 224 1 24 5 0

3 312

2228

::

1625455.

s23, 7d

y 5Dy

D5

|72 6315|

235

105 2 12623

522123

5 7

x 5Dx

D5

|6315

123|

235

189 2 18023

59

235 23

D 5 |72 123| 5 21 2 24 5 23

372

123

::

63154

59.

—CONTINUED—

58 1 1 1 36

2155

45215

5 23


215

y 5 |2125

122

4

212|

2155

s21d|224

12| 2 s1d|25 1

2| 1 s2d|25 224|

215

52 1 8 2 40

2155

230215

5 2


215

x 5 | 122

4

134

212|

2155

s1d|34 12| 2 s1d|22

412| 1 s2d|22

434|

215

5 22 1 1 2 14 5 215


D 5 |2125

134

212| 5 s21d|34 1

2| 2 s1d|25 12| 1 s2d|25 3

4|321

25

134

212

:::

122

44


59. —CONTINUED—

s2, 23, 3d 5220 2 18 2 7

2155

245215

5 3


215

z 5 |2125

124

122

4|215

5

s21d|34 224| 2 s1d|25 22

4| 1 s1d|25 34|

215

61.

There are many other correct solutions. Write equations so that satisfies each equation.s23, 24d

23x 1 2y 5 210

3x 2 y 5 6

63. VerbalModel:

Labels: Total cost

Cost per unit

Number of units

Initial cost

Total revenue

Price per unit

System ofequations:

items x 5 16,666.6 < 16,667

1.50x 5 25,000

5.25x 5 3.75x 1 25,000

R 5 C

R 5 5.25x C 5 3.75x 1 25,000

5 5.25

5 R

5 25,000

5 x

5 3.75

5 C

Number of units

?Price per unit

5Total

Revenue

Initial cost

1Number of units

?Cost per unit

5TotalCost

65. VerbalModel:

Labels: Gallons Solution 1

Gallons Solution 2




x 5 100 2 60 5 40

y 5 60

225y 5 21500

7500 2 75y 1 50y 5 6000

75s100 2 yd 1 50y 5 60s100d

x 5 100 2 y

0.75x 1 0.50y 5 0.60s100d x 1 y 5 100

5 y

5 x

0.60(100)5Value

Solution 21

ValueSolution 1

1005Gallons

Solution 21

GallonsSolution 1

67. Verbal model:

Labels: Length

Width

System ofequations:

meters in width

meters in length x 5 1.50s96d 5 144

y 5 96

5y 5 480

3y 1 2y 5 480

2s1.50yd 1 2y 5 480

x 5 1.50y 2x 1 2y 5 480

5 y

5 x

Width5 1.50 ?Length

Perimeter5Width1 2 ?Length2 ?


69. Verbal Model:

Labels: Number Tapes 1

Number Tapes 2

System ofequations:

tapes at $9.95

tapes at $14.95 y 5 650 2 400 5 250

x 5 400

25x 5 22000

9.95x 1 9717.50 2 14.95x 5 7717.50

9.95x 1 14.95s650 2 xd 5 7717.50

y 5 650 2 x

9.95x 1 14.95y 5 7717.50 x 1 y 5 650

5 y

5 x

5 7717.50ReceiptsTapes 2

1ReceiptsTapes 1

5 650NumberTapes 21

NumberTapes 1

71. Verbal Model:

Labels: Speed Plane 1

Speed Plane 2

Time

Distance miles

System ofequations:

mph

mph y 5 250 1 40 5 290

x 5 250

2x 5 500

2x 1 40 5 540

56 s2x 1 40d 5 450

56 x 1

56 sx 1 40d 5 450

y 5 x 1 40x ? 5

6 1 y ? 56 5 450

5 450

55060 5

56 hr

5 y

5 x

1 40SpeedPlane 1

5SpeedPlane 2

Distance5Time?SpeedPlane 2

1Time?SpeedPlane 1


73. Verbal model:

Labels: Number Number Number

System ofequations:

s16, 20, 32d

x 5 16 2y 1 3z 5 136

x 1 20 1 32 5 68 y 112 z 5 36

x 1 y 1 z 5 68

y 5 20 2y 1 3z 5 136

y 112 s32d 5 36 2y 1 z 5 72

z 5 32 x 1 y 1 z 5 68

2z 5 6422x 1 z 5 0

y 112 z 5 36 2x 1 y 5 4

x 1 y 1 z 5 68 x 1 y 1 z 5 68

z 5 2x

y 5 4 1 x x 1 y 1 z 5 68

3 5 z2 5 y1 5 x

Number1

5 2 ?Number

3

Number1

5 4 1Number

2

5 68Number

31Number

21Number

1

75.

y 5 3x2 1 11x 2 20c 5 |2514

2512

02614|

2425

840242

5 220

b 5Db

D5 |25

14

02614

111|

2425

2462242

5 11

a 5Da

D5 | 0

2614

2512

111|

2425

2126242

5 3

D 5 |2514

2512

111| 5 242

32514

2512

111

:::

026144

0

26

14

5

5

5

as25d2

as1d2

as2d2

1 bs25d 1

1 bs1d 1

1 bs2d 1

c

c

c

⇒

⇒

⇒

0

26

14

5

5

5

25a

a

4a

2

1

1

5b

b

2b

1 c

1 c

1 c


77.

Area5 112

s32d 5 16

|x1

x2

x3

y1

y2

y3

111| 5 |155 0

08

111| 5 20 1 0 2 s8d|15 1

1| 5 s28ds24d 5 32


79.

Area5 112

s14d 5 7

5 14

5 23 1 4 2 13


|x1

x2

x3

y1

y2

y3

111| 5 |143 2

252

111| 5 s1d|43 25

2| 2 s1d|13 22| 1 s1d|14 2


81.

x 2 2y 1 4 5 0

24x 1 8y 2 16 5 0

4y 2 16 2 4x 1 4y 5 0

s4dsy 2 4d 2 s1ds4x 2 4yd 5 0

2s24d|y4

11| 1 0 2 s1d|x

4y4| 5 0

| x24

4

y04

111| 5 0 83.

2x 1 6y 2 13 5 0

213 2 x 172 y 1 3x 1

52 y 5 0

s1ds213d 2 s1dsx 272 yd 1 s1ds3x 1

52 yd 5 0

s1d|25272

3

1| 2 s1d|x72 y1| 1 s1d| x

252

y3| 5 0

| x2

5272

y

3

1

1

1

1| 5 0


1. (a)

Not a solution

(b)

Solution

0 5 0 1 5 1

21 1 1 5 0 2 2 1 5 1

21 1 2s12d 5

?02s1d 2 2s1

2d 5?

1

s1, 12d

6 1 8 Þ 1

2s3d 2 2s24d 5?

1

s3, 24d 2.

s2, 4d

y 5 4

y 5 5s2d 2 6

x 5 2 y 5 5x 2 6

211x 5 222 2y 5 25x 1 6

4x 2 15x 1 18 5 244x 2 3y 5 24

4x 2 3s5x 2 6d 5 245x 2 y 5 6


CHAPTER 8 Systems of Equations

SECTION 8.1 Systems of Equations

1. Answers vary.

x

y

−1−3 −2 1 2

2

3

4

−1

−2

2. Answers vary.

x

y

−1 1 2 4 5

2

1

3

−1

−2

−3

3.

223 ? 3

2 5 21

m1 ? m2 5 21

32

4. The line with is steeperbecause this line’s slope is thegreater absolute value.

m 5 23 5.

y 55

11

211y 5 25

y 2 12y 1 6 5 1

y 2 3s4y 2 2d 5 1 6.

x 51411

211x 5 214

x 1 18 2 12x 5 4

x 1 6s3 2 2xd 5 4

7.

x 5150

7

7x 5 150

5x 1 2x 5 150

12 x 115 x 5 15 8.

x 5 64

x 2 4 5 60

110sx 2 4d 5 6 9.

y 5 234 x 1

54

4y 5 23x 1 5

3x 1 4y 2 5 5 0

10.

y 52

23x 1 2

23y 5 2x 2 6

22x 2 3y 1 6 5 0 11.

x

y

−1−2 2 3 4

2

1

−1

−2

−3

y 5 23x 1 2 12.

x

y

−3 −2−4 1 2

2

3

4

−1

−2

y 5 2x 1 2

22y 5 24x 2 4

4x 2 2y 5 24


13.

y 5 232 x 1 4

2y 5 23x 1 8

x

y

−1 1 2 4 5

2

4

1

3

−1

3x 1 2y 5 8 14.

x 5 23

x

y

−1−2−5 −4 1

2

1

3

−1

−2

−3

x 1 3 5 0

SECTION 8.2 Linear Systems in Two Variables

1.


2sx 1 yd 5 2x 1 2y 2.


x 5 11

x 2 4 1 4 5 7 1 4

x 2 4 5 7 3.

22 < x < 2

24 < 2x < 4

1 < 2x 1 5 < 9

4.

4 ≤ x < 16

0 ≤ x 2 4 < 12

0 ≤x 2 4

2< 6 5.

x > 2 x < 22

6x > 12 or 6x < 212

|6x| > 12 6.

22 < x < 3

3 > x > 22

26 < 22x < 4

25 < 1 2 2x < 5

|1 2 2x| < 5

7.

x < 3

4x < 12

4x 2 12 < 0 8.

x ≥ 54

4x ≥ 5

4x 1 4 ≥ 9

9.

Equation:

m19,555.56

m <88000.45

0.45m < 8800

0.45m 1 6200 < 15,000

C < 15,000

C 5 0.45m 1 6200

10.

Labels:

Equation:

x > $25,000

0.04x > 1000

0.04x 1 1500 > 2500

Gross sales 5 x

Totalcost

VerbalModel:

15Numberof miles

?Cost permile

Initialcost

PaymentPlan 1

VerbalModel:

5 2500

PaymentPlan 2

5 4% ? Grosssales

1 1500

PaymentPlan 2

> PaymentPlan 1


SECTION 8.3 Linear Systems in Three Variables

1. No, has only one solution.

x 5 212

2x 5 21

2x 1 8 5 7

2x 1 8 5 7 2.

Multiply both sides of the equationby the lowest common denominator,24.

t6

158

574

3. 4x2sx3d2 5 4x2 ? x6 5 4x8

4.

5 8x10y15

s2x2yd3sxy3d4 5 8x6y3 ? x 4y12 5.8x24

2x7 5 4x242s7d 5 4x211 54

x116. 1t 4

3 221

53t 4

7.

x 5 5 x 5 21

2x 5 10 2x 5 22

2x 2 4 5 6 or 2x 2 4 5 26

|2x 2 4| 5 6 8.

12 5 x

510 5 x

5 5 10x

5 2 2x 5 8x

14s5 2 2xd 5 2x

14s5 2 2xd 5 9x 2 7x

9.

t

d

1 2 3 4 5

15

30

45

60

75

d 5 15t

10.

Labels:

Equation: V 5 s3

Side 5 s

Volume 5 V

DistanceVerbal Model: 5 Rate ? Time VolumeVerbal Model: 5 ssided3

11.

A 5C 2

4p

A 5 p ?C 2

4p 2

A 5 p1 C2p2

2

A 5 pr2

Area 5 p ? sradiusd2

C

2p5 r

C 5 2pr

Circumference 5 2 ? p ? radius


SECTION 8.4 Matrices and Linear Systems

1.


2ab 2 2ab 5 0 2.

Multiplicative Identity Property

8t ? 1 5 8t 3.

Commutative Property of Addition

b 1 3a 5 3a 1 b

4.


3s2xd 5 s3 ? 2dx 5.

5 24 ?23

5 283

524

32

524

232 1 3

522 2 2

232 2 s23d

m 5y2 2 y1

x2 2 x1

x

y

−1−2−3−5 −4 1

2

1

3

−1

−2

−3

32

− −, 2 ((

( 3, 2)−

s23, 2d, s232, 22d

6.

534

568

50 2 s26d

8 2 0

m 5y2 2 y1

x2 2 x1

x

y

2 4 6 8

−2

−4

−6

−8

(8, 0)

(0, 6)−

s0, 26d, s8, 0d 7.

5 12 0

5 undefined

54 2

72

52 2

52

m 5y2 2 y1

x2 2 x1

x

y

−1 1 2 3 4 5

2

4

5

1

3

5

5 7

2

2 2

−1

, 4

,

((

((

s52, 72d, s5

2, 4d

8.

5 23013

5 2154

?813

52

184 1

34

88 1

58

52

154

138

52

92 2 s23

4d1 2 s25

8d

m 5y2 2 y1

x2 2 x1

x

y

5 3

9

8 4

2

− −,

1, −

(

(

(

(

−2 −1 1 2 3

−2

−3

−4

−5

s258, 23

4d, s1, 292d 9.

5 20.15

50.926

52.1 2 1.223 2 3

m 5y2 2 y1

x2 2 x1

x

y

−2 −1−3 1 2 3

2

1

3

4

−1

−2

( 3, 2.1)−

(3, 1.2)

s3, 1.2d, s23, 2.1d


10.

5 0

506

58 2 812 2 6

m 5y2 2 y1

x2 2 x1

x

y

(6, 8) (12, 8)

4 8 12

12

8

4

s12, 8d, s6, 8d

11.

Labels:

Equation:

7650 5 x

8415 5 1.10x

8415 5 x 1 0.10x

Number members before drive 5 x

Currentnumbermembers

VerbalModel:

1 10% ?5

Numbermembersbefore drive

Numbermembersbefore drive

12.

Labels:

Equation:

x 5 $940

x 5 0.04s23,500d

Price 5 $23,500

Amount increase 5 x

Amountincrease

VerbalModel:

5 4% ? Price

SECTION 8.5 Determinants and Linear Systems

1.

So a 5 pq.

spqdx2 1 spn 1 mqdx 1 mn

pqx2 1 pnx 1 mqx 1 mn

spx 1 mdsqx 1 nd 5 ax2 1 bx 1 c 2.

So b 5 pn 1 mq.



spx 1 mdsqx 1 nd 5 ax2 1 bx 1 c

3.

So c 5 mn.



spx 1 mdsqx 1 nd 5 ax2 1 bx 1 c 4. If then and or and q 5 21.p 5 21q 5 1p 5 1a 5 1

5.

x 5 24 x 5 1

3sx 1 4dsx 2 1d 5 0

3sx2 1 3x 2 4d 5 0

3x2 1 9x 2 12 5 0 6.

x 5 3 x 5 22

sx 2 3dsx 1 2d 5 0

x2 2 x 2 6 5 0 7.

x 552 x 5

52

s2x 2 5ds2x 2 5d 5 0

4x2 2 20x 1 25 5 0


8.

x 5 4 x 5 24

sx 2 4dsx 1 4d 5 0

x2 2 16 5 0 9.

Not real

x 5 24 x2 2 4x 1 16 5 0

sx 1 4dsx2 2 4x 1 16d 5 0

x3 1 64 5 0 10.

Not real

x 5 2 3x2 1 4 5 0

sx 2 2ds3x2 1 4d 5 0

3x2sx 2 2d 1 4sx 2 2d 5 0

3x3 2 6x2 1 4x 2 8 5 0

11.

Equation

320

r5 t

320 5 r ? t

12.

Equation:

592

x 1 7

5 1x 112

x 1 3x2 1 s1 1 5 1 1d

P 5 sx 1 1d 1 112

x 1 52 1 s3x 1 1d

DistanceVerbal Model: 5 Rate ? TimeVerbalModel:

1Perimeter 5Lengthside 1

Lengthside 2

1Lengthside 3

CHAPTER 9 Exponential and Logarithmic Functions

SECTION 9.1 Exponential Functions

1. Graph the line Test one point in each of thehalf-planes formed by this line. If the point satisfies theinequality, shade the entire half-plane to denote that everypoint in the region satisfies the inequality.

x 1 y 5 5. 2. and

The difference between the two graphs is that the firstcontains the boundary (because of the equal sign) and thesecond does not.

3x 2 5y < 153x 2 5y ≤ 15

3.

Test point:

True

0 > 0 2 2

s0, 0dy

x−2−3 −1 2 31

−3

−4

−2

−1

1

2

y > x 2 2 4.

Test point:

True

0 ≤ 5 2 0

s0, 0d

x

y

−1 1 2 43 5

2

4

1

3

−1

y ≤ 5 232 x

5.

Test point:

True

Test point:

False

0 < 0 2 1

s0, 0d

21 < 1

21 < 23s3d 2 1

s3, 21d

x

y

−1 2 3 4 5

2

1

3

−2

−3

y < 23 x 2 1 6.

Test point:

False

0 > 6 2 0

s0, 0dy

x−2 2 4 6 8

−2

2

6

4

8

x > 6 2 y

C H A P T E R 9Exponential and Logarithmic Functions

Section 9.1 Exponential Functions . . . . . . . . . . . . . . . . . . .475

Section 9.2 Inverse Functions . . . . . . . . . . . . . . . . . . . . . .482

Section 9.3 Logarithmic Functions . . . . . . . . . . . . . . . . . . .487

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .492

Section 9.4 Properties of Logarithms . . . . . . . . . . . . . . . . . .493

Section 9.5 Solving Exponential and Logarithmic Equations . . . . .497

Section 9.6 Applications . . . . . . . . . . . . . . . . . . . . . . . .503


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .514

Cumulative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517

475

C H A P T E R 9Exponential and Logarithmic Functions

Section 9.1 Exponential FunctionsSolutions to Odd-Numbered Exercises

7. because22 ? 22 ? 22 ? ex ? ex ? ex 5 28e3x.

3!28e 3x 5 22ex 9.

Keystrokes:

Scientific: 4 3

Graphing: 4 3 ENTER

4!3 < 11.036

!

!

5

>

11.

Keystrokes:

Scientific: 1 3

Graphing: 1 3

e1y3 < 1.396 13.

Keystrokes:

Scientific: 4 3 0.5 2

Graphing: 4 3 0.5 2

4s3e4d1y2 5 4 ? 31y2 ? e2 < 51.193

x d

x d

5 5

e x 4 ENTER

3

3 3

y x 3

> e x ENTER

15.

Keystrokes:

Scientific: 1 3

Graphing: 3

4e3

12e2 5e3

< 0.906 17. (a)

(b)

(c) f s1d 5 31 5 3

f s0d 5 30 5 1

f s22d 5 322 519

19. (a)

(b)

(c) gs!5 d 5 1.07!5 < 1.163

gs3d 5 1.073 < 1.225

gs21d 5 1.0721 < 0.935 21. (a)

(b)

(c) f spd 5 500s12dp

5 56.657

f s1d 5 500s12d1

5 250

f s0d 5 500s12d0

5 500

23. (a)

(b)

(c) f s10d 5 1000s1.05d2s10d 5 2653.298

f s5d 5 1000s1.05d2s5d 5 1628.895

f s0d 5 1000s1.05ds2ds0d 5 100025. (a)

(b)

(c) hs20d 55000

s1.06d8s20d < 0.447

hs10d 55000

s1.06d8s10d < 47.261

hs5d 55000

s1.06d8s5d < 486.11

27. (a)

(b)

(c) gs8d 5 10e20.5s8d 5 10e24 < 0.183

gs4d 5 10e20.5s4d 5 10e22 < 1.353

gs24d 5 10e20.5s24d 5 10e2 < 73.891 29. (a)

(b)

(c) gs50d 51000

2 1 e20.12s50d < 499.381

gs10d 51000

2 1 e20.12s10d < 434.557

gs0d 51000

2 1 e20.12s0d < 333.333

1. 2x ? 2x21 5 2x1sx21d 5 22x21 3.ex12

ex 5 ex122x 5 e2 5. s2exd3 5 8e3x

4 5

e 4 ENTER

4 ln x

ln x

ln xInv Inv

Inv

y x

476 Chapter 9 Exponential and Logarithmic Functions

31.

Table of values:

2x

y

4

3

2

1

112

x 0 1 2

0.1 0.3 1 3 9f sxd

2122

33.

Table of values:

2x

y

4

3

2

1

112

x 0 1 2

0.1 0.2 0.5 1.5 4.5hsxd

2122

35.

Table of values:

2x

y

2

1

1

1

12

x 0 1 2

1 72121.721.9gsxd

2122

37.

Table of values:

–8 –6 –4 2 4 6 8

−4

−2

2

4

6

8

10

12

x

y

x 0 1 5 6

0.004 1 49.8 3 10242.4 3 1024f sxd

21

39.

Table of values:

−8 −6 −4 2 4 6 8

−8

−6

−2

2

4

6

8

x

y

x 0 1 2

11212424.824.9f sxd

2122

41.

Table of values:

t2

y

3

2

1

112

t 0 1 2

0.1 0.5 1 0.5 0.1f std

2122

Section 9.1 Exponential Functions 477

43.

Table of values:

–8 –6 –4 4 6 8

−12

−10

−8

−6

−4

−2

2

4

x

y

x 0 1 2

2221.42120.725f sxd

2122

45.

Table of values:

3x

21

5

y

4

3

2

1

12

x 0 1 2

0.5 1 1.4 20.7hsxd

2122

47.

Table of values:

–8 –6 –4 –2 2 4 6 8

2

4

x

y

x 0 1 2

20.120.3212329f sxd

2122

49.

Table of values:

–8 –6 –4 –2 2 4 6 8

100

200

t

y

t 0 1 2

800 400 200 100 50gstd

2122

51.

(b) Basic graph

f sxd 5 2x 53.

(e) Basic graph reflected in the y-axis

f sxd 5 22x

55.

(f ) Basic graph shifted 1 unit right

f sxd 5 2x21 57.

(h) Basic graph reflected in -axis and shifted 2 unitsdown

y

f sxd 5 s12dx

2 2

59.

Keystrokes:

5 3

−9

−4

9

8

y 5 5xy3 61.

Keystrokes:

5 2 3

−9

−4

9

8

y 5 5sx22dy3

> x x 4 GRAPH2Y5 > x 4 d GRAPH Y5 d dX,T,u X,T,u


63.

Keystrokes:

500 1.06

−90

−200

30

1000

y 5 500s1.06dt 65.

Keystrokes:

3 0.2

−15

−4

15

16

y 5 3e0.2x

>x GRAPH GRAPHe x

67.

Keystrokes:

100 0.1

−20

−20

40

200

Pstd 5 100e20.1t 69.

Keystrokes:

6 3

−9

−4

9

8

y 5 6e2x2y3

GRAPHe xGRAPHe x x2c x2c x x2 4

71. Vertical shift 1 unit down

y

x−2−3 2 31

−2

−1

1

2

3

4

73. Horizontal shift 2 units left

y

x−1−4 −3 −2−5 1

−1

1

2

3

5

4

75. Reflection in the -axis

y

x−2−3 2 31

−3

−2

−4

−5

−1

1

x

77. (a) Algebraic (Linear)

(b) Algebraic (Radical)

(c) Exponential

(d) Algebraic (Quadratic)f sxd 5 2x2

f sxd 5 2x

f sxd 5 !2x

f sxd 5 2x 79. grams

Keystrokes:

16 0.5 8 3 Scientific

16 0.5 8 3 Graphing

y 5 16s12d80y30

5 2.520

3

3

y x x

x

d

d

4

4

5

> ENTER

81.n 1 4 12 365 Continuous

A $466.10 $487.54 $492.68 $495.22 $495.30

Compounded 1 time:

5 $466.10

A 5 10011 10.08

1 21s20d

Compounded 4 times:

5 $487.54

A 5 10011 10.08

4 24s20d

Compounded 12 times:

5 $492.68

A 5 10011 10.0812 2

12s20d


5 $495.22

A 5 10011 10.08365 2

365s20d Compounded continuously:

5 495.30

5 100e0.08s20d

A 5 Pert

Y5 d Y5

Y5 Y5 d

X,T,u X,T,u

X,T,uX,T,u


83.

Compounded 1 time:


Compounded continuously:

5 $4919.21

A 5 2000e0.09s10d

5 $4902.71

A 5 200011 10.0912 2

12s10d

5 $4734.73

A 5 200011 10.09

1 21s10d

Compounded 4 times:


5 $4918.66

A 5 200011 10.09365 2

365s10d

5 $4870.38

A 5 200011 10.09

4 24s10d

n 1 4 12 365 Continuous

A $4734.73 $4870.38 $4902.71 $4918.66 $4919.21

87.

Compounded 1 time:

$2541.75 5 P

5000

s1.07d10 5 P

5000 5 P11 10.07

1 21s10d

Compounded 4 times:

$2498.00 5 P

5000

s1.0175d40 5 P

5000 5 11 10.07

4 24s10d


$2483.09 5 P

5000

s1.0001918d3.650 5 P

5000 5 P11 10.07365 2

365s10d


P $2541.75 $2498.00 $2487.98 $2483.09 $2482.93

85.

Compounded 1 time:



5 $272,990.75

A 5 50000.10s40d

5 $268,503.32

A 5 500011 10.1012 2

12s40d

5 $226,296.28

A 5 500011 10.10

1 21s40d

Compounded 4 times:


5 $272,841.23

A 5 500011 10.10365 2

365s40d

5 $259,889.34

A 5 500011 10.10

4 24s40d


A $226,296.28 $259,889.34 $268,503.32 $272,841.23 $272,990.75


$2487.98 5 P

5000

s1.00583 d120 5 P

5000 5 P11 10.0712 2

12s10d

Compounded Continuously:

$2482.93 5 P

5000e0.7 5 P

5000 5 Pe0.07s10d


Compounded 1 time:

$18,429.30 5 P

1,000,000s1.105d40 5 P

1,000,000 5 P11 10.105

1 21s40d

Compounded 4 times:

$15,830.43 5 P

1,000,000

s1.02625d160 5 P

1,000,000 5 P11 10.105

4 24s40d


$15,272.04 5 P

1,000,000

s1.00875d480 5 P

1,000,000 5 P11 10.105

12 212s40d


$15,004.64 5 P

1,000,000

s1.002877d14,600 5 P

1,000,000 5 P11 10.105365 2

365s40d


$14,995.58 5 P

1,000,000

e4.2 5 P

1,000,000 5 Pe0.105s40d


P $18,429.30 $15,830.43 $15,272.04 $15,004.64 $14,995.58

91. (a)

(b)

< $20.13

5 25 2 0.4e2.5

p 5 25 2 0.4e0.02s125d

< $22.04

5 25 2 0.4e2

p 5 25 2 0.4e0.02s100d 93. (a)

(b)

< $161,269.89

5 64,000s2d4y3

vs20d 5 64,000s2d20y15

< $80,634.95

5 64,000s2d1y3

vs5d 5 64,000s2d5y15 95. (a)

(b)

Vs2d 5 16,000s34d2

5 9000

t12

V

12,000

16,000

108642

4,000

8,000

Vstd 5 16,000s34dt

97. (a) The balances in the accounts after years are modeled by and A2 5 500e0.08t.A1 5 500e0.06tt

(b) Keystrokes:

500 0.06

500 0.08

(d) The difference between the functions increases at anincreasing rate.

00

50

10,000

A2

A1

y2

y1 Y5 e x ENTER

e x GRAPH

(c)

Keystrokes:

500 0.08

0.06

00

50

10,000

A2

A2 − A1

A1

y1

5 500se0.08t 2 e0.06t d A2 2 A1 5 500e0.08t 2 500e0.06t

Y5 x e x 2 e x

GRAPH

89.

X,T,u

X,T,u

X,T,u

X,T,u d


99. (a) Keystrokes:

1950 50 1.6

20

00

100

2400

GRAPH

Y5 1 e x x2c 2

103. (a)

(c) As gets larger and larger, approaches e.11 11x2

x

x

(b) Keystrokes:

1 1

Yes, the graph is approaching a horizontal asymptote.

x

y

2010−10−20

−2

2

x 1 10 100 1000 10,000

2 2.5937 2.7048 2.7169 2.718111 11x2

xY5 x 1 4 > GRAPH

105. Polynomial functions have terms with variable bases and constant exponents. Exponential functions have terms with constantbases and variable exponents.

107. is an increasing function and is a decreasing function.gsxd 5 s13dx

f sxd 5 3 x

X,T,u

X,T,u X,T,u

X,T,u

(b)

(c) The parachutist will reach the ground at 97.5seconds.

h 5 1950 1 50e21.6s75d 2 20s75d 5 450 feet

h 5 1950 1 50e21.6s50d 2 20s50d 5 950 feet

h 5 1950 1 50e21.6s25d 2 20s25d 5 1450 feet

h 5 1950 1 50e21.6s0d 2 20s0d 5 2000 feet

109. False. is an irrational number.

is rational because its equivalent decimal form is a repeating decimal.271,80199,990

e

101. (a) Graph model:

Plot data:

Keystrokes:

Enter each x entry in L 1 followed by .

Enter each y entry in L 2 followed by .

or set window.

(b) Keystrokes: 10,958 .15The model fits the data.

−5 25

−1,000

11,000

STAT EDIT 1

ENTER

ENTER

ex

STAT PLOT ENTER ENTER ZOOM 9

Y5 x2c

(d) At an altitude of 8 kilometers, P is 3300 kilograms persquare meter. Use table.

Keystrokes: 8

(e) If P is 2000 kilograms per square meter, altitude is 11.3kilometers. Graph and find the intersectionpoint.

y2 5 2000

TABLE ENTER

d

X,T,u

(c)

h 0 5 10 15 20

P 10,332 5583 2376 1240 517

Approx. 10,958 5176 2445 1155 546


Section 9.2 Inverse Functions

1. (a)

(b)

(c)

(d) sg 8 f ds7d 5 2s7d 2 10 5 4

s f 8 gds4d 5 2s4d 2 7 5 1

sg 8 f dsxd 5 2sx 2 3d 2 4 5 2x 2 6 2 4 5 2x 2 10

s f 8 gdsxd 5 s2x 2 4d 2 3 5 2x 2 7 3. (a)

(b)

(c)

(d)

5 2

5 2s9d 2 60 1 44

sg 8 f ds23d 5 2s23d2 1 20s23d 1 44

s f 8 gds2d 5 2s2d2 2 1 5 2s4d 2 1 5 7

5 2x2 1 20x 1 44

5 2x2 1 20x 1 50 2 6

sg 8 f dsxd 5 2sx 1 5d2 2 6 5 2sx2 1 10x 1 25d 2 6

s f 8 gdsxd 5 s2x2 2 6d 1 5 5 2x2 2 1

5. (a)

(b)

(c)

(d) sg 8 f ds2d 5 3|2 2 3| 5 3

s f 8 gds1d 5 |3 2 3| 5 0

sg 8 f dsxd 5 3|x 2 3|s f 8 gdsxd 5 |3x 2 3| 7. (a)

(b)

(c)

(d) sg 8 f ds8d 5 !8 2 4 1 5 5 2 1 5 5 7

s f 8 gds3d 5 !3 1 1 5 2

sg 8 f dsxd 5 !x 2 4 1 5

s f 8 gdsxd 5 !x 1 5 2 4 5 !x 1 1

9. (a)

(b)

(c)

(d) sg 8 f ds2d 5 2s2 2 3d2 5 2s21d2 5 2

s f 8 gds21d 5s21d2

2 2 3s21d2 51

2 2 35

121

5 21

sg 8 f dsxd 52

1 1x 2 32

2 5 2sx 2 3d2

s f 8 gdsxd 51

2x2 2 3

?x2

x2 5x2

2 2 3x2 11. (a)

(b)

(c)

5 22

5 gf21g sg 8 f ds1d 5 gf f s1dg

gs21d 5 22

f s1d 5 21

13. (a)

(b) sg 8 f ds22d 5 gf f s22dg 5 gf3g 5 1

s f 8 gds23d 5 f fgs23dg 5 f f1g 5 21 15. (a)

(b)

(c) sg 8 f ds3d 5 gf f s3dg 5 gf10g 5 1

gs10d 5 1

f s3d 5 10

17. (a)

(b) s f 8 gds2d 5 f fgs2dg 5 f f3g 5 10

sg 8 f ds4d 5 gf f s4dg 5 gf17g 5 0 19.

(a)Domain:

(b)Domain: s2`, `dg 8 f 5 2sx 1 1d 2 5 5 2x 1 2 2 5 5 2x 2 3

s2`, `df 8 g 5 s2x 2 5d 1 1 5 2x 2 4

gsxd 5 2x 2 5f sxd 5 x 1 1,

21.

(a) Domain:

(b) Domain: f0, `dg 8 f 5 !x 2 2

f2, `df 8 g 5 !x 2 2

gsxd 5 x 2 2f sxd 5 !x, 23.

(a)Domain:

(b)Domain: s2`, `dg 8 f 5 !sx2 2 1d 1 3 5 !x2 1 2

f23, `d5 x 1 3 2 1 5 x 1 2f 8 g 5 s!x 1 3d2 2 1

gsxd 5 !x 1 3f sxd 5 x2 2 1,

Section 9.2 Inverse Functions 483

25.

(a) Domain:

(b) Domain: s2`, 25dg 8 f 5! xx 1 5

2 1

f1, `df 8 g 5!x 2 1

!x 2 1 1 5

gsxd 5 !x 2 1f sxd 5x

x 1 5, 27.

No, it does not have an inverse because it is possible tofind a horizontal line that intersects the graph of at morethan one point.

f

f sxd 5 x2 2 2

29.

Yes, it does have an inverse because no horizontal lineintersects the graph of at more than one point.f

f sxd 5 x2, x ≥ 0 31.

No, it does not have an inverse because it is possible tofind a horizontal line that intersects the graph of at morethan one point.

g

gsxd 5 !25 2 x2

33. Keystrokes:

3 1

One-to-one

−4

−10

4

10

Y5 > 2 GRAPH

35. Keystrokes:

4 5

One-to-one

−8

−8

16

8

Y5 GRAPHMATH x 2 c

41. Keystrokes:

4 1

Not one-to-one

−6

−2

6

6

43.

gs f sxdd 510x10

5 x

f sgsxdd 5 101 110

x2 5 x

Y5 GRAPHx c

37. Keystrokes:

4 6

Not one-to-one

−3

−10

3

10

Y5 > 2 GRAPH

39. Keystrokes:

5

One-to-one

−9

−6

9

6

Y5 GRAPH4

4 x2 1

45.

gs f sxdd 5 sx 1 15d 2 15 5 x

f sgsxdd 5 sx 2 15d 1 15 5 x 47.

5 12f1 2 1 1 2xg 5

12f2xg 5 x

gs f sxdd 512f1 2 s1 2 2xdg

5 1 2 s1 2 xd 5 1 2 1 1 x 5 x

f sgsxdd 5 1 2 2f12s1 2 xdg

X,T,u X,T,u

X,T,uX,T,u

X,T,u


49.

gs f sxdd 513f2 2 s2 2 3xdg 5

13f3xg 5 x

f sgsxdd 5 2 2 3f13 s2 2 xdg 5 2 2 s2 2 xd 5 x 51.

gs f sxdd 5 s 3!x 1 1 d3 2 1 5 x 1 1 2 1 5 x

f sgsxdd 5 3!x3 2 1 1 1 5 3!x3 5 x

53.

gs f sxdd 511x

5 x

f sgsxdd 51

1x

5 x 55.

f 21s f sxdd 5 f 21s5xd 55x5

5 x

f s f 21sxdd 5 f 1x52 5 51x

52 5 x

f 21sxd 5x5

57.

f 21s f sxdd 5 f 21s12xd 5 2s1

2xd 5 x

f s f 21sxdd 5 f s2xd 512s2xd 5 x

f 21sxd 5 2x 59.

f 21s f sxdd 5 f 21sx 1 10d 5 x 1 10 2 10 5 x

f s f 21sxdd 5 f sx 2 10d 5 x 2 10 1 10 5 x

f 21sxd 5 x 2 10

61.

f 21s f sxdd 5 f 21s3 2 xd 5 3 2 s3 2 xd 5 3 2 3 1 x 5 x

f s f 21sxdd 5 f s3 2 xd 5 3 2 s3 2 xd 5 3 2 3 1 x 5 x

f 21sxd 5 3 2 x 63.

f 21s f sxdd 5 f 21sx7d 5 7!x7 5 x

f s f 21sxdd 5 f s 7!x d 5 s 7!x d75 x

f 21sxd 5 7!x

65.

f 21s f sxdd 5 f 21s 3!x d 5 s 3!x d3 5 x

f s f 21sxdd 5 f sx3d 5 3!x3 5 x

f 21sxd 5 x3 67.

f 21sxd 5x8

x8

5 y

x 5 8y

y 5 8x

f sxd 5 8x 69.

g21sxd 5 x 2 25

x 2 25 5 y

x 5 y 1 25

y 5 x 1 25

gsxd 5 x 1 25

71.

3 2 x4

or x 2 324

5 g21sxd

x 2 324

5 y

x 2 3 5 24y

x 5 3 2 4y

y 5 3 2 4x

gsxd 5 3 2 4x 73.

4t 2 8 5 g21std

4st 2 2d 5 y

t 2 2 514 y

t 514 y 1 2

y 514t 1 2

gstd 514t 1 2 75.

x2 5 h21sxd, x ≥ 0

x2 5 y

x 5 !y

y 5 !x

hsxd 5 !x

77.

3!t 1 1 5 f 21std

3!t 1 1 5 y

t 1 1 5 y3

t 5 y3 2 1

y 5 t3 2 1

f std 5 t3 2 1 79.

g21ssd 55s

2 4, s Þ 0

y 1 4 55s

s 55

y 1 4

y 55

s 1 4

gssd 55

s 1 481.

x2 2 3 5 f 21sxd, x ≥ 0

x2 2 3 5 y

x2 5 y 1 3

x 5 !y 1 3

y 5 !x 1 3

f sxd 5 !x 1 3

Section 9.2 Inverse Functions 485

83.

x

y

−8 −2 4 86

6

4

8

2

−4

−8

f

f −1

s0, 24ds24, 0d

s4, 0ds0, 4d

f 21sxd 5 x 2 4f sxd 5 x 1 4, 85.

x

y

−4 −1 2 3 41

2

3

4

1

f

f −1

s0, 13ds13, 0d

s21, 0ds0, 21d

f 21sxd 513sx 1 1df sxd 5 3x 2 1,

87.

y

x−1 2 3 4

−1

3

2

4 f

f −1

s0, 1ds1, 0d

s21, 0ds0, 21d

f 21sxd 5 !x 1 1f sxd 5 x2 2 1, 89. (b) 91. (d)

93. Keystrokes:

1 3

3

−9

−6

9

6

y2

y1 Y5 x 4 c ENTER

GRAPH

95. Keystrokes:

1

1 4 0

00

18

12

y2

y1 Y5 x c

x

ENTER

GRAPH

! 1

x2 2 4 TEST

97. Keystrokes:

1 8 3

2 4

−12

−8

12

8

y2

y1 Y5 x 4 c ENTERMATH

MATH GRAPH

99. Keystrokes:

3 4

4 3

−12

−8

6

4

y2

y1 Y5

x 4c

ENTER

GRAPH

1

2

X,T,u

X,T,u

X,T,u

X,T,u X,T,u

X,T,u

X,T,uX,T,u

X,T,u

d


101.

!x 1 2 5 f 21sxd, x ≥ 0

!x 1 2 5 y

!x 5 y 2 2

x 5 sy 2 2d2

y 5 sx 2 2d2

f sxd 5 sx 2 2d2, x ≥ 2 103.

x 2 1 5 f 21sxd, x ≥ 1

x 2 1 5 y

x 2 1 5 |y| x 5 |y| 1 1

y 5 |x| 1 1

f sxd 5 |x| 1 1, x ≥ 0 105.

6x

y

66

4

5

5432

3

2

1

1

x 0 1 3 4

6 4 2 0f 21

107.

3x

y

4

2

3

1

1

1

2

3

34 2 2

x 2 3

1 32122f 21

2224

109. (a)

f 21sxd 53 2 x

2

y 53 2 x

2

2y 5 3 2 x

x 5 3 2 2y

y 5 3 2 2x (b)

sf 21d21sxd 5 3 2 2x

y 5 3 2 2x

2x 5 3 2 y

x 53 2 y

2

y 53 2 x

2

111. (a)

(b)

total cost

number of pounds at $0.50 per poundy:

x:

4s75 2 xd 5 y

24sx 2 75d 5 y

x 2 7520.25

5 y

x 2 75 5 20.25y

x 5 20.25y 1 75

y 5 20.25x 1 75

y 5 0.50x 1 75 2 0.75x

y 5 0.50x 1 0.75s100 2 xd

y 5 0.50x 1 0.75s100 2 xd

Total cost 5 Cost of $0.50 commodity 1 Cost of $0.75 commodity

(c)

If you buy only the cheaper commodity, your costwill be $50. If you buy only the more expensivecommodity, your cost will be $75. Any combinationwill lie between $50 and $75.

50 ≤ x ≤ 75

(d)

Thus, 60 pounds of the $0.50 per pound commodity is purchased.

y 5 60

y 5 4s15d

y 5 4s75 2 60d

113. (a)

(b)

This part represents the bonus because it gives 2% of sales over $200,000.

gs f sxdd 5 0.02sx 2 200,000d, x > 200,000

f sgsxdd 5 0.02x 2 200,000

Section 9.3 Logarithmic Functions 487

Section 9.3 Logarithmic Functions

115. (a)

(c)

5% discount before the $2000 rebate is given.

The 5% discount is given after the $2000 rebate isapplied.

5 Ss p 2 2000d 5 0.95s p 2 2000d

sS 8 Rds pd 5 SfRs pdg

sR 8 Sds pd 5 RfSs pdg 5 Rs0.95pd 5 0.95p 2 2000

R 5 p 2 2000

117. True, the -coordinate of a point on the graph of becomes the -coordinate of a point on the graph of f 21.y

fx 119. False: Domain

Domain f0, `d f 21sxd 5 x2 1 1

f1, `d f sxd 5 !x 2 1

121. If and then andsg 8 f dsxd 5 4x2.

s f 8 gdsxd 5 2x2gsxd 5 x2,f sxd 5 2x 123. (a) In the equation for replace by

(b) Interchange the roles of and

(c) If the new equation represents as a function of solve the new equation for

(d) Replace by f 21sxd.y

y.x,y

y.x

y.f sxdf sxd,

125. Graphically, a function has an inverse function if and only if no horizontal line intersects the graph of at more than one point. This is equivalent to saying that the function is one-to-one.ff

f

1.

52 5 25

log5 25 5 2 3.

422 51

16

log4 1

16 5 22 5.

325 51

243

log3 1

243 5 25

7.

361y2 5 6

log36 6 512 9.

82y3 5 4

log8 4 523 11.

21.3 < 2.462

log2 2.462 < 1.3

13.

log7 49 5 2

72 5 49 15.

log3 19 5 22

322 519 17.

log8 4 523

82y3 5 4

19.

log25 15 5 2

12

2521y2 515 21.

log4 1 5 0

40 5 1 23.

log5 9.518 < 1.4

51.4 < 9.518

25. because 23 5 8.log2 8 5 3 27. because 101 5 10.log10 10 5 1 29. because103 5 1000.log10 1000 5 3

31. because 222 514.log2

14 5 22 33. because 423 5

164.log4

164 5 23

(b)

(d)

yields the smaller cost because the dealer discount isbased on a larger amount.R 8 S

sS 8 Rds26,000d 5 0.95s26,000 2 2000d 5 $22,800

sR 8 Sds26,000d 5 0.95s26,000d 2 2000 5 $22,700

S 5 0.95p

S 5 p 2 0.05p

35. because 1024 51

10,000.log10

110,000

5 24 37. is not possible because there is no power towhich 2 can be raised to obtain 23.log2s23d



7. Compounded 1 time per year:

Compounded 12 times per year:


< $3361.27

5 750e0.075s20d

A 5 Pert

< $3345.61

A 5 75011 10.075

12 212s20d

< $3185.89

A 5 75011 10.075

1 21s20d



< $3360.75

A 5 75011 10.075365 2

365s20d

< $3314.90

A 5 75011 10.075

4 24s20d

8. A 5 2.23es0.04ds5d 5 $2.72

1. (a)

(b)

(c)

(d)

58!3

9

f s1.5d 5 1432

1.5

< 1.54

f s21d 5 1432

21

534

f s0d 5 1432

0

5 1

f s2d 5 1432

2

5169

2.

Domain:

Range: s0, `ds2`, `d

gsxd 5 220.5x 3.

–8 –6 –4 2 4 6 8−2

2

4

6

8

10

12

14

x

y

9. (a)

(b)

(c)

(d) sg 8 f ds4d 5 gf f s4dg 5 gf5g 5 53 5 125

5 2s28d 2 3 5 219 s fgds22d 5 f fgs22dg 5 f f28g

sg 8 f dsxd 5 gf f sxdg 5 s2x 2 3d3

s f 8 gdsxd 5 f fgsxdg 5 2x3 2 3

10.

gf f (xdg 515

f3 2 s3 2 5xdg 515

f3 2 3 1 5xg 515

f5xg 5 x

f fgsxdg 5 3 2 5315

s3 2 xd4 5 3 2 1s3 2 xd 5 3 2 3 1 x 5 x

−8 −4−4−6−8

−10−12

4

46

x

y

f

g

4.

–8 –6 –4 2 4 6 8−2

2

4

6

x

y 5.

4 8 12 16 20 24 28−4

4

8

12

t

y 6.

–100 –60 20 60 100−20

80100120140160180

x

y

Section 9.4 Properties of Logarithms 493

Section 9.4 Properties of Logarithms

11.

x 2 3

105 h21sxd

x 2 3

105 y

x 2 3 5 10y

x 5 10y 1 3

y 5 10x 1 3

hsxd 5 10x 1 3 12.

3!2t 2 4 5 g21std

3!2t 2 4 5 y

2t 2 4 5 y3

t 2 2 512

y3

t 512

y3 1 2

y 512

t3 1 2

gstd 512

t3 1 2 13.

422 51

16

log41 1162 5 22

14.

log3 81 5 4

34 5 81 15. because 53 5 125.log5 125 5 3

16. and are inverse functions because the graphs of andreflect about the line

–4 –2 2 4 6 8 10

−4

2

4

6

8

10

x

y

g

f

y 5 x.gfgf 17. Keystrokes:

.5

−2

−4

10

4

Y5 LN GRAPH

18. Keystrokes:

3

−2

−1

10

7

Y5 LN GRAPH2

19.

The graph of hasbeen shifted 3 units right and 1 unitup, so k 5 1.h 5 2,

f sxd 5 log5 x

f sxd 5 log5sx 2 2d 1 1 20. log6 450 5log 450log 6

< 3.4096

1. log5 52 5 2 ? log5 5 5 2 ? 1 5 2 3.

5 29 ? log2 2 5 29 ? 1 5 29

log2s18d3

5 log2s223d35 log2 2

29

5. because 61y2 5 61y2. log6 !6 5 log6 61y2 5

12 7. or ln 80 5 ln 1 5 0ln 80 5 0 ? ln 8 5 0

9. ln e4 5 4 ln e 5 4s1d 5 4 11. because42 5 16.log4 2 1 log4 8 5 log4 2 ? 8 5 log4 16 5 2

X,T,u

X,T,u



7. Compounded 1 time per year:



< $3361.27

5 750e0.075s20d

A 5 Pert

< $3345.61

A 5 75011 10.075

12 212s20d

< $3185.89

A 5 75011 10.075

1 21s20d



< $3360.75

A 5 75011 10.075365 2

365s20d

< $3314.90

A 5 75011 10.075

4 24s20d

8. A 5 2.23es0.04ds5d 5 $2.72

1. (a)

(b)

(c)

(d)

58!3

9

f s1.5d 5 1432

1.5

< 1.54

f s21d 5 1432

21

534

f s0d 5 1432

0

5 1

f s2d 5 1432

2

5169

2.

Domain:

Range: s0, `ds2`, `d

gsxd 5 220.5x 3.

–8 –6 –4 2 4 6 8−2

2

4

6

8

10

12

14

x

y

9. (a)

(b)

(c)

(d) sg 8 f ds4d 5 gf f s4dg 5 gf5g 5 53 5 125

5 2s28d 2 3 5 219 s fgds22d 5 f fgs22dg 5 f f28g

sg 8 f dsxd 5 gf f sxdg 5 s2x 2 3d3

s f 8 gdsxd 5 f fgsxdg 5 2x3 2 3

10.

gf f (xdg 515

f3 2 s3 2 5xdg 515

f3 2 3 1 5xg 515

f5xg 5 x

f fgsxdg 5 3 2 5315

s3 2 xd4 5 3 2 1s3 2 xd 5 3 2 3 1 x 5 x

−8 −4−4−6−8

−10−12

4

46

x

y

f

g

4.

–8 –6 –4 2 4 6 8−2

2

4

6

x

y 5.

4 8 12 16 20 24 28−4

4

8

12

t

y 6.

–100 –60 20 60 100−20

80100120140160180

x

y


Section 9.4 Properties of Logarithms

11.

x 2 3

105 h21sxd

x 2 3

105 y

x 2 3 5 10y

x 5 10y 1 3

y 5 10x 1 3

hsxd 5 10x 1 3 12.

3!2t 2 4 5 g21std

3!2t 2 4 5 y

2t 2 4 5 y3

t 2 2 512

y3

t 512

y3 1 2

y 512

t3 1 2

gstd 512

t3 1 2 13.

422 51

16

log41 1162 5 22

14.

log3 81 5 4

34 5 81 15. because 53 5 125.log5 125 5 3

16. and are inverse functions because the graphs of andreflect about the line

–4 –2 2 4 6 8 10

−4

2

4

6

8

10

x

y

g

f

y 5 x.gfgf 17. Keystrokes:

.5

−2

−4

10

4

Y5 LN GRAPH

18. Keystrokes:

3

−2

−1

10

7

Y5 LN GRAPH2

19.

The graph of hasbeen shifted 3 units right and 1 unitup, so k 5 1.h 5 2,

f sxd 5 log5 x

f sxd 5 log5sx 2 2d 1 1 20. log6 450 5log 450log 6

< 3.4096

1. log5 52 5 2 ? log5 5 5 2 ? 1 5 2 3.

5 29 ? log2 2 5 29 ? 1 5 29

log2s18d3

5 log2s223d35 log2 2

29

5. because 61y2 5 61y2. log6 !6 5 log6 61y2 5

12 7. or ln 80 5 ln 1 5 0ln 80 5 0 ? ln 8 5 0

9. ln e4 5 4 ln e 5 4s1d 5 4 11. because42 5 16.log4 2 1 log4 8 5 log4 2 ? 8 5 log4 16 5 2

X,T,u

X,T,u


13.because 82 5 64.log8 4 1 log8 16 5 log8 4 ? 16 5 log8 64 5 2 15. because 41 5 4.log4 8 2 log4 2 5 log4

82 5 log4 4 5 1

17.because 62 5 36.log6 72 2 log6 2 5 log6

722 5 log6 36 5 2 19.

because 223 5 223.log2 5 2 log2 40 5 log2

540 5 log2

18 5 log2 2

23 5 23

21.

5 12 ln e 5 12 ? 1 5 12

ln e8 1 ln e4 5 ln e8 ? e4 5 ln e12 23. ln e3

e2 5 ln e 5 1

25. log4 4 5 log4 2 1 log4 2 5 0.5000 1 0.5000 5 1 27.

5 0.5000 1 0.7925 < 1.2925

log4 6 5 log4 2 ? 3 5 log4 2 1 log4 3

29. log4 32 5 log4 3 2 log4 2 5 0.7925 2 0.5000 < 0.2925 31. log4 !2 5

12 log4 2 5

12s0.5000d 5 0.25

33.

5 0.7925 1 4s0.5000d < 2.7925

log4s3 ? 24d 5 log4 3 1 4 log4 2

37. log10 9 5 log10 32 5 2 log10 3 < 2s0.477d < 0.954 39.

< 1.556

< 0.477 1 1.079

log10 36 5 log10s3 ? 12d 5 log10 3 1 log10 12

35. log4 30 5 log4 1 5 0

41.

< 0.778

< 12s1.556d

log10 !36 5 log10 361y2 512 log10 36 43. log3 11x 5 log3 11 1 log3 x

45. log7 x2 5 2 log7 x 47. log5 x

22 5 22 log5 x

49.

5 12slog4 3 1 log4 xd

log4 !3x 5 log4s3xd1y2 512 log4s3xd 51. ln 3y 5 ln 3 1 ln y

53. log2 z

175 log2 z 2 log2 17 55. ln

5x 2 2

5 ln 5 2 lnsx 2 2d

57.

5 2 ln x 1 lnsy 2 2d

ln x2sy 2 2d 5 ln x2 1 lnsy 2 2d 59.

5 6 log4 x 1 2 log4sx 2 7d

log4fx6sx 2 7d2g 5 log4 x6 1 log4sx 2 7d2

61. log3 3!x 1 1 5

13 log3sx 1 1d 63. ln !xsx 1 2d 5

12 fln x 1 lnsx 1 2dg

65.

5 2flnsx 1 1d 2 lnsx 2 1dg

ln1x 1 1x 2 12

2

5 2 ln1x 1 1x 2 12 67.

513

f2 ln x 2 lnsx 1 1dg

513

fln x2 2 lnsx 1 1dg

ln 3! x2

x 1 15 ln1 x2

x 1 121y3

513

ln1 x2

x 1 12


69.

5 3 ln a 1 lnsb 2 4d 2 2 ln c

ln a3sb 2 4d

c2 5 ln a3 1 lnsb 2 4d 2 ln c2 71.

5 ln x 113

ln y 2 4sln w 1 ln zd

5 ln x 1 ln y1y3 2 4 lnswzd

ln x 3!yswzd4 5 ln x 1 ln 3!y 2 lnswzd4

73.

5 log6 a 112 log6 b 1 3 log6sc 2 dd

5 log6 a 1 log6 b1y2 1 3 log6sc 2 dd

log6fa!bsc 2 dd3g 5 log6 a 1 log6 !b 1 log6sc 2 dd3

75.

5 lnsx 1 yd 115

lnsw 1 2d 2 sln 3 1 ln td

5 lnsx 1 yd 1 lnsw 1 2d1y5 2 sln 3 1 ln td

ln3sx 1 yd 5!w 1 2

3t 4 5 lnsx 1 yd 1 ln 5!w 1 2 2 lns3td

77. log12 x 2 log12 3 5 log12 x3

79. log2 3 1 log2 x 5 log2 3x 81. log10 4 2 log10 x 5 log10 4x

83. b > 04 ln b 5 ln b4, 85.

5 log5 1

4x2 , x > 0

22 log5 2x 5 log5s2xd22 87.13

lns2x 1 1d 5 ln 3!2x 1 1

89.

5 log3 2!y

log3 2 112

log3 y 5 log3 2 1 log3 !y 91. x > 0, y > 0, z > 0 2 ln x 1 3 ln y 2 ln z 5 ln x2y3

z,

93.

5 ln 32y3

x, x > 0, y > 0

5 ln 32 2 ln x 1 ln y3

5 ln 2 2 ln x 1 3 ln y 5 ln 25 2 ln x 1 ln y3 95. or x > 0, y > 0ln x 4y4,4sln x 1 ln yd 5 lnsxyd4

97.

5 ln x2

sx 1 1d2 , x > 0

2fln x 2 lnsx 1 1dg 5 2 ln x

x 1 15 ln1 x

x 1 122

99.

5 log4 sx 1 8d

x3 , x > 0

log4sx 1 8d 2 3 log4 x 5 log4sx 1 8d 2 log4 x3

101.

5 log5 !x 1 2x 2 3

12

log5sx 1 2d 2 log5sx 2 3d 5 log5sx 1 2d1y2 2 logsx 2 3d

103.

5 log6 sc 1 dd5

!m 2 n

5 log6sc 1 dd 212

log6sm 2 nd 5 log6sc 1 dd5 2 log6sm 2 nd1y2


105.

5 log2 5!x3

y4, y > 0

5151log2

x3

y42

15

s3 log2 x 2 4 log2 yd 515

slog2 x3 2 log2 y

4d

107.

5 log6 5!x 2 3

x2sx 1 1d3, x > 3

15

log6sx 2 3d 2 2 log6 x 2 3 logsx 1 1d 5 log6sx 2 3d1y5 2 log6 x2 2 log6sx 1 1d3

109.

5 ln 3 1 2

5 ln 3 1 2 ln e

ln 3e2 5 ln 3 1 ln e2 111.

5 1 112

log5 2

512

f2 1 log5 2g

512

f2 log5 5 1 log5 2g

log5 !50 512

flog5s52 ? 2dg 113.

5 1 2 2 log4 x

5 1 2 log4 x2

log4 4x2 5 log4 4 2 log4 x

2

115.

Keystrokes:

10 1

2 10 1

Graph and y3.y2

y3

y2

y1

−4

−8

20

8

Y5 x

x x

4 x 1

1

c c

c c

ENTER

ENTERLN

LN LN2

117.

Keystrokes:

2

2 2y2

y1

−4

−4

20

12

Y5 x x

x

1

1 1

c c

c

ENTERLN

LN LN GRAPH

119. Choose two values for and such as and and show the two expressions are not equal.

0.6826062 Þ 20.5108256 5 20.5108256

ln 3ln 5

Þ ln 35

5 ln 3 2 ln 5

y 5 5,x 5 3y,x

121.

5 10flog10 I 1 16g

5 10flog10 I 2 s216dg

5 10flog10 I 2 log10 10216g

B 5 10 log101 I102162 or

5 60 decibels

5 10f210 1 16g

B 5 10flog10 10210 1 16g

or

1 Þ 0

1 Þ ln 1

ln eln e

Þ ln ee

123. E 5 1.4slog10 C2 2 log10 C1d 5 1.41log10 C2

C12 125. True,

5 s2 2 xds1d 5 2 2 x

ln e22x 5 s2 2 xd ln e

y1

GRAPH

X,T,u x2 x2

x2X,T,u

X,T,u x2 X,T,u

X,T,uX,T,u

Section 9.5 Solving Exponential and Logarithmic Equations497

Section 9.5 Solving Exponential and Logarithmic Equations

127. True,

5 2

5 log8 64

log8 4 1 log8 16 5 log8 4 ? 16 129. False, log3su ? vd 5 log3 u 1 log3 v

131. True,

5 1 1 f sxd

5 1 1 loga x

f saxd 5 loga ax 5 loga a 1 loga x 133. False; 0 is not in the domain of f.

135. False; f sx 2 3d 5 lnsx 2 3d Þ ln x 2 ln 3. 137. False; if then f svd 5 ln u2 5 2 ln u 5 2 f sud.v 5 u2,

1. (a)

not a solution

323 Þ 27

32s1d25 5?

27 3. (a)

solution

45 5 45

eln 45 5?

45

e251 ln 4515 5?

45(b)

solution

33 5 27

32s4d25 5?

27 (b)

not a solution

ee 45Þ 45

e251e 4515 5?

45

5. (a)

not a solution

log9 162 Þ32

log9s6 ? 27d 5? 3

2(b)

solution

log9 27 532

log9s6 ? 92d 5

? 32 7.

so x 5 5

2x 5 25

9.

x 5 8

so x 1 4 5 12

3x145312 11.

x 5 8

so x 2 1 5 7

3x21 5 37 13.

x 523

so 3x 5 2

43x 5 42

43x 5 16

15.

x 5 2

2x 5 4

so 2x 2 1 5 3

62x21 5 63

62x21 5 216 17.

so x 5 23

5x 5 523

5x 51

125 19.

x 5 26

so x 1 2 5 24

2x12 5 224

2x12 51

16

21.

2 5 x

6 5 3x

2x 1 6 5 5x

so 2sx 1 3d 5 5x

s22dx13 5 s25dx

4x13 5 32x 23.

x 5225

so 5x 5 22

ln 5x 5 ln 22 25.

x 5 6

so 3x 5 18

log6 3x 5 log6 18

27.

x 5 9

2x 5 18

so 2x 2 3 5 15

lns2x 2 3d 5 ln 15 29.

x 5 4

so x 1 3 5 7

log2sx 1 3d 5 log2 7 31.

No solution since expressions oneither side are undefined forx 5 1.

1 5 x

2 5 2x

so 2x 2 3 5 4x 2 5

log5s2x 2 3d 5 log5s4x 2 5d


33.

x 5 27

2x 5 7

2 2 x 5 32

log3s2 2 xd 5 2 35.

5 2x 2 1

5 s2x 2 1ds1dln e2x21 5 s2x 2 1d ln e 37. x > 010log10 2x 5 2x,

39.

x < 5.49

x 5log 45log 2

log2 2x 5 log2 45

2x 5 45 41.

x 5log 3.6log 3

< 1.17

log3 3x 5 log3 3.6

3x 5 3.6 43.

y < 0.86

y 5log 52

2

2y 5 log 52

log 102y 5 log 52

102y 5 52

45.

y < 0.83

y 5log 1263 log 7

y 5log7 126

3

3y 5 log7 126

log7 73y 5 log7 126

73y 5 126 47.

x < 22.37

x 5log 6log 3

2 4

x 1 4 5 log3 6

log3 3x14 5 log3 6

3x14 5 6 49.

x < 23.60

x 5 log 250 2 6

x 1 6 5 log 250

log 10x16 5 log 250

10x16 5 250

51.

x < 2.64

x 5 ln 14

ln ex 5 ln 14

ex 5 14

3ex 5 4253.

x < 3.00

x 5 ln 20

ln ex 5 ln 20

ex 5 20

14

ex 5 5 55.

x 5ln 40

3< 1.23

3x 5 ln 40

ln e3x 5 ln 40

e3x 5 40

12

e3x 5 20

57.

x < 35.35

x 5log 4

log 1.04

x 5 log1.04 4

log1.04 1.04x 5 log1.04 4

s1.04dx 5 4

250s1.04dx 5 1000 59.

x < 6.80

x 5 2 ln 30

x2

5 ln 30

ln exy2 5 ln 30

exy2 5 30

300e xy2 5 9000 61.

x < 12.22

x 5log 25,000

0.12 log 1000

x 5log1000 25,000

0.12

0.12x 5 log1000 25,000

log1000 10000.12x 5 log1000 25,000

10000.12x 5 25,000

63.

x < 3.28

x 5log 1500

log 42 2

x 1 2 5log 1500

log 4

log4 4x12 5 log4 1500

4x12 5 1500

15

4x12 5 300 65.

No solution

is not possible.log2s25d

log2 2x21 5 log2s25d

2x21 5 25

6 1 2x21 5 1 67.

x < 21.04

x 5 2 2 ln 21

2x 5 ln 21 2 2

2 2 x 5 ln 21

ln e22x 5 ln 21

e22x 5 21

7 1 e22x 5 28


69.

x 5 2ln 1

12< 2.48

2x 5 ln 1

12

ln e2x 5 ln 1

12

e2x 51

12

212e2x 5 21

8 2 12e2x 5 7 71.

x < 0.90

x 5ln 6

2

2x 5 ln 6

ln e2x 5 ln 6

e2x 5 6

4 1 e2x 5 10 73.

x < 0.38

x 5ln 14

7

7x 5 ln 14

ln e7x 5 ln 14

e7x 5 14

32 1 e7x 5 46

75.

x < 0.39

x 5 ln 4 2 1

x 1 1 5 ln 4

ln ex11 5 ln 4

ex11 5 4

25ex11 5 220

23 2 5e x11 5 3 77.

x < 8.99

x 5 3 ln 20

x3

5 ln 20

ln exy3 5 ln 20

exy3 5 20

1 1 exy3 5 21

4s1 1 exy3d 5 84 79.

9.73 < t

log1.03 43

5 t

log1.03 43

5 log1.03 1.03t

43

5 s1.03dt

80006000

5 s1.03dt

8000

s1.03dt 5 6000

81.

lns1

2d20.15

5 t < 4.62

ln1122 5 20.15t

ln1122 5 ln e20.15t

212

5 2e20.15t

32

2 2 5 2e20.15t

300200

5 2 2 e20.15t

300

2 2 e20.15t 5 200 83.

x 5 1000.00

10log10 x 5 103

log10 x 5 3 85.

x 5 22.63

x 5 24.5

2log2 x 5 24.5

log2 x 5 4.5

87.

x 5 2187.00

x 5 37

3log3 x 5 37

log3 x 5 7

4 log3 x 5 28 89.

x < 6.52

x 5 e15y8

eln x 5 e15y8

ln x 53016

16 ln x 5 30 91.

x 51004

5 25.00

x 5102

4

4x 5 102

10log10 4x 5 102

log10 4x 5 2


93.

x < 10.04

x 5e3

2

2x 5 e3

eln 2x 5 e3

ln 2x 5 3 95.

x < ±20.09

x 5 ±!e6

x2 5 e6

eln x25 e6

ln x2 5 6 97.

x 5 3.00

x 5 41.5 2 5

x 1 5 5 41.5

4log4sx15d 5 41.5

log4sx 1 5d 532

2 log4sx 1 5d 5 3

99.

x < 19.63

x 5 81.5 2 3

x 1 3 5 81.5

8log8sx13d 5 83/2

log8sx 1 3d 532

2 log8sx 1 3d 5 3 101.

x < 12.18

x 5 e2.5

eln x 5 e2.5

ln x 552

22 ln x 5 25

1 2 2 ln x 5 24 103.

x 5 2000.00

x 5 2s10d3

x2

5 103

10log10sxy2d 5 103

log10 x2

5 3

3 log10 x2

5 9

21 1 3 log10 x2

5 8

105.

x 5 3.20

x 5165

5x 5 16

4log4 5x 5 42

log4 xs5d 5 2

log4 x 1 log4 5 5 2 107.

x 5 4.00

3x 5 12

3x 1 24 5 36

6log6 3sx18d 5 62

log6sx 1 8ds3d 5 2

log6sx 1 8d 1 log6 3 5 2 109.

0.75 5 x

34

5 x

3 5 4x

x 1 3 5 5x

x 1 3

x5 5

5log5 fsx13dyxg 5 51

log51x 1 3x 2 5 1

log5sx 1 3d 2 log5 x 5 1

111.

x 5 5, x 5 22 swhich is extraneousd

sx 2 5dsx 1 2d 5 0

x2 2 3x 2 10 5 0

xsx 2 3d 5 10

10log10xsx23d 5 101

log10 xsx 2 3d 5 1

log10 x 1 log10sx 2 3d 5 1 113.

and (which is extraneous)24.46x < 2.46

522 ± !48

2

x 522 ± !4 2 4s1ds211d

2s1d 522 ± !4 1 44

2

x2 1 2x 2 11 5 0

x2 1 2x 2 3 5 23

log2sx 2 1dsx 1 3d 5 3

log2sx 2 1d 1 log2sx 1 3d 5 3


115.

and (which is extraneous)20.29x < 2.29

56 ± !36 1 24

65

6 ± !606

x 52s26d ± !s26d2 2 4s3ds22d

2s3d

3x2 2 6x 2 2 5 0

3x2 2 6x 5 2

3xsx 2 2d 5 2

4log4 3xsx22d 5 41y2

log4 3xsx 2 2d 512

log4 3x 1 log4sx 2 2d 512

117.

(which is extraneous)

x 5 6.00

x 5 28

sx 1 8dsx 2 6d 5 0

x2 1 2x 2 48 5 0

x2 1 2x 5 48

x2 1 2x

35 16

2log2sx212xy3d 5 24

log2 xsx 1 2d

35 4

log2 x 1 log2sx 1 2d 2 log2 3 5 4

119. Keystrokes:

10 2 5

-intercept

−10

−6

8

6

s1.40, 0d1.3974 < 1.40

x

Y5 > x 4 c 2 GRAPH

121. Keystrokes:

6 .4 13

-intercept

−5

−25

40

5

s21.82, 0d21.822846 < 21.82

x

Y5 x c 2 GRAPHLN

123. Keystrokes:

2

Point of intersection:

−6

−1

6

7

s0.69, 2dy2

y1 Y5

125. Keystrokes:

3

2 3

Point of intersection:

−6

−4

12

8

s1.48, 3dy2

y1 Y5

x c GRAPHLN

ENTER

e x GRAPH

ENTER

1

127. Formula:

Labels:

Equation:

0.09 < r < 9%

0.18 < 2r

ln 1.197217 5 lnse2r d 1.197217 5 e2r

11,972.17

10,0005 e2r

11,972.17 5 10,000ers2d

Annual interest rate 5 r

Time 5 t 5 2 years

Amount 5 A 5 $11,972,17

Principal 5 P 5 $10,000

A 5 Pert 129.

7.70 years < t

ln 20.09

5 t

ln 2 5 0.09t

ln 2 5 lnse0.09t d 2 5 e0.09t

50002500

5 e0.09t

5000 5 2500e0.09t

X,T,u X,T,u

X,T,uX,T,u


131.

3.1623 3 1029 5 I watts per square centimeter

1028.5 5 I

s107.5ds10216d 5 I

107.5 5I

10216

107.5 5 10log10sIy10216d

7.5 5 log101 I102162

75 5 10 log101 I102162

B 5 10 log101 I102162 133.

205 < m

e5.322580645 5 m

5.322580645 5 ln m

213.2 5 22.48 ln m

2.5 5 15.7 2 2.48 ln m

135. (a)

t < 3.64 months

t 5 102y3 2 1

t 1 1 5 102y3

23 5 log10st 1 1d 812 5 log10st 1 1d 8 5 12 log10st 1 1d 8 5 log10st 1 1d12

28 5 2log10st 1 1d12

72 5 80 2 log10st 1 1d12 (b) Keystrokes:

80 12 1

72

(c) Answers will vary.

−150

8

90

y2

y1 Y5 2 LOG x 1 c ENTER

GRAPH

137. (a)

K < 20.1572

K 514

ln 8

15

Ks4d 5 ln 328

608

Ks4d 5 ln 328 2 08

608 2 08

Kt 5 ln T 2 ST0 2 S

(b)

t < 3.25 hours

t 51

20.1572 ln

428

708

t 51

20.1572 ln

328 2 s2108d608 2 s2108d

t 51K

ln T 2 ST0 2 S

(c)

t < 2.84 hours

t 51

20.1572 ln

328

508

t 51

20.1572 ln

328 2 08

508 2 08

t 51K

ln T 2 ST0 2 S

139. (c)

—CONTINUED—

7.2% < r

0.0719184469 5 r

0.005993204 5r

12

1.005993204 5 1 1r

12

s1.24d1y36 5 1 1r

12

1.24 5 11 1r

12236

6200 5 500011 1r

12212s3d

Formula: A 5 P11 1rn2

nt

X,T,u

Section 9.6 Applications 503

Section 9.6 Applications

139. —CONTINUED—

(d)

634

years < t

ln 1.50.06

5 t

ln 1.5 5 0.06t

ln 1.5 5 ln e0.06t

1.5 5 e0.06t

7500 5 5000e0.06t

Formula: A 5 Pert (e)

Effective yield 582.431000

5 0.08243 < 8.24%

A 5 $1082.43

A 5 1000s1.02d4

A 5 100011 10.08

4 24s1d

Formula: A 5 P11 1rn2

nt

(f)

11.6 years < t

ln 20.06

5 t

ln 2 5 0.06t

ln 2 5 ln e0.06t

2 5 e0.06t

10,000 5 5000e0.06t

Formula: A 5 Pert

141. requires logarithms because canbe rewritten as and the exponents set equal.2x21 5 25

2x21 5 322x21 5 30 143. To solve an exponential equation, first isolate the expo-nential expression, then take the logarithms of both sidesof the equation, and solve for the variable.

To solve a logarithmic equation, first isolate the logarith-mic expression, then exponentiate both sides of the equa-tion, and solve for the variable.

1.

7% < r

0.07 < r

0.0058333 5r

12

1.0058333 5 1 1r

12

s2.00966d1y120 5 1 1r

12

2.00966 5 11 1r

122120

1004.83 5 50011 1r

12212s10d

A 5 P11 1rn2

nt

3.

9% < r

0.0899981 5 r

0.0002466 5r

365

1.0002466 5 1 1r

365

s36.581d1y14,600 5 1 1r

365

36.581 5 11 1r

365214,600

36,581.00 5 100011 1r

3652365s40d

A 5 P11 1rn2

nt

23.1 years < t

ln 40.06

5 t

ln 4 5 0.06t

ln 4 5 ln e0.06t

4 5 e0.06t

20,000 5 5000e0.06t


5.

8% < r

0.08 < r

ln 11.023173

305 r

ln 11.023173 5 30r

ln 11.023173 5 ln e

11.023173 5 ers30d

8267.38 5 750ers30d

A 5 Pert 7.

6% < r

0.059 < r

0.00164384 5r

365

1.000164384 5 1 1r

365

s4.481136d1y9125 5 1 1r

365

4.481136 5 11 1r

36529125

22,405.68 5 500011 1r

3652365s25d

A 5 P11 1rn2

nt

9.

8.75 years < t

log 2

log 1.024 4 5 t

log 2

log 1.025 4t

log1.02 2 5 log1.02 1.024t

2 5 s1.02d4t

12,000 5 600011 10.08

4 24t

A 5 P11 1rn2

nt

11.

6.60 years < t

log 2

log 1.00028774 365 5 t

log 2

log 1.00028775 365t

log1.0002877 2 5 log1.0002877 1.0002877365t

2 5 s1.0002877d365t

4000 5 200011 10.105365 2

365t

A 5 P11 1rn2

nt

13.

9.24 years < t

ln 2

0.0755 t

ln 2 5 0.075t

ln 2 5 ln e0.075t

2 5 e0.075t

3000 5 1500e0.075t

A 5 Pert 15.

14.21 years < t

log1.05 2 5 t

log1.05 2 5 log1.05 1.05t

2 5 1.05t

600 5 30011 10.05

1 21std

A 5 P11 1rn2

nt

17.

Continuous compounding

1587.75 5 1587.75

1587.75 5 750e0.075s10d

1587.75 5 75011 10.075

n 2ns10d

19.

Quarterly compounding

141.48 5 141.48

141.48 5 10011 10.07

4 24s5d

141.48 5 10011 10.07

n 2ns5d

21.

5 0.08329 < 8.33%


A 5 $1083.29

A 5 1000e0.08s1d

A 5 Pert 23.

5 0.07229 < 7.23%


A 5 $1072.29

A 5 100011 10.0712 2

12s1d

A 5 P11 1rn2

nt


37.

$951.23 < P

1000

s1.000136986d365 5 P

1000 5 P11 10.05365 2

365s1d

A 5 P11 1rn2

nt

39.

A < $5496.57

A 530se0.08s10d 2 1d

e0.08y12 2 1

A 5Psert 2 1dery12 2 1

41.

A < $320,250.81

A 550se0.10s40d 2 1d

e0.10y12 2 1


43.

Total interest5 $17,729.42 2 7200 < $10,529.42

A < $17,729.42

A 530se0.08s20d 2 1d

e0.08s20d 2 1


45.

3 5 C

3 5 Ceks0d

y 5 Cekt

ln

83

25 k < 0.4904

ln 83

5 2k

ln 83

5 ln e2k

83

5 e2k

8 5 3eks2d 47.

400 5 C

400 5 Ceks0d

y 5 Cekt

ln

12

35 k < 20.2310

ln 12

5 3k

ln 12

5 ln e3k

12

5 e3k

200 5 400eks3d

31.

$1652.99 < P

10,000

e1.8 5 P

10,000 5 Pe0.09s20d

A 5 Pert 33.

$626.46 < P

750

s1.0001644d1095 5 P

750 5 P11 10.06365 2

365s3d

A 5 P11 1rn2

nt

35.

$3080.15 < P

25,000

s1.005833d360 5 P

25,000 5 P11 10.0712 2

12s30d

A 5 P11 1rn2

nt

25.

< 6.136%

5 0.06136


A 5 $1061.36

A 5 100011 10.06

4 24s1d

A 5 P11 1rn2

nt

27.

5 0.083 5 8.300%


A 5 $1083.00

A 5 100011 10.0812 2

12s1d

A 5 P11 1rn2

nt29. No. Each time the amount is divid-

ed by the principal, the result isalways 2.


49.

12.2 5 C

12.2 5 Ceks0d

y 5 Cekt

61.

20.00012 < k

ln 0.55730

5 k

ln 0.5 5 5730k

ln 0.5 5 ln e5730k

0.5 5 e5730k

0.5C 5 Ceks5730d

y 5 Cekt

4.51 grams < C

4

e20.12 5 C

4 5 Ce20.12

4 5 Ce20.00012s1000d

57. (a) is larger in Exercise 51, because the population ofShanghai is increasing faster than the population ofOsaka.

k (b) corresponds to gives the annual percentage rateof growth.

kr;k

0.0076 < k

1

21 ln

143122

5 k

ln 143122

5 21k

ln 143122

5 ln e21k

14.312.2

5 e21k

14.3 5 12.2eks21d

y < 14.9 million

y 5 12.2e0.0076s26d

y 5 12.2e0.0076t 51.

14.7 5 C

14.7 5 Ceks0d

y 5 Cekt

0.0221 < k

121

ln 234147

5 k

ln 234147

5 21k

ln 234147

5 ln e21k

23.414.7

5 e21k

23.4 5 14.7eks21d

y < 26.1 million

y 5 14.7e0.0221s26d

y 5 14.7e0.0221t

53.

10.5 5 C

10.5 5 Ceks0d

y 5 Cekt

0.0005 < k

lns106

105d21

5 k

ln 106105

5 ln e21k

10.610.5

5 e21k

10.6 5 10.5eks21d

y < 10.6 million

y 5 10.5e0.0005s26d

y 5 10.5e0.0005t 55.

15.5 5 C

15.5 5 Ceks0d

y 5 Cekt

0.0092 < k

lns188

155d21

5 k

ln11881552 5 ln e21k

18.815.5

5 e21k

18.8 5 15.5ks21d

y < 19.7 million

y 5 15.5e0.0092s26d

y 5 15.5e0.0092t

59.

6 5 C

6 5 Ceks0d

y 5 Cekt

20.00043 < k

ln 0.51620

5 k

ln 0.5 5 ln e1620k

0.5 5 e1620k

3 5 6eks1620d

y < 3.91 grams

y 5 6e20.00043s1000d

63.

4.2 5 C

4.2 5 Ceks0d

y 5 Cekt

20.00003 < k

ln 0.524,360

5 k

ln 0.5 5 ln e24,360k

0.5 5 e24,360k

2.1 5 4.2eks24,360d

y < 4.08 grams

y 5 4.2e20.00003s1000d


69.

y 5 22,000e20.2876821s3d < $9281.25

20.2876821 5 k

ln 16,50022,000

5 k

ln 16,50022,000

5 ln ek

16,50022,000

5 ek

16,500 5 22,000e ks1d

75.

pH 5 2log10s9.2 3 1028d < 7.04

pH 5 2log10fH1g

77.

fruit:

The of fruit is times as great.107H1

0.0031623 5 H1

1022.5 5 10log10

fH1g

22.5 5 log10fH1g

2.5 5 2log10fH1g

pH 5 2log10fH1g

tablet:

3.1623 3 10210 5 H1

1029.5 5 10log10

fH1g

29.5 5 log10fH1g

9.5 5 2log10fH1g 5 10,000,071

H1 of fruitH1 of tablet

50.0031623

3.1623 3 10210

65.

5 5 C

5 5 Ceks0d

y 5 Cekt

20.00043 < k

ln 0.51620

5 k

ln 0.5 5 1620k

ln 0.5 5 ln e1620k

0.5 5 e1620k

2.5 5 5eks1620d

y < 3.25 grams

y 5 5e20.00043s1000d 67.

5 5 C

5 5 Ceks0d

y 5 Cekt

20.00012 < k

ln 0.55730

5 k

ln 0.5 5 5730k

ln 0.5 5 ln e5730k

0.5 5 e5730k

2.5 5 5eks5730d

y < 4.43 grams

y 5 5e20.00012s1000d

71.

Alaska:

The earthquake in Alaska was 63 times as great.

108.4 5 I

108.4 5 10log10

I

8.4 5 log10 I

R 5 log10 I

San Fernando Valley:

106.6 5 I

106.6 5 10log10

I

6.6 5 log10 I

Ratio of two intensitiies:

5 101.8 < 63 5 108.426.6

I for AlaskaI for San Fernando Valley

5108.4

106.6

73.

Mexico City:

The earthquake in Mexico City was 40 times as great.

108.1 5 I

108.1 5 10log10 I

8.1 5 log10 I

R 5 log10 I

Nepal:

106.5 5 I

106.5 5 10log10 I

6.5 5 log10 I

Ratio of two intensities:

5 101.6 < 40

5 108.126.5 5 101.6 < 40

I for Mexico CityI for Nepal

5108.1

106.5


79. (a) Keystrokes:

5000 1 4 6

(b)

(c) ps9d 55000

1 1 4e29y6 < 2642

ps0d 55000

1 1 4e20y6 55000

55 1000

0500

10

3500

Y5 4 x 1 x x2c 4

x x

GRAPH

(d)

t < 5.88 years

t 5 sln 0.375ds26d

2t6

5 ln 0.375

ln e2ty6 5 ln 0.375

e2ty6 5 0.375

4e2ty6 5 1.5

1 1 4e2ty6 5 2.5

2000 55000

1 1 4e2ty6

81. (a)

S 5 10s1 2 e20.0575xd

20.0575 < k

ln 0.75

55 k

ln 0.75 5 5k

ln 0.75 5 ln e5k

0.75 5 e5k

20.75 5 2e5k

0.25 5 1 2 e5k

2.5 5 10s1 2 eks5dd

S 5 10s1 2 ekxd (b)

Thus, 3314 units must be sold.

< 3.314

5 10s0.3313536611d

5 10s1 2 e20.4025d

S 5 10s1 2 e20.0575s7dd

83. If the equation models exponential decay, because decay is decreasing so must be negative.kk < 0y 5 Cekt

85. The effective yield of an investment collecting compound interest is the simple interest rate that would yield the same balanceat the end of 1 year. To compute the effective yield, divide the interest earned in 1 year by the amount invested.

87. If the reading on the Richter scale is increased by 1, the intensity of the earthquake is increased by a factor of 10.


1. (a)

(b)

(c) f s2d 5 22 5 4

f s1d 5 21 5 2

f s23d 5 223 518 3. (a)

(b)

(c) gs6d 5 e26y3 5 e22 < 0.135

gspd 5 e2py3 < 0.351

gs23d 5 e2s23dy3 5 e1 < 2.718

5. (c) Basic graph 7. (a) Basic graph reflected in the axisx-

X,T,uex


9.

Table of values:

x

y

−1−2−3−4 1 2 3

6

5

4

3

2

1

−1

11.

Table of values:

x

y

−2−3−4 1 2 3

5

4

3

2

1

−2

13.

Table of values:

x

y

−1−2−3−4 1 2 3

6

5

4

3

1

−1

0 1

0 2223y

21x0 1

1 313y

21x 0 1

1 3 9y

21x

15. Table of values:

3x

21

5

y

4

3

2

1

12

17. Table of values:

–2 –1 2 3 4

1

2

3

4

x

y

0 2

1 3 13y

22x 0 2

121273y

22x

19. Keystrokes:

5 4

−10

−4

20

16

23. (a)

so

(b)

5 1

5 1 2 4 1 4

so sg 8 f ds21d 5 s21d2 1 4s21d 1 4

5 x2 1 4x 1 4

sg 8 f dsxd 5 sx 1 2d2

s f 8 gds2d 5 22 1 2 5 6

s f 8 gdsxd 5 x2 1 2 25. (a)

so

(b)

so sg 8 f ds21d 5 21

5 x

5 x 1 1 2 1

sg 8 f dsxd 5 s!x 1 1 d22 1

s f 8 gds5d 5 |5| 5 5

5 |x| 5 !x2

s f 8 gdsxd 5 !x2 2 1 1 1

27. (a)

Domain:

(b)

Domain: f4, `d

g 8 f 5 2!x 2 4

f2, `d

s f 8 gd 5 !2x 2 4

21. Keystrokes:

2

−12

−2

12

14

X,T,u X,T,uY5 dx x2c 4 GRAPH Y5 GRAPHx 1 dex ex


29. No, does not have an inverse. is not one-to-one.ff sxd 31. Yes, does have an inverse. is one-to-one.fhsxd

33.

x 2 4

35 f 21sxd 5

13

sx 2 4d

x 2 4

35 y

x 2 4 5 3y

x 5 3y 1 4

y 5 3x 1 4

f sxd 5 3x 1 4 35.

sx ≥ 0d

x2 5 f 21sxd

x2 5 y

x 5 !y

y 5 !x 37.

3!t 2 4 5 f 21std

3!t 2 4 5 y

t 2 4 5 y3

t 5 y3 1 4

y 5 t3 1 4

f std 5 t 3 1 4

39. log4 64 5 3 41. e1 5 e 43. because103 5 1000.log10 1000 5 3

45. because 322 519.log3

19 5 22 47. ln e7 5 7 ln e 5 7 49. ln 1 5 0

51. (a)

(b)

(c) f s0.5d 5 log3 0.5 5log 0.5log 3

< 20.631

f s27d 5 log3 27 5 3

f s1d 5 log 31 5 0 53. (a)

(b)

(c) f s10d 5 ln 10 < 2.303

f 1132 5 ln

13

< 21.099

f sed 5 ln 3 5 1

55. (a)

(b)

(c) gs7.5d 5 ln e3s7.5d 5 ln e22.5 5 22.5

gs0d 5 ln e3s0d 5 0

gs22d 5 ln e3s22d 5 26

57.

Table of values:

y

x−1 21 3 4

−2

−1

2

1

3

59.

Table of values:

4x

32

y

1

1

2

3

4

61.

Table of values:2 6 8 10 12 14

−6

−4

−2

2

4

6

8

x

y

x 1 3

y 2122

x 5 6

y 0 1

x 1 3

y 0 1


63. Table of values:y

x2 64 8 10

−4

−2

4

2

6

65. Table of values:y

x−2 42 6 8

−2

4

2

6

8 x 1 e

y 5 4

x 4 5

y 0 0.7

67. log4 9 5log 9log 4

< 1.585 69. log12 200 5log 200log 12

< 2.132

71.

< 1.79588

< 2s0.6826d 1 0.43068

5 2 log5 3 1 log 2

log5 18 5 log5 32 1 log5 2 73.

< 20.43068

< 0 2 s0.43068d

log5 12 5 log5 1 2 log5 2

75.

< 1.02931

< 23f2s0.43068d 1 0.6826g

log5s12d2y3 523f2 log5 2 1 log5 3g 77. log4 6x 4 5 log4 6 1 4 log4 x

79. log5 !x 1 2 512

log5sx 1 2d 81. ln x 1 2x 2 2

5 lnsx 1 2d 2 lnsx 2 2d

83.

512

fln 2 1 ln xg 1 5 lnsx 1 3d

5 lns2xd1y2 1 5 lnsx 1 3d

lnf!2xsx 1 3d5g 5 ln !2x 1 lnsx 1 3d5 85. 223

ln 3y 5 lns3yd22y3 5 ln1 13y2

2y3

87.

5 log8s32x3d

log8 16x 1 log8 2x2 5 log8s16x ? 2x2d 89.

5 ln1 32x2

2

5 ln 9

4x2, x > 0

22sln 2x 2 ln 3d 5 ln12x3 2

22

91.

t < k 5 log21 kk 2 t2

4

,

4flog2 k 2 log2sk 2 tdg 5 43log21 kk 2 t24 93.

z > 0y > 0,x > 0, 5 ln x3y 4z,

3 ln x 1 4 ln y 1 ln z 5 ln x3 1 ln y4 1 ln z

95. False

5 2 1 log2 x

log2 4x 5 log2 4 1 log2 x

97. True

log10 102x 5 2x log10 10 5 2x

99. True

5 2 2 log4 x

log4 16x

5 log4 16 2 log4 x


101.

x 5 6

2x 5 26

2x 5 64 103.

x 5 1

x 2 3 5 22

4x23 5 422

4x23 5116 105.

x 5 243

3log3 x 5 35

log3 x 5 5

107.

x 5 50

2x 5 100

log2 2x 5 log2 100 109.

x 5 4

2x 5 8

2x 1 1 5 9

3log3s2x11d 5 32

log3s2x 1 1d 5 2 111.

x < 5.66

x 5log 500log 3

log3 3x 5 log3 500

3x 5 500

113.

x < 1408.10

x 5 e7.25

eln x 5 e7.25

ln x 5 7.25 115.

x < 6.23

x 5 2 ln 22.5

0.5x 5 ln 22.5

ln e0.5x 5 ln 22.5

e0.5x 5 22.5

2e0.5x 5 45 117.

No solution; there is no powerthat will raise 4 to 21

2.

4x 5 212

24x 512

24x 532 2 1

1 2 4x 51812

12s1 2 4xd 5 18

119.

x 5101.5

2< 15.81

2x 5 101.5

log10 2x 5 1.5 121.

x 5 26 5 64

2log2 x 5 26

log2 x 5 6

13

log2 x 5 2

13

log2 x 1 5 5 7 123.

x 583

< 2.67

3x 5 8

2log2 3x 5 23

log2 xs3d 5 3

log2 x 1 log2 3 5 3

125.

5% < r

0.0499 5 r

0.0124997 5r4

1.0124997 5 1 1r4

s1.6436d1y40 5 1 1r4

1.6436 5 11 1r42

40

410.90 5 25011 1r42

4s10d

A 5 P11 1rn2

nt

127.

7.5% < r

0.074999 5 r

0.000205479 5r

365

1.000205479 5 1 1r

365

s3.07986d1y5475 5 1 1r

365

3.07986 5 11 1r

36525475

15399.30 5 500011 1r

3652365s15d

A 5 P11 1rn2

nt


129.

ln 16.4464667

405 r < 7%

ln 16.4464667 5 40r

ln 16.4464667 5 ln e40r

16.44464667 5 e40r

24,666.97 5 1500ers40d

A 5 Pert 131.


5 0.0565 < 5.65%

A 5 $1056.54

A 5 100011 10.055365 2

365s1d

A 5 P11 1rn2

nt

133.


2 0.07714 < 7.71%

A 5 $1077.14

A 5 100011 10.075

4 24s1d

A 5 P11 1rn2

nt

135.


5 0.07788 < 7.79%

A 5 $1077.88

A 5 1000e0.075s1d

A 5 Pert

137.

3.5 5 C

3.5 5 Ceks0d

y 5 Cekt

20.00043 < k

ln 0.51620

5 k

ln 0.5 5 1620k

ln 0.5 5 ln e1620k

0.5 5 e1620k

1.75 5 3.5eks1620d

y < 2.282 grams

y 5 3.5e20.00043s1000d

139.

20.00012 < k

ln 0.55730

5 k

ln 0.5 5 5730k

ln 0.5 5 ln e5730k

0.5 5 eks5730d

0.5C 5 Ceks5730d

y 5 Cekt

2.934 grams < C

2.6

e20.12 5 C

2.6 5 Ce20.12

2.6 5 Ce20.00012s1000d

141.

5 5 C

5 5 Ceks0d

y 5 Cekt

20.000028 < k

ln 0.524,360

5 k

ln 0.5 5 24,360k

ln 0.5 5 ln e24,360k

0.5 5 e24,360k

2.5 5 5eks24,360d

y < 4.860 grams

y 5 5e20.000028s1000d


143.

3.8 years < t

log

30.0024.95

log 1.055 t

log1.05 30.0024.95

5 log1.05 1.05t

30.0024.95

5 1.05t

30.00 5 24.95s1.05dt 145.

ln 2

0.0555 t < 12.6 years

ln 2 5 0.055t

ln 2 5 ln e0.055t

2 5 e0.055t

1500 5 750e0.055t

A 5 Pert 147.

per square centimeter 5 3.16 3 1024 watts

1023.5 5 I

1012.5s10216d 5 I

1012.5 5I

10216

1012.5 5 10log10sIy10216d

12.5 5 log101 I102162

125 5 10 log101 I102162

B 5 10 log101 I102162

149. Keystrokes:

600 1 2 .2

The limiting size of the population in this habitat is 600.

900

360

0

151. (a) Keystrokes: (b) cubic feet per minute per person

78.56 11.6314 Trace to

30

5000

0

x 5 250

V 5 14.3


1. (a)

5 81

5 54s32d

f s21d 5 54s23d21

(b)

5 54

f s0d 5 54s23d0

(c)

< 44.09

f s12d 5 54s2

3d1y2 (d)

5 24

5 54s47d

f s2d 5 54s23d2

2.

–6 3 6 9 12 15−3

3

6

9

12

15

18

x

y 3. (a)

Domain:

(b)

Domain: s2`, `d 5 9x2 2 24x 1 17

5 9x2 2 24x 1 16 1 1

g 8 f 5 gs f sxdd 5 gs3x 2 4d 5 s3x 2 4d2 1 1

s2`, `d 5 3x2 2 1

5 3x2 1 3 2 4

f 8 g 5 f sgsxdd 5 f sx2 1 1d 5 3sx2 1 1d 2 4

X,T,u

Y5 2 LN X,T,u GRAPH

Y5 4 x 1 x x2c d d GRAPHex


4.

y 5 5x 1 6

f sxd 5 5x 1 6

x 2 6 5 5y

x 5 5y 1 6x 2 6

55 y f 21sxd 5

15

sx 2 6d

5.

5 x

5 x 2 6 1 6

5 22s212 x 1 3d 1 6

gs f sxdd 5 gs212 x 1 3d

5 x

5 x 2 3 1 3

5 212s22x 1 6d 1 3

f sgsxdd 5 f s22x 1 6d 6. and are inverse functions.

2 4 6 8 10 12

2

4

6

8

10

12

x

y

f

g

gf

7. log4 5x2

!y5 log4 5 1 2 log4 x 2

12

log4 y 8. ln x 2 ln y 5 ln xy4, y > 0

9.

5 3 1 log5 6

log5s53 ? 6d 5 3 log5 5 1 log5 6 10.

x 5 64

4log4 x 5 43

log4 x 5 3 11.

y < 0.973

y 5log 832

3

3y 5 log 832

log 103y 5 log 832

103y 5 832

12.

t < 13.733

t 5ln 30.08

0.08t 5 ln 3

ln e0.08t 5 ln 3

e0.08t 5 3

400e0.08t 5 1200 13.

x < 15.516

x 5e10y3 1 3

2

2x 2 3 5 e10y3

elns2x23d 5 e10y3

lns2x 2 3d 5103

3 lns2x 2 3d 5 10 14.

so x 5 2

3x 5 32

3x 5 9

23x 5 29

2 2 3x 5 27

8s2 2 3xd 5 256

15.

x 5 8

4x 5 32

2log2 4x 5 25

log2 xs4d 5 5

log2 x 1 log2 4 5 5 16.

x < 109.196

x 5 2e4

x2

5 e4

elnsxy2d 5 e4

ln x2

5 4

ln x 2 ln 2 5 4 17.

so x 5 0

e x 5 e0

e x 5 1

e x 1 9 5 10

30se x 1 9d 5 300


18. (a)

(b)

5 $8110.40

A 5 2000e0.07s20d

5 $8012.78

A 5 200011 10.07

4 24s20d

19.

$10,806.08 5 P

100,000

s1.0225d100 5 P

100,000 5 P11 10.09

4 24s25d

20.

7% < r

0.07 < r

ln 2.01376

105 r

ln 2.01376 5 10r

ln 2.01376 5 ln e10r

2.01376 5 e10r

1006.88 5 500ers10d

21.

18,000 5 C

18,000 5 Ceks0d

y 5 Cekt

20.2513144 5 k

ln 1418

5 k

ln 1418

5 ln ek

14,00018,000

5 ek

14,000 5 18,000eks1d

5 $8469.14

y 5 18,00020.2513144s3d

22. ps0d 52400

1 1 3e20y4 5 600 23. ps4d 52400

1 1 3e24y4 < 1141 24.

t 5 24 ln 13

< 4.4 years

2t4

5 ln 13

ln e2ty4 5 ln 13

e2ty4 513

3e2ty4 5 1

1 1 3e2ty4 524001200

1200 52400

1 1 3e2ty4

Cumulative Test For Chapters 7–9


1. V varies directly as the square root of x and inversely as y.

V 5k!x

y

2.

216 5 k

264 5 ks2d2

v 5 kt2

3.

d 5 128 feet

d 52

25s40d2

225

5 k

50625

5 k

50 5 ks25d2

d 5 ks2 4.

300 5 k

N 5 50 prey 300 5k

0 1 1

N 5300

5 1 1 N 5

kt 1 1

5.

–6 –4 4 6 8 10

−6

−4

−2

2

4

x

y

5x 1 2y > 10 or y > 252

x 1 5 6.

y ≥ 2x 1 2

y 5 2x 1 2

y 2 2 5 2x

y 2 2 5 2sx 2 0d

m 52 2 0

0 2 s21d 521

5 2

7.

23

5 a

69

5 a

6 5 9a

4 5 as9d 2 2

4 5 as0 2 3d2 2 2

y 523sx 2 3d2 2 2y 5 asx 2 hd2 1 k 8.

center

–4 –2 1 2 4

−4

−2

−1

1

2

4

x

y

r 5 !8 < 2.85 s0, 0d

x2 1 y2 5 8


9.

–3 –2 2 3

−3

1

3

x

y

2

−1

−2

y 5 2x2

2

2y 5 2x2

x2 1 2y 5 0 10.

–3 –2 2 3

−3

1

3

x

y

x2

11

y2

45 1 11.

–4 –3 –2 2 3 4

−4

−3

2

3

4

x

y

x2

11

y2

45 1

12. equation at circular arch

Maximum height of truck: feet8 1 3 5 11

5 !9 5 3

y 5 !25 2 16

y 5 !25 2 x2

x

y

4, 25 − x2 )(

(5, 0)

8 pillar′ 8 pillar′8′

10′

x2 1 y2 5 25

13.

y-intercept:

x-intercept:

none

vertical asymptote:

horizontal asymptote: since the degree of the numerator is less than the degree of the denominator

y 5 0

x 5 2

x 2 2 5 0

0 5 4

0 54

x 2 2

y 54

0 2 25 22

–4 4 6 8 10 12

8

6

4

2

2

4

6

8

x

yy 5

4x 2 2

14.

y-intercept:

x-intercept:

vertical asymptote: none


y 5 4

x 1 1 Þ 0

0 5 x

0 5 4x2

0 54x2

x2 1 1

y 54s0d2

0 1 15 0

y

x1 2 3 4−1−2−3−4

−2

−3

2

5

3

y 54x2

x2 1 1


15. vertical asymptote:

horizontal asymptote:

f sxd 52x

x 2 3

y 5 2

x 5 316.

horizontal asymptote is since the degrees are equaland the leading coefficient of the numerator is 10 and theleading coefficient of the denominator is 1. As x increases,the average cost approaches $10.

y 5 10

Csxd 510x 1 13

x

17.

Keystrokes:

1

2 5y2

y1

s2, 1d

y 5 x 2 1

y 5 22x 1 5 2y 5 2x 1 1

−10

−10

10

10

2x 1 y 5 5 x 2 y 5 1

2x 1 y 5 5

x 2 y 5 1

Y5 X,T,u 2 ENTER

x2c X,T,u 1 GRAPH

18.

s3, 22d

5 3

x 5 5s22d 1 13 20y 1 52 1 2y 5 8

y 5 22 4s5y 1 13d 1 2y 5 8

22y 5 244 x 5 5y 1 13

x 2 5y 5 13

4x 1 2y 5 8

19.

23y 2 11z 5 223

y 1265

z 5465

x 2 2y 2 6z 5 212

23y 2 11z 5 223

5y 1 26z 5 46

x 2 2y 2 6z 5 212

22x 1 y 1 z 5 1

4x 2 3y 1 2z 5 22

x 2 2y 2 6z 5 212

x 2 2y 2 6z 5 212

22x 1 y 1 z 5 1

4x 2 3y 1 2z 5 22

s2, 4, 1d x 5 2

x 2 8 2 6 5 212

x 2 2s4d 2 6s1d 5 212

y 5205

5 4

y 1265

5465

z 5 1

y 1265

z 5465

x 2 2y 2 6z 5 212

235

z 5235

y 1265

z 5465

x 2 2y 2 6z 5 212

20.

1215

, 2225 2

x 5|23 4

25|5

5 2225

x 5| 425

211|

55 2

15

D 5 |23 211| 5 s2d 2 s23d 5 5

3x 1 y 5 25

2x 2 y 5 4


21.

s21, 6, 22d

x 5 21

x 1 30 5 29

x 1 5s6d 5 29

y 5 6

y 5305

y 235 5

275

y 1310s22d 5

275z 5 22

511R33

100

510

0310

1

:::

29275

224

4R2 1 R33100

510

0310115

:::

29275

22254

2110R23

100

51

24

0310

1

:::

29275

2264

22R1 1 R23100

521024

023

1

:::

292542264

3120

50

24

023

1

:::

294

2264 2 4y 1 z 5 226

2x 2 3z 5 4

x 1 5y 5 29 22.

Area 5 212s211d 5

112

5 211

5 24 2 1 2 6

5 21s2 1 2d 2 1s2 2 1d 1 1s24 2 2d

5 21| 222

11| 2 1|21 1

1| 1 1|21 222|

|x1

x2

x3

y1

y2

y3

111| 5 |21

21

12

22

111|

sx3, y3d 5 s1, 22d

sx2, y2d 5 s2, 2d

sx1, y1d 5 s21, 1d

23.

if system is inconsistent

0 Þ 235

24x 1 8y 5 232

4x 2 8y 5 23

s22d2x 2 4y 5 16s22d

4x 2 8y 5 23

K 5 24

2x 1 Ky 5 16

4x 2 8y 5 23

26. f and g are inverse functions, so the graphs are reflections in the line y 5 x.

x

y

−1

1

3

2

−1 1 2 3

f

g

24.

–1 4 5x

−3

1

3

y

2

−1

−2

32

g sxd 5 log3sx 2 1d 25. log41

16 5 22 because 422 51

16


27.

5 log2sxyd3

z

3slog2x 1 log2yd 2 log2z 5 log2sxyd3 2 log2z 28.

5 ln5 1 lnx 2 2lnsx 1 1d

ln5x

sx 1 1d2 5 ln5 1 lnx 2 lnsx 1 1d2

29. (a)

(c)

t < 18.01

t 5log4

log1.08

log1.081.08t 5 log1.084

1.08t 5 4

1.08t 52000500

500s1.08dt 5 2000

3 5 x

9 5 x2

19

51x2

19

5 x22

xlogxs19d 5 x22

logx1192 5 22 (b)

(d)

x < 0.87

x 5

ln173

2

2x 5 ln173

lne2x 5 ln173

e2x 5173

1 1 e2x 5203

3s1 1 e2xd 5 20

x < 12.18

x 5 e52

elnx 5 e52

lnx 552

lnx 5104

4lnx 5 10

30.

Cs5d 5 $29.63

Cs5d 5 24.95s1.035d5

Cstd 5 Ps1.035dt 31.

5 8.329%

5 0.08329

effective yield 583.291000

A 5 $1083.29

A 5 1000e0.08s1d

A 5 Pert 32.

15.40 years 5 t

ln40.09

5 t

ln4 5 0.09t

ln4 5 lne0.09t

4 5 e0.09t

4000 5 1000e0.09t

A 5 Pert


8.

x 5 4 x 5 24

sx 2 4dsx 1 4d 5 0

x2 2 16 5 0 9.

Not real

x 5 24 x2 2 4x 1 16 5 0

sx 1 4dsx2 2 4x 1 16d 5 0

x3 1 64 5 0 10.

Not real

x 5 2 3x2 1 4 5 0

sx 2 2ds3x2 1 4d 5 0

3x2sx 2 2d 1 4sx 2 2d 5 0

3x3 2 6x2 1 4x 2 8 5 0

11.

Equation

320

r5 t

320 5 r ? t

12.

Equation:

592

x 1 7

5 1x 112

x 1 3x2 1 s1 1 5 1 1d

P 5 sx 1 1d 1 112

x 1 52 1 s3x 1 1d

DistanceVerbal Model: 5 Rate ? TimeVerbalModel:

1Perimeter 5Lengthside 1

Lengthside 2

1Lengthside 3

CHAPTER 9 Exponential and Logarithmic Functions

SECTION 9.1 Exponential Functions

1. Graph the line Test one point in each of thehalf-planes formed by this line. If the point satisfies theinequality, shade the entire half-plane to denote that everypoint in the region satisfies the inequality.

x 1 y 5 5. 2. and

The difference between the two graphs is that the firstcontains the boundary (because of the equal sign) and thesecond does not.

3x 2 5y < 153x 2 5y ≤ 15

3.

Test point:

True

0 > 0 2 2

s0, 0dy

x−2−3 −1 2 31

−3

−4

−2

−1

1

2

y > x 2 2 4.

Test point:

True

0 ≤ 5 2 0

s0, 0d

x

y

−1 1 2 43 5

2

4

1

3

−1

y ≤ 5 232 x

5.

Test point:

True

Test point:

False

0 < 0 2 1

s0, 0d

21 < 1

21 < 23s3d 2 1

s3, 21d

x

y

−1 2 3 4 5

2

1

3

−2

−3

y < 23 x 2 1 6.

Test point:

False

0 > 6 2 0

s0, 0dy

x−2 2 4 6 8

−2

2

6

4

8

x > 6 2 y


7.

Test point:

True

24 ≤ 22

s0, 24dy

x−2−3 −1 2 31

−3

−4

−5

−1

1

y ≤ 22 8.

Test point:

True

8 > 7

s8, 0d

x

y

−2 4 6 82 10

4

2

6

−4

−2

−6

x > 7

9.

Test point:

True

2 ≥ 0

2 ≥ 223s3d 1 2

s3, 2dy

x−1 1 2 3 4

−1

1

3

2

4

y ≥ 223 x 1 2

3y ≥ 22x 1 6

2x 1 3y ≥ 6 10.

Test point:

True

0 > 0 252

s0, 0dy

x−2−3 −1 2 31

−3

−2

−1

1

2

y > 52 x 2

52

22y < 5 2 5x

5x 2 2y < 5

11.

Labels:

Equation:

x 1 3 < 21.6

x < 18.6 and 21.61 (reject)

x 517 ± !409

2

x 517 ± !289 1 120

2

x 52s217d ± !s217d2 2 4s1ds230d

2s1d

0 5 x2 2 17x 2 30

10x 1 30 1 10x 5 x2 1 3x

10sx 1 3d 1 10x 5 xsx 1 3d

10xsx 1 3d11x

11

x 1 32 5 1 110210xsx 1 3d

1x

11

x 1 35

110

Rate together 51

10

Rate for person 2 51

x 1 3

Rate for person 1 51x

Rate forperson 1

VerbalModel:

1Rate forperson 2

5Rate together


12. Formula:

Labels:

Equation:

c < 67.1 feet

c 5 !4500 5 !900 ? 5 5 30!5

c2 5 4500

c2 5 3600 1 900

c2 5 602 1 302

b 5 30 feet

a 5 60 feet

c 5 hypotenuse

c2 5 a2 1 b2

SECTION 9.2 Inverse Functions

1.

is not a function of because for some values of therecorrespond two values of For example, and

are solution points.s4, 22ds4, 2dy.

xxy

±!x 5 y

x 5 y2

x 2 y2 5 0 2.

is a function of because for each value of there corresponds exactly one value of y.

xxy

y 512|x| 2 2

22y 5 2|x| 1 4

|x| 2 2y 5 4

3.

The domain of is The domain of is is undefined at x 5 ±2.g

s22, 2d.gf22, 2g.f

f sxd 5 !4 2 x2, gsxd 56

!4 2 x24.

Range:

hs16d 5 8 2 !16 5 4

hs9d 5 8 2 !9 5 5

hs4d 5 8 2 !4 5 6

hs0d 5 8 2 !0 5 8

H4, 5, 6, 8J

hsxd 5 8 2 !x over H0, 4, 9, 16J

5.

5 22x2 2 4

2s5x2 2 1d 1 s3x2 2 5d 5 25x2 1 1 1 3x2 2 5 6.

5 30x3 1 40x2

s22xds25xds3x 1 4d 5 10x2s3x 1 4d

7. (multiply by FOIL)su 2 4vdsu 1 4vd 5 u2 2 16v2 8.

5 9a2 2 12ab 1 4b2

5 9a2 2 6ab 2 6ab 1 4b2

s3a 2 2bd2 5 s3a 2 2bds3a 2 2bd

9.

5 t 3 2 6t 2 1 12t 2 8

5 t 3 2 4t 2 1 4t 2 2t 2 1 8t 2 8

5 st 2 2 4t 1 4dst 2 2d

st 2 2d3 5 st 2 2d2st 2 2d 10.

5x2

22

x4

6x3 2 3x2

12x5

6x3

12x2

3x2

12x


11.

802

645 h 5 100 feet

802 5 64h

80 5 !64h

80 5 !2s32dh

v 5 !2gh 12.

Labels:

Equation:

13 minutes

12 5 x

4.20 5 0.35x

5.15 5 0.95 1 0.35x

Additional minute cost 5 0.35x

First minute cost 5 0.95

Total cost 5 5.15

Totalcost

VerbalModel:

5First minute cost

1Additional minute cost

SECTION 9.3 Logarithmic Functions

1.

Horizontal shift 4 unitsright

y

x−2 42 86

−2

4

2

6

8

gsxd 5 sx 2 4d2 2.

Reflection in the axisx-y

x−2−4−6 2 64

−4

−6

−8

−10

2

hsxd 5 2x2

3.

Vertical shift 1 unit upy

x−2−4−6 2 64

−2

4

6

8

10

jsxd 5 x2 1 1 4.

Horizontal shift 3 unitsleft


y

x−2−4−6−8 2 4

−4

−6

2

4

6

ksxd 5 sx 1 3d2 2 5

5. 2x3 2 6x 5 2xsx2 2 3d 6.

5 s2 2 yds6 1 yd

5 s4 2 y 2 2ds4 1 y 1 2d

16 2 sy 1 2d2 5 f4 2 sy 1 2dg f4 1 sy 1 2dg

7.

5 st 1 5d2

t 2 1 10t 1 25 5 st 1 5dst 1 5d 8.

5 s5 2 uds1 1 u2d

5 2 u 1 5u2 2 u3 5 1s5 2 ud 1 u2s5 2 ud


9.

Intercepts:

x 5 6, s6, 0d

12 x 5 3

0 5 3 212 x

y 5 3 212s0d 5 3, (0, 3d

y

x−2 42 6

−4

−2

4

2

6

y 5 3 212 x 10.

Intercepts:

x 5 2

3x 5 6

3x 2 4s0d 5 6

y 5 232

3s0d 2 4y 5 6

y

x−2 −1 2 3 4

−3

−4

−1

1

2

3x 2 4y 5 6

11.

Intercepts: Vertex:

s1, 0d, s5, 0d

x 5 1 x 5 5

s3, 24d0 5 sx 2 1dsx 2 5d

5 sx 2 3d2 2 40 5 x2 2 6x 1 5

y 5 sx2 2 6x 1 9d 1 5 2 9y 5 5, s0, 5d

y

x−2 42 6 8

−4

−2

4

2

6

y 5 x2 2 6x 1 5

12.

Intercepts: Vertex:

s3, 0d, s1, 0d

x 5 3, 1

x 5 2 ± 1

x 2 2 5 ±1

sx 2 2d2 5 1

0 5 2sx 2 2d2 1 1

s2, 1d y 5 2s0 2 2d2 1 1 5 23, s0, 23d

y

x−1 2 4

−1

−2

−3

1

y 5 2sx 2 2d2 1 1

SECTION 9.4 Properties of Logarithms

1. Multiplication Property: n!u n!v 5 n!uv 2. Division Property:n!un!v

5 n!uv

3. and cannot be added because the indices aredifferent.

3!2x!2x 4. is not in simplest form. The radical in the denomi-nator must be rationalized.

1!2x

51

!2x?!2x!2x

5!2x2x

1y!2x

5.

5 19!3x

5 s25 2 6d!3x

5 25!3x 2 6!3x

25!3x 2 3!12x 5 25!3x 2 3 ? 2!3x 6. (multiply by FOIL)s!x 1 3ds!x 2 3d 5 x 2 9


7.

5 !5u

5 s2 2 1d!5u

5 2!5u 2 !5u

!us!20 2 !5 d 5 !20u 2 !5u 8.

5 4t 1 12!t 1 9

5 4t 1 6!t 1 6!t 1 9

s2!t 1 3d25 s2!t 1 3ds2!t 1 3d

9.

5 25x!2

550x!2

2

50x!2

550x!2

?!2!2

10.

5 6s!t 1 2 2 !t d

512s!t 1 2 2 !t d

2

512s!t 1 2 2 !t d

t 1 2 2 t

12

!t 1 2 1 !t5

12!t 1 2 1 !t

?!t 1 2 2 !t!t 1 2 2 !t

11.

22 units < x

21.9952 5 x

10.9976 5 0.5x

10.4976 5 0.5x 2 0.5

3.242 5 0.5sx 2 1d

3.24 5 !0.5sx 2 1d

23.24 5 2!0.5sx 2 1d

26.76 5 30 2 !0.5sx 2 1d

p 5 30 2 !0.5sx 2 1d 12.

Labels:

Equation:

$2300 5 x

1955 5 x ? 0.85

5 85%

100% 2 Discount rate 5 100% 2 15%

List price 5 x

Sale price 5 1955

Saleprice

VerbalModel:

5Listprice

? 100% 2 Discount rate

SECTION 9.5 Solving Exponential and Logarithmic Equations

1.

It is not possible for this system to have exactly two solutions. A system of linear equations has no solutions,one solution, or an infinite number of solutions.

x 1 y 5 4

7x 2 2y 5 8 2.

This system has no solution because the equations represent parallel lines and have no point of intersection.

22x 1 y 5 1

8x 2 4y 5 5

3.

2 5 x

20 5 10x

2x 1 2 5 12x 2 18

3s23 x 1

23d 5 s4x 2 6d3

23 x 123 5 4x 2 6 4.

(can use quadratic formula also)

x 5 5 ± 2!2

x 2 5 5 ±!8

sx 2 5d2 5 8

x2 2 10x 1 25 5 217 1 25

x2 2 10x 5 217

x2 2 10x 1 17 5 0


5.

212

5 x

236

5 x

23 5 6x

5 2 8 5 6x

2x1 52x

24x2 5 s3d2x

52x

24x

5 3 6.

x 553

3x 5 5

sx 2 5d 1 2x 5 0

xsx 2 5d11x

12

x 2 52 5 s0dxsx 2 5d

1x

12

x 2 55 0

7.

x 5 7 x 5 1

x 2 4 5 3 or x 2 4 5 23

|x 2 4| 5 3 8. Check:

7 5 7 x 5 47

!49 5?

7 x 1 2 5 49

!47 1 2 5?

7 s!x 1 2 d25 72

!x 1 2 5 7

9.

Function:

t

d

2 4 6

300

200

100

d 5 73 ? t

10.

V 5 25ph

V 5 p s5d2h

h

V

5 10 15 20

1600

1200

800

400

V 5 pr2hDistanceVerbal Model: 5 ?Rate Time

11.

V 5 10pr 2

V 5 pr 2s10d

r

V

4 8 12 16

8000

6000

4000

2000

V 5 pr2h 12.

F 5 25x

25 5 k

100 5 ks4d

x

F

1 2 3 4

100

75

50

25

F 5 kx

SECTION 9.6 Applications

1.

Direct variation as power

nth

y 5 kx2 2.

Inverse variation

y 5kx

3.

Joint variation

z 5 kxy 4.

Joint variation

z 5kxy


5.

s3, 3d

x 5 3

x 2 3 5 0

y 5 3

23y 5 29

2x 2 2y 5 29

x 2 y 5 0

x 1 2y 5 9

x 2 y 5 0 6.

s103 , 53d

x 5103

x 5203 ? 1

2

2x 5203

2x 5453 2

253

2x 5 15 2253

2x 1 5s53d 5 15

y 5 53

3y 5 5

26x 2 12y 5 240

6x 1 15y 5 45

22s3x 1 6y 5 20d 2 2

3s2x 1 5y 5 15d3

3x 1 6y 5 20

2x 1 5y 5 15 7.

s212, 14d

y 514

y 5 s212d2

x 5 212

s2x 1 1dsx 2 2d 5 0

2x2 2 3x 2 2 5 0

23x 1 2x2 5 2

23x 1 2y 5 2

y 5 x2

s2, 4d

y 5 4

y 5 22

x 5 2

8.

s0, 0d

x 5 0

x 2 0 5 0

y 5 0

y2s2 2 yd 5 0

2y2 2 y3 5 0

2x 1 2y2 5 0

x 2 y3 5 0

x 2 2y2 5 0

x 2 y3 5 0 9.

s1, 2, 1d z 5 1

22z 5 22

1 1 2s2d 2 2z 5 3

x 5 1

x 2 2 5 21

y 5 2

5y 5 10

3y 2 2z 5 4

x 2 y 5 21

2y 1 2z 5 6

3y 2 2z 5 4

x 2 y 5 21

3x 2 y 1 2z 5 3

x 1 2y 2 2z 5 3

x 2 y 5 21 10.

s4, 21, 3d

x 5 4

x 2 3 5 1

z 5 3

4z 5 12

3s21d 1 4z 5 9

3y 1 4z 5 9

y 5 21

x 2 z 5 1

22R1 1 R2 23R1 1 R3

3x 1 3y 1 z 5 12

2x 1 y 2 2z 5 1

x 2 z 5 1

R1 ↔ R2

3x 1 3y 1 z 5 12

x 2 z 5 1

2x 1 y 2 2z 5 1

s8, 2d x 5 8

x 2 23 5 0

y 5 2


CHAPTER 10 Sequences, Series, and Probability

SECTION 10.1 Sequences and Series

1. Multiplicative Property of Equality

(Multiply both sides of the equation by the reciprocal ofthe coefficient of the variable.)

x 5 25

217 ? 27x 5 35 ? 2

17

27x 5 35

2. Additive Property of Equality

(Add the opposite of 63 on both sides of the equation.)

7x 5 228

7x 1 63 2 63 5 35 2 63

7x 1 63 5 35

3. is a solution of the equation ifthe equation is true when is substituted for t.23

t 2 1 4t 1 3 5 0t 5 23 4.

The first step in solving this equation is to multiply bothsides of the equation by the lowest common denominatorxsx 1 1d.

3x

21

x 1 15 10

5. sx 1 10d22 51

sx 1 10d2 6.

5 18sx 2 3d3

18sx 2 3d5

sx 2 3d2 5 18sx 2 3d522 7. sa2d24 5 a28 51a8

8. s8x3d1y3 5 81y3x3?1y3 5 2x 9.

5 8x!2x

!128x3 5 !64 ? 2 ? x2 ? x 10.

55s!x 1 2d

x 2 4

55s!x 1 2ds!x d2

2 22

5

!x 2 25

5!x 2 2

?!x 1 2!x 1 2

11. (a) Graph opens down because

(b)

(c)

s2, 4d

y 5 222 1 4s2d 5 24 1 8 5 4

x 524

2s21d 5 2

x 52b2a

s0, 0d s4, 0d

x 5 0 x 5 4

2x 5 0 x 2 4 5 0

0 5 2xsx 2 4d

0 5 2x2 1 4x

a < 0. 12. Keystrokes: 4

−1

−2

5

5

Y5 x2c X,T,u X,T,ux2 1 GRAPH

C H A P T E R 1 0Sequences, Series, and Probability

Section 10.1 Sequences and Series . . . . . . . . . . . . . . . . . . . .523

Section 10.2 Arithmetic Sequences . . . . . . . . . . . . . . . . . . .528

Section 10.3 Geometric Sequences and Series . . . . . . . . . . . . . .533

Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .537

Section 10.4 The Binomial Theorem . . . . . . . . . . . . . . . . . . .539

Section 10.5 Counting Principles . . . . . . . . . . . . . . . . . . . . .541

Section 10.6 Probability . . . . . . . . . . . . . . . . . . . . . . . . .543


Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .547

523

C H A P T E R 1 0Sequences, Series, and Probability

Section 10.1 Sequences and SeriesSolutions to Odd-Numbered Exercises

1.

2, 4, 6, 8, 10, . . . , 2n, . . .

a5 5 2s5d 5 10

a4 5 2s4d 5 8

a3 5 2s3d 5 6

a2 5 2s2d 5 4

a1 5 2s1d 5 2 3.

22, 4, 26, 8, 210, . . . , s21dn2n, . . .

a5 5 s21d5 ? 2s5) 5 210

a4 5 s21d4 ? 2s4d 5 8

a3 5 s21d3 ? 2s3d 5 26

a2 5 (21d2 ? 2s2d 5 4

a1 5 s21d1 ? 2s1d 5 22

5.

12

, 14

, 18

, 116

, 132

, . . . , 1122

n

, . . .

a1 5 1122

5

51

32

a4 5 1122

4

51

16

a3 5 1122

3

518

a2 5 1122

2

514

a1 5 1122

1

512

7.

14

, 18

, 21

16,

132

, 21

64, . . . , 12

122

n11

, . . .

a5 5 12122

6

51

64

a4 5 12122

5

5 21

32

a3 5 12122

4

51

16

a2 5 12122

3

5 218

a1 5 12122

2

514

9.

a5 5 s20.2d521 5 s20.2d4 5 0.0016

a4 5 s20.2d421 5 s20.2d3 5 20.008

a3 5 s20.2d321 5 s20.2d2 5 0.04

a2 5 s20.2d221 5 s20.2d1 5 20.2

a1 5 s20.2d121 5 s20.2d0 5 1 11.

12

, 13

, 14

, 15

, 16

, . . . , 1

n 1 1, . . .

a5 51

5 1 15

16

a4 51

4 1 15

15

a3 51

3 1 15

14

a2 51

2 1 15

13

a1 51

1 1 15

12

524 Chapter 10 Sequences, Series, and Probability

13.

25

, 12

, 611

, 1017

, . . . , 2n

3n 1 2, . . .

a5 52s5d

3s5d 1 25

1017

a4 52s4d

3s4d 1 25

814

547

a3 52s3d

3s3d 1 25

611

a2 52s2d

3s2d 1 25

48

512

a1 52s1d

3s1d 1 25

25

15.

21, 14

, 219

, 116

, 21

25, . . . ,

s21dn

n2 , . . .

a5 5s21d5

52 5 21

25

a4 5s21d4

42 51

16

a3 5s21d3

32 5 219

a2 5s21d2

22 514

a1 5s21d1

12 5 21

17.

92

, 194

, 398

, 7916

, 15932

, . . . , 5 212n, . . .

a5 5 5 2125 5

15932

a4 5 5 2124 5

7916

a3 5 5 2123 5

398

a2 5 5 2122 5

194

a1 5 5 2121 5

92 19.

a5 5s5 1 1d!

5!5

6!5!

56 ? 5!

5!5 6

a4 5s4 1 1d!

4!5

5!4!

55 ? 4!

4!5 5

a3 5s3 1 1d!

3!5

4!3!

54 ? 3!

3!5 4

a2 5s2 1 1d!

2!5

3!2!

53 ? 2!

2!5 3

a1 5s1 1 1d!

1!5

2!1!

52 ? 1

15 2

21.

a5 52 1 s22d5

5!5

2305 ? 4 ? 3 ? 2 ? 1

5214

a4 52 1 s22d4

4!5

184 ? 3 ? 2 ? 1

534

a3 52 1 s22d3

3!5

263 ? 2 ? 1

5 21

a2 52 1 s22d2

2!5

62 ? 1

5 3

a1 52 1 s22d1

1!5 0 23.

5 272

5 21f72g

a15 5 s21d15f5s15d 2 3g

25. a8 582 2 2

s8 2 1d! 5627!

562

7 ? 6 ? 5 ? 4 ? 3 ? 2 ? 15

312520

27.5!4!

55 ? 4 ? 3 ? 2 ? 1

4 ? 3 ? 2 ? 15 5

29.10!12!

510!

12 ? 11 ? 10!5

1132

31.

5 53130

525 ? 24 ? 23 ? 22 ? 21

5 ? 4 ? 3 ? 2 ? 15 5 ? 6 ? 23 ? 11 ? 7

25!20! 5!

525 ? 24 ? 23 ? 22 ? 21 ? 20!

20! 5!

Section 10.1 Sequences and Series525

43. Keystrokes(calculator in sequence and dot mode):

.8 1

0

−4

10

4


.5

00

10

10

33.n!

sn 1 1d! 5n ? 1

sn 1 1dn ? 15

1n 1 1 35.

5 sn 1 1dn

sn 1 1d!sn 2 1d! 5

sn 1 1dnsn 2 1d!sn 2 1d! 37.

s2nd!s2n 2 1d! 5

s2nds2n 2 1d!s2n 2 1d! 5 2n

39. (c) 41. (b)


3 4

0

−5

10

5

49. n: 1 2 3 4 5

Terms: 3 6 9 12 15

Apparent pattern:Each term is three times n.

an 5 3n

51. n: 1 2 3 4 5

Terms: 1 4 7 10 13

Apparent pattern:Each term is three times n minus two.

an 5 3n 2 2

53. n: 1 2 3 4 5

Terms: 0 3 8 15 24

Apparent pattern:Each term is the square of n minus one.

an 5 n2 2 1

55. n: 1 2 3 4 5

Terms: 2 6 10

Apparent pattern:The terms have alternating signs withthose in the even position being negative. Each term isdouble n.

an 5 s21dn112n

2824

57. n: 1 2 3 4 5

Terms:

Apparent pattern:The numerator is 1 more than n and thedenominator is 2 more than n.

an 5n 1 1n 1 2

67

56

45

34

23

59. n: 1 2 3 4

Terms:

Apparent pattern: The numerator is 1 and each denomina-tor is two to the nth power. The terms have alternatingsigns with those in the even position being negative.

an 5s21dn11

2n

21

1618

214

12

61. n: 1 2 3 4

Terms: 1

Apparent pattern: The numerator is 1 and the denomina-tor is two to the power.

an 51

2n21

n 2 1

18

14

12

2x x>Y5 x2c d dn nTRACE TRACEY5

Y5 TRACEn4x2 d


63. n: 1 2 3 4 5

Terms:

Apparent pattern:The sum of one and .

an 5 1 11n

1n

1 115

1 114

1 113

1 112

1 111

65. n: 1 2 3 4 5

Terms: 1

Apparent pattern:The numerator is one and the denomi-nator is n factorial.

an 51n!

1120

124

16

12

67.

5 63

5 3 1 6 1 9 1 12 1 15 1 18

o6

k513k 5 3s1d 1 3s2d 1 3s3d 1 3s4d 1 3s5d 1 3s6d

69.

5 77

5 5 1 7 1 9 1 11 1 13 1 15 1 17

o6

i50s2i 1 5d 5 f2s0d 1 5g 1 f2s1d 1 5g 1 f2s2d 1 5g 1 f2s3d 1 5g 1 f2s4d 1 5g 1 f2s5d 1 5g 1 f2s6d 1 5g

71.

5 100

5 8 1 14 1 20 1 26 1 32

5 s18 2 10d 1 s24 2 10d 1 s30 2 10d 1 s36 2 10d 1 s42 2 10d

o7

j53s6j 2 10d 5 s6 ? 3 2 10d 1 s6 ? 4 2 10d 1 s6 ? 5 2 10d 1 s6 ? 6 2 10d 1 s6 ? 7 2 10d

73.

530193600

536003600

2900

36001

4003600

2225

36001

1443600

5 1 214

119

21

161

125

o5

j51

s21d j11

j25

s21d111

12 1s21d211

22 1s21d311

32 1s21d411

42 1s21d511

52

75.

543760

< 7.283

5 2 132

143

154

165

542

164

186

1108

11210

o6

m52

2m2sm 2 1d 5

2s2d2s2 2 1d 1

2s3d2s3 2 1d 1

2s4d2s4 2 1d 1

2s5d2s5 2 1d 1

2s6d2s6 2 1d

77. o6

k51s28d 5 s28d 1 s28d 1 s28d 1 s28d 1 s28d 1 s28d 5 248

Section 10.1 Sequences and Series527

83. Keystrokes:

3 1 6 1

o6

n513n2 5 273

79.

5 1 219

589

5 1 1 1212

1122 1 12

13

1132 1 12

14

1142 1 12

15

1152 1 12

16

1162 1 12

17

1172 1 12

18

1182 2

19

317

21

7 1 14 1 318

21

8 1 14

o8

i5111

i2

1i 1 12 5 31

12

11 1 14 1 31

22

12 1 14 1 31

32

13 1 14 1 31

42

14 1 14 1 31

52

15 1 14 1 31

62

16 1 14 1

81.

5182243

5243 2 81 1 27 2 9 1 3 2 1

243

5 1 1 12132 1

19

1 121

272 11

811 12

12432

o5

n50 12

132

n

5 12132

0

1 12132

1

1 12132

2

1 12132

3

1 12132

4

1 12132

5

85. Keystrokes:

2 6 1

o6

j52 s j! 2 jd 5 852

87. Keystrokes:

6 0 4 1

o4

j50

6j!

5 16.25

89. Keystrokes:

0 6 1

o6

k51ln k 5 6.5793

91. o5

k51k 93. o

5

k512k 95. o

10

k51

12k

97. o20

k51

1k2

99. o9

k50

1s23dk 101. o

20

k51

4k 1 3

103. o11

k51

kk 1 1

105. o20

k51

2kk 1 3

ENTER

ENTER

x2MATH 5

MATH 5 MATH

LIST

LIST

LIST

LIST X,T,u

X,T,u X,T,u

X,T,u

X,T,u

X,T,u d

d

2

, , , , ,

PRB 4

MATHX,T,u PRB 4

OPS 5

OPS 5

MATH 5LIST LIST OPS 5

MATH 5LIST LIST OPS 5

ENTERX,T,u d

ENTERX,T,u d

LN

4

, , , ,

, , , ,

, , , ,


115.

a5 1 2a6 5 1088 1 2408 5 3488 < 3608

a6 5180s6 2 2d

65 1208

a5 5180s5 2 2d

55 1088 117.

d11 5180s11 2 6d

115 81.88

d10 5180s10 2 6d

105 728

d9 5180s9 2 6d

95 608

d8 5180s8 2 6d

85 458

d7 5180s7 2 6d

75 25.78

113. (a)

(b) A40 5 500s1 1 0.07d40 5 $7487.23

A8 5 500s1 1 0.07d8 5 $859.09

A7 5 500s1 1 0.07d7 5 $802.89

A6 5 500s1 1 0.07d6 5 $750.37

A5 5 500s1 1 0.07d5 5 $701.28

A4 5 500s1 1 0.07d4 5 $655.40

A3 5 500s1 1 0.07d3 5 $612.52

A2 5 500s1 1 0.07d2 5 $572.45

A1 5 500s1 1 0.07d1 5 $535.00 (c) Keystrokes(calculator in sequence and dot mode):

500 1 0.07

(d) Yes. Investment earning compound interest increases at an increasing rate.

00

40

8000

Y5 1x > TRACEnd

111.

55.67

5 0.8

x 50.5 1 0.8 1 1.1 1 0.8 1 0.7 1 0.7 1 1.0

7

107. o6

k50k! 109.

5185

5 3.6

x 53 1 7 1 2 1 1 1 5

5

119. An infinite sequence has an unlimited number of terms such as .an 5 3n

121. The nth term of a sequence is . When n is odd, the terms are negative.

an 5 s21dn n

123. True.

o4

k513k 5 30 5 3 o

4

k51k 5 3s10d

o4

k513k 5 3 o

4

k51k.

Section 10.2 Arithmetic Sequences

1.

5 2 2 5 3, 8 2 5 5 3, 11 2 8 5 3

d 5 3 3.

94 2 100 5 26, 88 2 94 5 26, 82 2 88 5 26

d 5 26

Section 10.2 Arithmetic Sequences529

21. The sequence is not arithmetic.

The difference is NOT the same.

43 2

23 5

23

23 2

13 5

13



!3 2 !2 5 !3 2 !2 < .31

!2 2 1 5 !2 2 1 < .41



ln 12 2 ln 8 5 ln 12 2 ln 8 < .41

ln 8 2 ln 4 5 ln 8 2 ln 4 < .69

27.

a5 5 3s5d 1 4 5 19

a4 5 3s4d 1 4 5 16

a3 5 3s3d 1 4 5 13

a2 5 3s2d 1 4 5 10

a1 5 3s1d 1 4 5 7

29.

a5 5 22s5d 1 8 5 22

a4 5 22s4d 1 8 5 0

a3 5 22s3d 1 8 5 2

a2 5 22s2d 1 8 5 4

a1 5 22s1d 1 8 5 6 31.

a5 552s5d 2 1 5

232

a4 552s4d 2 1 5 9

a3 552s3d 2 1 5

132

a2 552s2d 2 1 5 4

a1 552s1d 2 1 5

32 33.

a5 535s5d 1 1 5

205 5 4

a4 535s4d 1 1 5

175

a3 535s3d 1 1 5

145

a2 535s2d 1 1 5

115

a1 535s1d 1 1 5

85

35.

a5 5 214s5 2 1d 1 4 5 3

a4 5 214s4 2 1d 1 4 5

134

a3 5 214s3 2 1d 1 4 5

72

a2 5 214s2 2 1d 1 4 5

154

a1 5 214s1 2 1d 1 4 5 4 37.

an 512n 1

52

an 5 3 112n 2

12

an 5 3 1 sn 2 1d12

an 5 a1 1 sn 2 1dd 39.

an 5 225n 1 1025

an 5 1000 2 25n 1 25

an 5 1000 1 sn 2 1)s225d

an 5 a1 1 sn 2 1dd

9.94 2

72 5 2

54, 1 2

94 5 2

54, 2

14 2 1 5 2

54

d 5 254 11. The sequence is arithmetic.

4 2 2 5 2, 6 2 4 5 2, 8 2 6 5 2

d 5 2

13. arithmetic;

8 2 10 5 22, 6 2 8 5 22, 4 2 6 5 22, 2 2 4 5 22

d 5 22 15. The sequence is arithmetic.

16 2 32 5 216, 0 2 16 5 216, 216 2 0 5 216

d 5 216

17. The sequence is arithmetic.

4 2 3.2 5 0.8, 4.8 2 4 5 0.8, 5.6 2 4.8 5 0.8

d 5 0.8

19. The sequence is arithmetic.

72 2 2 5

32, 5 2

72 5

32, 13

2 2 5 532

d 532

5.

226 2 214 5 212, 238 2 226 5 212

22 2 10 5 212, 214 2 22 5 212,

d 5 212 7.53 2 1 5

23, 7

3 253 5

23, 3 2

73 5

23

d 523


41.

so

28 5 a1

20 5 a1 2 8

20 5 a1 1 s3 2 1ds24d

an 5 a1 1 sn 2 1dd

an 5 24n 1 32

an 5 28 2 4n 1 4

an 5 28 1 sn 2 1ds24d

43.

an 532n 1

32

an 5 3 132n 2

32

an 5 3 1 sn 2 1d32

an 5 a1 1 sn 2 1dd

45.

so

52 5

104 5 d

10 5 4d

15 5 5 1 4d

15 5 5 1 s5 2 1dd

an 5 a1 1 sn 2 1dd

an 552n 1

52

an 5 5 152n 2

52

an 5 5 1 sn 2 1d52

47.

an 5 4n 1 4

an 5 8 1 sn 2 1ds4d

8 5 a1

16 5 a1 1 s3 2 1d4

an 5 a1 1 sn 2 1dd

49.

an 5 210n 1 60

an 5 50 1 sn 2 1ds210d

210 5 d

220 5 2d

30 5 50 1 s3 2 1dd

an 5 a1 1 sn 2 1dd 51.

an 5 212

n 1 11

an 5212

212

n 112

an 5212

1 sn 2 1d12122

212

5 a1

10 5 a1212

10 5 a1 1 s2 2 1d12122

an 5 a1 1 sn 2 1dd

d 58 2 10

45 2

12

53.

an 5 20.05n 1 0.40

an 5 0.35 2 0.05n 1 0.05

an 5 0.35 1 sn 2 1ds20.05d

an 5 a1 1 sn 2 1dd

d 50.30 2 0.35

15 20.05

55.

a2 5 25 1 s5 2 1ds3d 5 37

a2 5 25 1 s4 2 1ds3d 5 34

a2 5 25 1 s3 2 1ds3d 5 31

a2 5 25 1 s2 2 1ds3d 5 28

a1 5 25 and d 5 3

an 5 a1 1 sn 2 1dd 57.

a5 5 a411 5 a4 2 3 5 0 2 3 5 23

a4 5 a311 5 a3 2 3 5 3 2 3 5 0

a3 5 a211 5 a2 2 3 5 6 2 3 5 3

a2 5 a111 5 a1 2 3 5 9 2 3 5 6

a1 5 9

an 5 a1 1 sn 2 1dd

59.

a5 5 a411 5 a4 1 6 5 8 1 6 5 14

a4 5 a311 5 a3 1 6 5 2 1 6 5 8

a3 5 a211 5 a2 1 6 5 24 1 6 5 2

a2 5 a111 5 a1 1 6 5 210 1 6 5 24

a1 5 210

an 5 a1 1 sn 2 1dd 61.

a5 5 100 1 s5 2 1ds220d 5 20

a4 5 100 1 s4 2 1ds220d 5 40

a3 5 100 1 s3 2 1ds220d 5 60

a2 5 100 1 s2 2 1ds220d 5 80

a1 5 100 and d 5 220

an 5 a1 1 sn 2 1dd

63.

5 210

o20

k51k 5 2011 1 20

2 2 65.

5 1425

o50

k51sk 1 3d 5 5014 1 53

2 2 67.

5 255

o10

k51s5k 2 2d 5 1013 1 48

2 2

Section 10.2 Arithmetic Sequences531

69.

5 62,625

o500

n51

n2

5 500112 1 250

2 2 71.

5 35

o30

n5111

3n 2 42 5 3012

113 1 62 2 73.

5 522

o12

n51s7n 2 2d 5 1215 1 82

2 2

75.

5 1850

o25

n51s6n 2 4d 5 2512 1 146

2 2 77.

5 900

o8

n51s225 2 25nd 5 81200 1 25

2 2 79.

5 12,200

o50

n51s12n 2 62d 5 501250 1 538

2 2

81.

5 243

o12

n51s3.5n 2 2.5d 5 1211 1 39.5

2 2 83.

5 23

o10

n51s0.4n 1 0.1d 5 1010.5 1 4.1

2 2

85. (b) 87. (e) 89. (c)


2 21

00

10

25


.6 1.5

00

10

10


2.5 8

0

−10

10

20

97. Keystrokes:

750 30 1 25 1

o25

j51s750 2 30jd 5 9000

99. Keystrokes:

300 8 3 1 60 1

o60

i51s300 2

83id 5 13,120

101. Keystrokes:

2.15 5.4 1 50 1

o50

n51s2.15n 1 5.4d 5 3011.25

n TRACEY5 1x2c n TRACEY5 1

n TRACEY5 2

ENTERMATH 5LIST LIST X,T,u X,T,u d2OPS 5

ENTERMATH 5LIST LIST X,T,u X,T,u d2OPS 5

ENTERMATH 5LIST LIST X,T,u X,T,u dOPS 5

4

1

, , , ,

, , , ,

, , , ,


103. o75

n515 7511 1 75

2 2 5 2850 105. o50

n512n 5 5012 1 100

2 2 5 2550

107.

Total salary5 6136,000 1 46,0002 2 5 $246,000

36,000, 38,000, 40,000, 42,000, 44,000, 46,000 109. Sequence

Charge $25.43 to make at least $15,000

15,000

5905 25.43

Total costTotal seats

5 Cost per ticket

an 5 19 1 n

5 590 seatsan 5 20 1 sn 2 1d1

o20

n51s19 1 nd 5 20120 1 39

2 2an 5 a1 1 sn 2 1dd

d 5 1n 5 205 20, 21, 22, . . .

111. Sequence

5 632 bales

o8

n51s97 2 4nd 5 8193 1 65

2 25 93, 89, 85, 81, . . . 113. Sequence

Total chimes 5 78 1 36 5 114 chimes

3 chimes each hour 3 12 hours 5 36 chimes

an 5 n

an 5 1 1 n 2 1

5 78 chimesan 5 1 1 sn 2 1ds1d

o12

n51n 5 1211 1 12

2 2an 5 a1 1 sn 2 1dd

5 1, 2, 3, 4, . . .

115. Sequence

5 1024 feet

o8

n51s32n 2 16d 5 8116 1 240

2 2an 5 240

an 5 32n 2 16an 5 16 1 224

5 16 1 32n 2 32an 5 16 1 s8 2 1d32

an 5 16 1 sn 2 1d32an 5 a1 1 sn 2 1dd

d 5 32n 5 85 16, 48, 80, . . . 117. (a)

(b) The sums of positive odd integers yield perfect squares.

(c)

5 n12n2 2 5 n2

on

k51f1 1 sk 2 1d2g 5 n11 1 s2n 2 1d

2 21 1 3 1 5 1 7 1 9 1 11 1 13 5 49

1 1 3 1 5 1 7 1 9 1 11 5 36

1 1 3 1 5 1 7 1 9 5 25

1 1 3 1 5 1 7 5 16

1 1 3 1 5 5 9

1 1 3 5 4

119.

9 5 a1

12 5 a1 1 s2 2 1d3

d 5 15 2 12 5 3

an 5 a1 1 sn 2 1dd 121. A recursion formula gives the relationship between the terms and .anan11

123. Sequence

5 15,150

o200

n5100n 5 1011100 1 200

2 25 100, 101, 102, . . . , 200 (Note:

if n begins at 1.

To start at 100, use n.)

an 5 n 1 99

an 5 100 1 sn 2 1d1

an 5 a1 1 sn 2 1dd


119.

5 70.875 square inches

Total area 5 o6

n513611

22n21

5 361s12d6 2 112 2 1 2

an 5 361122

n21

r 5a2

a15

1836

512

a2 5 s3!2d2 5 18

a1 5 62 5 36 121.

5 666.21 feet

5 100 1 566.21

Total distance 5 100 1 o10

i512s100ds0.75dn

< 566.21

5 150s3.774745941d

o10

i512s100ds0.75dn 5 2s100ds0.75d30.7510 2 1

0.75 2 1 4

123. (a)

an 5 2n

Sequence 5 2, 4, 8, 16, . . . (b)

5 1.4757 3 1020

5 21266 2 12 2 1 2

Total ancestors 5 o66

i512n (c) It is likely that you have had no

common ancestors in the last 2000years.

125. The general formula for the nth term of a geometric sequence is .an 5 a1r

n21

127. An example of a geometric sequence whose terms

alternate in sign is an 5 s223dn21

.

129. An increasing annuity is an investment plan where equal deposits are made in an account at equal time intervals.


1.

a5 5 3211

42521

51

8

a4 5 3211

42421

51

2

a3 5 3211

42321

5 2

a2 5 3211

42221

5 8

a1 5 3211

42121

5 32 2.

a5 5s23d5 ? 5

5 1 45 2135

a4 5s23d4 ? 4

4 1 45

812

a3 5s23d3 ? 3

3 1 45 2

817

a2 5s23d2 ? 2

2 1 45 3

a1 5s23d1 ? 1

1 1 45 2

35

3. o4

k5110k 5 4110 1 40

2 2 5 100 4. o10

i514 5 1014 1 4

2 2 5 40

117. (a)

(b)

< 69.4%

5 .694069887

P 5 s0.999d365

P 5 s0.999dn (c) Keystrokes(calculator in sequence and dot mode):

.999

00

750

1.0

700 days

TRACEY5 n>


6. o8

n51812

122 5 8s24d 5 232

7. o20

k51

23k

8. o25

k51

s21dk21

k39. d 5

12 10. d 5 26

11. r 562 5 3 12. r 5

12 13.

23 5 d

29 5 3d

11 5 20 1 s4 2 1dd

an 5 a1 1 sn 2 1dd

an 5 23n 1 23

an 5 20 2 3n 1 3

an 5 20 1 sn 2 1ds23d

14.

an 5 3212142

n21

an 5 a1rn21 15.

5 4075

o50

n51s3n 1 5d 5 5018 1 155

2 2 16.

5 9030

o300

n51 n5

5 300115 1 60

2 2

17.

< 25.947

5 912.960982.333 2

5 912566561 2 1

213

2

o8

i51912

32i21

5 91s23d8

2 123 2 1 2

19.

5 3s3d 5 9

o`

i50312

32i

5 31 11 2 2

32

18.

< 18,392.796

5 50012.2071.06 2

o20

j51500s1.06d j21 5 50011.0620 2 1

1.06 2 1 2

20.

545 1

432 5

1615

5451 1

1 2 142

o`

i50 451

142

i

21. Geometric sequence with

< 20.026

a12 5 625s2.4d1221

an 5 625s2.4dn21

an 5 a1rn21

a1 5 625 and r 5 2.4. 22.

bn 5 1012122

n21

⇒ lower graph

an 5 101122

n21

⇒ upper graph

23.

a10 5 5.58

an 5 25.75 1 s10 2 1ds22.25d

an 5 25.75 1 sn 2 1ds22.25d

arithmetic with a1 5 25.75, d 5 22.25

Sequence 5 25.75, 23.5, 21.25, 19, . . . 24. bn 5 ln an is arithmetic.

5.

5 87

5 30 1 20 1 15 1 12 1 10

o5

j51

60j 1 1

5602

1603

1604

1605

1606

Section 10.4 The Binomial Theorem 539

Section 10.4 The Binomial Theorem

1. 6C4 5 6C2 56 ? 5

2 ? 15 15 3. 10C5 5

10 ? 9 ? 8 ? 7 ? 65 ? 4 ? 3 ? 2 ? 1

5 252 5. 20C20 5 1

7. 18C18 5 1 9. 50C48 5 50C2 550 ? 492 ? 1

5 1225 11. 25C4 525 ? 24 ? 23 ? 22

4 ? 3 ? 2 ? 15 12,650

13. Keystrokes:

30 6 30C6 5 593,775

15. Keystrokes:

12 7 12C7 5 792

17. Keystrokes:

52 5 52C5 5 2,598,960

19. Keystrokes:

200 195 200C195 5 2,535,650,040

21. Keystrokes:

25 12 25C12 5 5,200,300

23.

entry 2

Row 6: 1 6 15 20 15 6 1

6C2 5 15

25.

entry 3

Row 7: 1 7 21 35 35 21 7 1

7C3 5 35 27.

entry 4

Row 8: 1 8 28 56 70 56 28 8 1

8C4 5 70

29.

5 a3 1 6a2 1 12a 1 8

sa 1 2d3 5 s1da3 1 s3da2s2d 1 s3das22d 1 1s23d

31. sx 1 yd8 5 1x8 1 8x7y 1 28x6y2 1 56x5y3 1 70x4y4 1 56x3y5 1 28x2y6 1 8xy7 1 1y8

33.

5 32x5 2 80x4 1 80x3 2 40x2 1 10x 2 1

s2x 2 1d5 5 1s2xd5 1 5s2xd4s21d 1 10s2xd3s21d2 1 10s2xd2s21d3 1 5s2xds21d4 1 s21d5

35.

5 64y6 1 192y5z 1 240y4z2 1 160y3z3 1 60y2z4 1 12yz5 1 z6

s2y 1 zd6 5 s1ds2yd6 1 6s2yd5z 1 15s2yd4z2 1 20s2yd3z3 1 15s2yd2z4 1 6s2ydz5 1 1z6

37.

5 x8 1 8x6 1 24x4 1 32x2 1 16

sx2 1 2d4 5 1sx2d41 4sx2d3s2d 1 6sx2d2s2d2 1 4sx2ds2d3 1 1s2d4

39.

5 x6 1 18x5 1 135x4 1 540x3 1 1215x2 1 1458x 1 729

sx 1 3d6 5 1x6 1 6x5s3d 1 15x4s3d2 1 20x3s3d3 1 15x2s3d4 1 6xs3d5 1 1s3d6

41.

5 x6 2 24x5 1 240x4 2 1280x3 1 3840x2 2 6144x 1 4096

sx 2 4d6 5 s1dx6 2 s6dx5s4d 1 s15dx4s42d 2 s20dx3s43d 1 s15dx2s44d 2 s6dxs45d 1 s1d46

43. sx 1 yd4 5 1x4 1 4x3y 1 6x2y2 1 4xy3 1 1y4 45.

5 u3 2 6u2v 1 12uv 2 8v3

su 2 2vd3 5 1u3 2 3u2s2vd 1 3us2vd2 2 1s2vd3

MATH ENTERPRB 3

MATH ENTERPRB 3

MATH ENTERPRB 3 MATH ENTERPRB 3

MATH ENTERPRB 3


47.

5 81a4 1 216a3b 1 216a2b2 1 96ab3 1 16b4

s3a 1 2bd4 5 1s3ad4 1 4s3ad3s2bd 1 6s3ad2s2bd2 1 4s3ads2bd3 1 1s2bd4

49.

5 32x10 2 80x8y 1 80x6y2 2 40x4y3 1 10x2y4 2 y5

s2x2 2 yd5 5 1s2x2d5 1 5s2x2d4s2yd 1 10s2x2d3s2yd2 1 10s2x2d2s2yd3 1 5s2x2ds2yd4 1 1s2yd5

51.

10C3 510 ? 9 ? 83 ? 2 ? 1

5 120

10C3 x713

n 5 10, n 2 r 5 7, r 5 3, x 5 x, y 5 1

nCr xn2ryr 53.

215C11 5 215C4 5 215 ? 14 ? 13 ? 12

4 ? 3 ? 2 ? 15 21365

15C11x4s2yd11 5 215C11x4y11

n 5 15, n 2 r 5 4, r 5 11, x 5 x, y 5 s2ydnCr xn2ryr

55.

s2d312C9 5 8s220d 5 1760

12C9 5 12C3 512 ? 11 ? 10

3 ? 2 ? 15 220

12C9s2xd3y9

n 5 12, n 2 r 5 3, r 5 9, x 5 2x, y 5 y

nCr xn2ryr 57.

s23d24C2 5 9s6d 5 54

4C2 54 ? 3

s2 ? 1d 5 6

4C2sx2)2s23d2

n 5 4, n 2 r 5 2, r 5 2, x 5 x2, y 5 s23dnCr xn2ryr

59.

8C4 58 ? 7 ? 6 ? 54 ? 3 ? 2 ? 1

5 70

8C4s!x d4s1d

n 5 8, n 2 r 5 4, r 5 4, x 5 !x, y 5 1

nCr xn2ryr

61.

< 1.172

< 1 1 0.16 1 0.0112 1 0.000448

5 s1d8 1 8s1d7s0.02d 1 28s1d6s0.02d2 1 56s1d5s0.02d3 1 . . .

s1.02d8 5 s1 1 0.02d8

63.

< 510,568.785

< 531,441 2 21,257.64 1 389.7234 2 4.33026 1 0.03247695 2 0.0001732104

5 1s3d12 2 12s3d11s0.01d 1 66s3d10s0.01d2 2 220s3d9s0.01d3 1 495s3d8s0.01d4 2 792s3d7s0.01d5 1 . . .

s2.99d12 5 s3 2 0.01d12

65.

5 132 1

532 1

1032 1

1032 1

532 1

132

s12 1

12d5

5 1s12d5

1 5s12d4s1

2d 1 10s12d3s1

2d21 10s1

2d2s12d3

1 5s12ds1

2d41 1s1

2d5

67.

5 1256 1

12256 1

54256 1

108256 1

81256

s14 1

34d4

5 1s14d4

1 4s14d3s3

4d 1 6s14d2s3

4d21 4s1

4ds34d3

1 1s34d4

Section 10.5 Counting Principles 541

69. The difference between consecutive entries increases by 1.2, 3, 4, 5

71. There are terms in the expansion of sx 1 ydn.n 1 1

73. The signs in the expansion of are all positive.The signs in the expansion of alternate.sx 2 ydn

sx 1 ydn 75. nCr 5 nCn2r

Section 10.5 Counting Principles

1. 5 waysH0, 2, 4, 6, 8J

3. First number Second number

1 9

2 8

3 7

4 6

5 5

6 4

7 3

8 2

9 1

9 ways


1 9

2 8

3 7

4 6

6 4

7 3

8 2

9 1

8 ways

7. 10 waysH1, 3, 5, 7, 9, 11, 13, 15, 17, 19J 9. 8 waysH2, 3, 5, 7, 11, 13, 17, 19J

11. 6 waysH3, 6, 9, 12, 15, 18J


1 7

2 6

3 5

4 4

5 3

6 2

7 1

7 ways


1 7

2 6

3 5

5 3

6 2

7 1

6 ways

17. 3 ? 2 5 6 ways 19. letter number

26 10 5 260 labels?

label 5

21. digit digit digit digit letter letter

10 10 10 10 26 26 5 6,760,000 plates?????

plate 5


23. (a) 9 ? 10 ? 10 5 900 numbers (b) 10 ? 9 ? 8 5 720 numbers (c) 4 ? 10 ? 10 5 400 numbers

25. 3 ? 3 ? 2 ? 1 5 18 ways 27. 3 ? 2 ? 1 ? 5 ? 4 ? 3 ? 2 ? 1 5 720 ways

29. A, B, C, D; A, B, D, C; A, C, B, D; A, C, D, B; A, D, B, C; A, D, C, B;

B, A, C, D; B, A, D, C; B, C, A, D; B, C, D, A; B, D, A, C; B, D, C, A;

C, A, B, D; C, A, D, B; C, B, A, D; C, B, D, A; C, D, A, B; C, D, B, A;

D, A, B, C; D, A, C, B; D, B, A, C; D, B, C, A; D, C, A, B; D, C, B, A

31. AB BA

AC CA

AD DA

BC CB

BD DB

CD DC

33. 6! 5 6 ? 5 ? 4 ? 3 ? 2 ? 1 5 720 ways

35. 40 ? 40 ? 40 5 64,000 ways 37. 8! 5 40,320 ways 39. 10P4 5 10 ? 9 ? 8 ? 7 5 5040

41.

{A, B}, {A, C}, {A, D}, {A, E}, {A, F}, {B, C},{B, D}, {B, E}, {B, F}, {C, D}, {C, E}, {C, F},{D, E}, {D, F}, {E, F}

6C2 56!

4! 2!5

6 ? 52 ? 1

5 15 subsets 43. 20C3 520!

17! 3!5

20 ? 19 ? 183 ? 2 ? 1

5 1140 ways

45.

5 126 ways

9C4 59!

5! 4!5

9 ? 8 ? 7 ? 64 ? 3 ? 2 ? 1

47.

5 220 ways

12C9 512!

3! 9!5

12 ? 11 ? 103 ? 2 ? 1

49.

5 3003 ways

15C5 515!

5! 10!

51. (a)

5 15 ways

6C4 56!

2! 4!5

6 ? 52 ? 1

(b)

5 6 ways

4C2 ? 2C2 5 6 ? 1

2C2 52!

0! 2!5 1

4C2 54!

2! 2!5

4 ? 32 ? 1

5 6

53. (a) 8C4 58!

4! 4!5

8 ? 7 ? 6 ? 54 ? 3 ? 2 ? 1

5 70 (b) 2C1 ? 2C1 ? 2C1 ? 2C1 5 2 ? 2 ? 2 ? 2 5 16

55. 7C2 57!

5! 2!5

7 ? 62 ? 1

5 21 57. Diagonals of Hexagon5 6C4 2 6C1 5 9

59. Diagonals of Decagon5 10C8 2 10C1 5 35 61. The Fundamental Counting Principle: Let be two events that can occur in ways and ways,respectively. The number of ways the two events canoccur is m1 ? m2.

m2m1

E1 and E2

Section 10.6 Probability 543

Section 10.6 Probability

1. {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}

number of outcomes5 26

3. {AB, AC, AD, AE, BC, BD, BE, CD, CE, DE}

number of outcomes5 10

5. {ABC, ACB, BAC, BCA, CAB, CBA} 7. {WWW, WWL, WLW, WLL, LWW, LWL, LLW, LLL}

9. 1 2 0.35 5 0.65 11. PsEd 5 1 2 p 5 1 2 0.82 5 0.18 13. PsEd 5nsEdnsSd 5

38

15. PsEd 5nsEdnsSd 5

78


2652

512


1252

53

13


16


56

25.

(F is event that person does havetype B.)

PsEd 5 1 2nsFdnsSd 5 1 2

110

59

10


24.3100

5 0.243 29. PsEd 5nsEdnsSd 5

60.9100

5 0.609

31. (a) PsEd 5nsEdnsSd 5

15

(b) PsEd 5nsEdnsSd 5

13

(c) PsEd 5nsEdnsSd 5 1

33. (a) Pscandidate A or candidate Bd 5 0.5 1 0.3 5 0.8 (b) PsCandidate 3d 5 1 2 0.5 2 0.3 5 0.2


70325

51465

37. (a)


139,440,000196,950,000

546486565

PsEd 5nsEdnsSd 5

57,510,000196,950,000

519176565

39.

(E is the probability that they do not meet.)

1 2 PsEd 5 1 2 0.5625 5 0.4375

PsEd 5nsEdnsSd 5

452

602 520253600

5 0.5625

63. Permutation: The ordering of five students for a picture.

Combination: The selection of three students from a group of five students for a class project.


41. (a)

(b) Because the probabilities are the same, it is equally likely that a newborn will be a boy or a girl.

Probability of a boy 524 5

12

Probability of a girl 524 5

12


11 ? 4 ? 3 ? 2 ? 1

51

2445. PsEd 5

nsEdnsSd 5

110 ? 10 ? 10 ? 10 ? 10

51

100,000


1

10C85

145


1

10C25

110

8! 2!

51

10 ? 92 ? 1

51

45

51. PsEd 5nsEdnsSd 5 4C4

10C45

110!

6! 4!

51

10 ? 9 ? 8 ? 74 ? 3 ? 2 ? 1

51

210


52C55

13!8! 5!52!

47! 5!

5

13 ? 12 ? 11 ? 10 ? 95 ? 4 ? 3 ? 2 ? 1

52 ? 51 ? 50 ? 49 ? 485 ? 4 ? 3 ? 2 ? 1

513 ? 11 ? 9

52 ? 51 ? 5 ? 49 ? 45

11 ? 34 ? 17 ? 5 ? 49 ? 4

533

66,640

55. (d) 8 ? 5 ? 3 5 120 (e) The drawing will be done withoutreplacement since each personreceives only one gift.

(f ) (a)


1105

PsEd 5nsEdnsSd 5

1150

57. The probability that the event does not occur is 1 234 5

14. 59. Over an extended period, it will rain 40% of the time

under the given weather conditions.

XX

X X

X

Y

XX

XY XY

Mal

e

Female


1.

a5 5 3s5d 1 5 5 20

a4 5 3s4d 1 5 5 17

a3 5 3s3d 1 5 5 14

a2 5 3s2d 1 5 5 11

a1 5 3s1d 1 5 5 8 3.

a5 5125 1

12

51

321

1632

51732

a4 5124 1

12

51

161

816

59

16

a3 5123 1

12

518

148

558

a2 5122 1

12

534

a1 5121 1

12

5 1

5. an 5 2n 2 1 7. an 5n

sn 1 1d2


9. (a) 11. (b) 13. (d)

15. o4

k517 5 7 1 7 1 7 1 7 5 28

17. o4

n5111

n2

1n 1 12 5

12

116

11

121

120

530 1 10 1 5 1 3

605

4860

545

19. o4

n51s5n 2 3d 21. o

6

n51

13n

23. d 5 22.5

25.

a5 5 132 2 5s5d 5 107

a4 5 132 2 5s4d 5 112

a3 5 132 2 5s3d 5 117

a2 5 132 2 5s2d 5 122

a1 5 132 2 5s1d 5 127 27.

a5 534s5d 1

12 5

174

a4 534s4d 1

12 5

72

a3 534s3d 1

12 5

114

a2 534s2d 1

12 5 2

a1 534s1d 1

12 5

54 29.

a5 5 14 1 3 5 17

a4 5 11 1 3 5 14

a3 5 8 1 3 5 11

a2 5 5 1 3 5 8

a1 5 5

31.

a5 5145

2 252 5

1402 5 70

a4 5150

2 252 5

1452

a3 5155

2 252 5

1502 5 75

a2 5 80 252 5

1602 2

52 5

1552

a1 5 80 33.

an 5 4n 1 6

6 5 c

10 5 4s1d 1 c

an 5 dn 1 c 35.

an 5 250n 1 1050

1050 5 c

1000 5 250s1d 1 c

an 5 dn 1 c

37. o12

k51s7k 2 5d 5 1212 1 79

2 2 5 486 39. o100

j51

j4

5 100114 1 25

2 2 5 1262.5

41. Keystrokes:

1.25 4 1 60 1

o60

i21s125i 1 4d 5 2527.5

43. r 532 45.

a5 5 10s3d521 5 810

a4 5 10s3d421 5 270

a3 5 10s3d321 5 90

a2 5 10s3d221 5 30

a1 5 10s3d121 5 10

an 5 10s3dn21

an 5 a1rn21

LIST MATH 5 LIST OPS 5 X,T,u X,T,u ENTER1 d, , , ,


47.

a5 5 100s212d521 5 6.25

a4 5 100s212d421 5 212.5

a3 5 100s212d321 5 25

a2 5 100s212d221 5 250

a1 5 100s212d121 5 100

an 5 100s212dn21

an 5 a1rn21 49.

a5 5 2s24d 5 48

a4 5 2s12d 5 24

a3 5 2s6d 5 12

a2 5 2s3d 5 6

a1 5 3 51.

an 5 1s223dn21

an 5 a1rn21

53.

an 5 24s2dn21

an 5 a1rn21 55.

an 5 1212122

n21

an 5 a1rn21 57. o

12

n512n 5 21212 2 1

2 2 1 2 5 8190

59. o8

k51512

342

k

5 2154 1s23

4d82 1

234 2 1 2 < 21.928 61. o

8

i51s1.25di21 5 111.258 2 1

1.25 2 1 2 < 19.842

63. o120

n51500s1.01dn 5 50511.01120 2 1

1.01 2 1 2 < 116,169.54 65. o`

i5117

82i21

51

1 2 78

5118

5 8

67. o`

k51412

32k21

54

1 2 23

5413

5 12

69. Keystrokes:

50 1.2 1 1 50 1

o50

k5150s1.2dk21 < 2.275 3 106

71. 8C3 58!

3! 5!5

8 ? 7 ? 6 ? 5!3 ? 2 ? 5!

5 5673. 12C0 5 1

75. Keystrokes:

40 4 40C4 5 91,390

77. Keystrokes:

25 6 25C6 5 177,100

X,T,u X,T,ud dLIST LIST

ENTER ENTER

ENTER

MATH PRB 3 MATH PRB 3

MATH 5 OPS 5 2>x x d

79.

5 x10 1 10x9 1 45x8 1 120x7 1 210x6 1 252x5 1 210x4 1 120x3 1 45x2 1 10x 1 1

1 10xs1d9 1 1s1d10

sx 1 1d10 5 1x10 1 10x9s1d 1 45x8s1d2 1 120x7s1d3 1 210x6s1d4 1 252x5s1d5 1 210x4s1d6 1 120x3s1d7 1 45x2s1d8

81.

5 81x4 2 216x3y 1 216x2y2 2 96xy3 1 16y4

s3x 2 2yd4 5 1s3xd4 1 4s3xd3s22yd 1 6s3xd2s22yd2 1 4s3xds22yd3 1 s22yd4

83.

5 u18 1 9u16v3 1 36u14v6 1 84u12v9 1 126u10v12 1 126u8v15 1 84u6v18 1 36u4v21 1 9u2v24 1 v27

1 9su2dsv3d81 sv3d9

su2 1 v3d9 5 1su2d9 1 9su2d8sv3d 1 36su2d7sv3d2 1 84su2d6sv3d3 1 126su2d5sv3d4 1 126su2d4sv3d5 1 84su2d3sv3d6 1 36su2d2sv3d7

, , , ,



1.

a 5 5 s223d521

51681

a4 5 s223d421 5 2

827

a3 5 s223d321

549

a2 5 s223d221

5 223

a1 5 s223d121

5 1

an 5 s223dn21

2. o4

j50s3j 1 1d 5 1 1 4 1 7 1 10 1 13 5 35

3. o5

n51s3 2 4nd 5 5121 1 217

2 2 5 245 4. o12

n51

23n 1 1

5.

a5 5 4s5d 1 8 5 28

a4 5 4s4d 1 8 5 24

a3 5 4s3d 1 8 5 20

a2 5 4s2d 1 8 5 16

a1 5 4s1d 1 8 5 12

5 12 1 4n 2 4 5 4n 1 8

an 5 12 1 sn 2 1d4an 5 a1 1 sn 2 1dd 6.

an 5 2100n 1 5100

an 5 5000 2 100n 1 100

an 5 5000 1 sn 2 1ds2100dan 5 a1 1 sn 2 1dd 7. o

50

n515 5013 1 150

2 2 5 3825

8. r 5 232

9.

an 5 41122

n21

an 5 a1rn21 10. o

8

n512s2nd 5 4128 2 1

2 2 1 2 5 1020

85.

10C5 5 252 ? s23d5 5 261,236

n 5 10, n 2 r 5 5, r 5 5, x 5 3, y 5 s23dnCr xn2ryr 87.

7C3s2d3 5 35 ? 8 5 280

n 5 7, r 5 3, n 2 r 5 4, x 5 x, y 5 s2ydnCr xn2ryr

89. o50

n514n 5 5014 1 200

2 2 5 5100 91. o12

n51s3n 1 19d 5 12122 1 55

2 2 5 462

93. (a)

(b)

< 154,328

a50 5 85,000s1.012d50

an 5 85,000s1.012dn 95. 2 ? 2 ? 2 5 8 97.

5 3003

15C5 515 ? 14 ? 13 ? 12 ? 11

5 ? 4 ? 3 ? 2 ? 1


26

513


14 ? 3 ? 2 ? 1

51

24


84C8< 0.346


14.

balance 5 f50s1.0066667d1g31.0066667300 2 11.0066667 2 1 4 5 $47,868.64

a1 5 50s1.0066667d1

a300 5 11 10.0812 2

12s25d550s1.0066667d300

A 5 P11 1rn2

nt

16.

5 x5 2 10x4 1 40x3 2 80x2 1 80x 2 32

sx 2 2d5 5 1sx5d 2 5x4s2d 1 10x3s2d2 2 10x2s2d3 1 5xs2d4 2 1s2d5

15. 20C3 520 ? 19 ? 18

3 ? 2 ? 15 1140

17. The coefficient of in expansion of is 56,since 8C3 5 56.

sx 1 yd8x3y5 18. digit digit digit

26 10 10 10 5 26,000 plates???5

plates 5 letter

19. 25C4 525!

4! 21!5

25 ? 24 ? 23 ? 224 ? 3 ? 2 ? 1

5 12,650 20. 1 2 0.75 5 0.25


652

53

2622. PsEd 5

nsEdnsSd 5

1

4C25

14!

2! 2!

51

4 ? 32 ? 1

516

11. o10

n51311

22n

5321

110

2 2 112 2 1 2 5

30691024

12. o`

i5111

22i

512

1 2 12

51212

5 1 13. o`

i51412

32i21

54

1 2 23

5413

5 12


CHAPTER 10 Sequences, Series, and Probability

SECTION 10.1 Sequences and Series

1. Multiplicative Property of Equality

(Multiply both sides of the equation by the reciprocal ofthe coefficient of the variable.)

x 5 25

217 ? 27x 5 35 ? 2

17

27x 5 35

2. Additive Property of Equality

(Add the opposite of 63 on both sides of the equation.)

7x 5 228

7x 1 63 2 63 5 35 2 63

7x 1 63 5 35

3. is a solution of the equation ifthe equation is true when is substituted for t.23

t 2 1 4t 1 3 5 0t 5 23 4.

The first step in solving this equation is to multiply bothsides of the equation by the lowest common denominatorxsx 1 1d.

3x

21

x 1 15 10

5. sx 1 10d22 51

sx 1 10d2 6.

5 18sx 2 3d3

18sx 2 3d5

sx 2 3d2 5 18sx 2 3d522 7. sa2d24 5 a28 51a8

8. s8x3d1y3 5 81y3x3?1y3 5 2x 9.

5 8x!2x

!128x3 5 !64 ? 2 ? x2 ? x 10.

55s!x 1 2d

x 2 4

55s!x 1 2ds!x d2

2 22

5

!x 2 25

5!x 2 2

?!x 1 2!x 1 2

11. (a) Graph opens down because

(b)

(c)

s2, 4d

y 5 222 1 4s2d 5 24 1 8 5 4

x 524

2s21d 5 2

x 52b2a

s0, 0d s4, 0d

x 5 0 x 5 4

2x 5 0 x 2 4 5 0

0 5 2xsx 2 4d

0 5 2x2 1 4x

a < 0. 12. Keystrokes: 4

−1

−2

5

5

Y5 x2c X,T,u X,T,ux2 1 GRAPH


11. (a)

Equation:

(b) Keystrokes: 2 3

(c) Let and find the intersection of the twographs. x < 10.8

y2 5 200

250

120

0

f sxd 5 s2x 2 3d ? x

AreaVerbal Model: 5 ?Length Width

Y5 x 2

x

GRAPH

12. (a)

Equation:

(b) Keystrokes: .5 4

(c) Let and find the intersection of the twographs. x < 22.1

y2 5 200

250

250

0

f sxd 512

? x ? sx 2 4d

Y5 x 2

x

GRAPH

HeightAreaVerbal Model: 512

? ?Base

SECTION 10.2 Arithmetic Sequences

1. An algebraic expression is a collection of letters (called variables) and real numbers (called constants) combined with the oper-ations of addition, subtraction, multiplication, and division.

2. The terms of an algebraic expression are those parts separated by addition or subtraction.

3. A trinomial of degree 3 is any polynomial with 3 terms and whose highest exponent on a variable is 3, such as 2x3 2 3x2 1 2.

4. A monomial of degree 4 is any polynomial with only one term and the highest exponent on the variable is 4, such as 7x 4.

5.

Domain: s2`, `df sxd 5 x3 2 2x 6.

Domain: s2`, `dgsxd 5 3!x

X,T,u X,T,u

X,T,u X,T,u

7.

Domain:

Test intervals: Negative:

Positive:

Negative:

Positive: f24, 4gf4, `d

f24, 4gs2`, 24g

s4 2 xds4 1 xd ≥ 0

16 2 x2 ≥ 0

f24, 4ghsxd 5 !16 2 x2 8.

Domain:

6 Þ x x Þ 26

6 2 x Þ 0 6 1 x Þ 0

s6 2 xds6 1 xd Þ 0

36 2 x2 Þ 0

s2`, 26d < s26, 6d < s6, `d

Asxd 53

36 2 x2

9.

Domain:

t > 2

t 2 2 > 0

s2, `dgstd 5 lnst 2 2d 10.

Domain: s2`, `df ssd 5 630e20.2s


11. Formula:

A < $30,798.61

A 5 10,00011 10.075365 2

365s15d

A 5 P11 1rn2

nt

12. Formula:

A 5 $5395.40

A 5 400011 10.0612 2

12s5d

A 5 P11 1rn2

nt

SECTION 10.3 Geometric Sequences and Series

4.

intercept: Let and solve for

s24, 0d

24 5 x

0 5 x 1 4

0 5 s!x 1 4 d2

0 5 !x 1 4

0 5 2!x 1 4

x.y 5 0x-

f sxd 5 2!x 1 4

intercept: Let and solve for

s0, 4d

y 5 4

5 2 ? 2

5 2!4

y 5 2!0 1 4

y.x 5 0y-

1. The point is 6 units to the left ofthe axis and 4 units above the

axis.x-y-

2. y

108642

2 4 6 8 10

−4−6−8

−10

−2−4−6−8−10

(10, 5)−

(10, 5)

( 10, 5)− −

( 10, 5)−

3. The graph of is the set ofordered pairs where isin the domain of f.

xsx, f sxdd,f

5.

x > 53

3x > 5

3x 2 5 > 0 6.

y < 6

y < 9 ? 23

32 y < 9

32 y 1 11 < 20

7.

35 < x < 60

70 < 2x < 120

100 < 2x 1 30 < 150 8.

212 < x < 30

30 > x > 212

230 < 2x < 12

25 < 2x6

< 2


9.

Critical numbers:

Positive:

Negative:

Positive:

Solution: s2`, 1d < 152

, `215

2, `2

11, 522

s2`, 1d

x 5 1, 52

s2x 2 5dsx 2 1d > 0

2x2 2 7x 1 5 > 0 10.

Critical numbers:

Test intervals:

Negative:

Positive:

Solution: s21, 0d < 152

, `2

s21, 0d < 152

, `2

s2`, 21d < 10, 522

x 5 21, 0, 52

s2x 2 5dsx 1 1d

x> 0

2x2 2 5 2 3x

x> 0

2x 25x

2 3 > 0

2x 25x

> 3

11. Formula:

Equation:

a < 13.4 inches

a 5 !180.5

a2 5 180.5

2a2 5 361

a2 1 a2 5 192

a2 1 b2 5 c2 12. Formula:

Equation:

47.2 < c

2225 5 c2

625 1 1600 5 c2

252 1 402 5 c2

a2 1 b2 5 c2

SECTION 10.4 The Binomial Theorem

1. It is not possible to find the determinant of this matrixbecause it is not square.

2. The three elementary row operations are:

(1) interchange two rows.

(2) multiply a row by a nonzero constant.

(3) add a multiple of one row to another row.

3. This matrix is in row-echelon form. 4.

5 2200

5 250 2 150

det A 5 |106

2525| 5 10s25d 2 6s25d

5.

5 32

5 18 1 14

det A 5 | 322

76| 5 3s6d 2 s22ds7d

6.

(using second row)

5 260

5 215 2 45

5 5s23d 2 3s15d

5 0 1 5|36 11| 2 3|36 22

1| det A 5 |306 22

51

131|


7.

(using third row)

5 2126

5 280 2 46

5 5s216d 1 2s223d

5 5|32 522| 2 s22d|43 5

22| 1 0

det A 5 |435 32

22

522

0|

8.

(using second row)

Area 5 212s2116d 5 58

5 2116

5 2120 1 4

5 210s12d 2 1s24d

5 210| 824

11| 1 0 2 1|25

38

24| |x1

x2

x3

y1

y2

y3

111| 5 |25

103

80

24

111|

sx3, y3d 5 s3, 24dsx2, y2d 5 s10, 0d,sx1, y1d 5 s25, 8d,

9.

y 5 20.07x2 1 1.3x 1 2

c 5 | 0100400

01020

280|

220005

0 1 0 1 2|100400

1020|

220005

2s22000d22000

5 2

b 5 | 0100400

280

111|

220005

0 2 2|100400

11| 1 1|100

40080|

220005

s22ds2300d 2 320022000

52260022000

5 1.3

a 5 |280 01020

111|

220005

2|1020

11| 2 0 1 1|80 10

20|22000

52s210d 1 160

220005

14022000

5 20.07

D 5 | 0100400

01020

111| 5 1|100

4001020| 5 s1ds22000d 5 22000

30

100400

01020

111

:::

2804

0 5 as20d2 1 bs20d 1 c ⇒ 0 5 400a 1 20b 1 c

8 5 as10d2 1 bs10d 1 c ⇒ 8 5 100a 1 10b 1 c

2 5 as0d2 1 bs0d 1 c ⇒ 2 1 c


10.

(using first row)

(divide by )

y 5 4x 2 9

4x 2 y 2 9 5 0 or

22 28x 1 2y 1 18 5 0

xs28d 2 ys22d 1 1s18d 5 0

x|217

11| 2 y|24 1

1| 1 1|24 217| 5 0

|x24

y21

7

111| 5 0

sx1, y1d 5 s2, 21d, sx2, y2d 5 s4, 7d

SECTION 10.5 Counting Principles

1. is exponential since it has a constant baseand a variable exponent.gsxd 5 2s5xd 2. using the law of exponents

am ? an 5 am1n

e2 ? e2x 25 e21s2x 2d 5 e22x 2

3. in exponential form is 43 5 64.log4 64 5 3 4. in exponential form is 324 5181.log3

181 5 24

5. in exponential form is e0 5 1.ln 1 5 0 6. in exponential form is e1.6094 . . . < 5.ln 5 < 1.6094 . . .

7.

x < 3.56

x 5log 50log 3

log3 3x 5 log3 50

3x 5 50 8.

x < 4.16

x 5 2 ln 8

x2

5 ln 8

ln e xy2 5 ln 8

e xy2 5 8

9.

x 5 69

x 2 5 5 26

log2sx 2 5d 5 6 10.

x < 22,023.47

x 5 e10 2 3

x 1 3 5 e10

lnsx 1 3d 5 10

11. (a) Keystrokes: 22,000 0.8 (b) Let and find the intersection.

20,000

15,000

10,000

5,000

1 2 3 4 5 6t

V

t 5 1.7y2 5 15,000Y5 x x > GRAPHX,T, u


12.

10 5 C

10 5 Ce ks0d

y 5 Ce kt

20.00012097 < k

ln

12

57305 k

ln 12

5 5730k

ln 12

5 ln e5730k

12

5 e5730k

5 5 10e ks5730d

y 5 10e kt

y < 6.96 grams

y 5 10e20.00012097s3000d

y 5 10e20.00012097t

SECTION 10.6 Probability

1. loga 1 5 0 2. loga a 5 1 3. loga ax 5 x

4. logasuvd 5 loga u 1 loga v 5. loga uv

5 loga u 2 loga v 6. loga un 5 n loga u

7. log2sx2yd 5 log2 x2 1 log2 y 5 2 log2 x 1 log2 y 8. log2 !x2 1 1 5 log2sx2 1 1d1y2 5

12 log2sx2 1 1d

9. ln 7

x 2 35 ln 7 2 lnsx 2 3d 10. ln1u 1 2

u 2 222

5 2 ln1u 1 2u 2 22 5 2flnsu 1 2d 2 lnsu 2 2dg

11. (a) Keystrokes:

10,000 1 4

3

(b) Let and find the intersection of the twographs. years.

(c) Trace along the graph. The maximum level of annualsales is 10,000.

x < 4y2 5 5000

10,000

100

0

y1

12.


5 0.0565 5 5.65%

A 5 $1056.54

A 5 1000e0.055s1d

A 5 Pert

Y5 4 x 1 e x x X,T, u 4

x x

GRAPH

x2c

A P P E N D I C E S

Appendix A Introduction to Graphing Utilities . . . . . . . . . . . . .550

Appendix B Further Concepts in Geometry

Appendices B.1 Exploring Congruence and Similarity . . . . . . . .552

Appendices B.2 Angles . . . . . . . . . . . . . . . . . . . . . . . .554

Appendix C Further Concepts in Statistics . . . . . . . . . . . . . . .558

Appendix D Introduction to Logic

Appendices D.1 Statements and Truth Tables . . . . . . . . . . . .561

Appendices D.2 Implications, Quantifiers, and Venn Diagrams . . .563

Appendices D.3 Logical Arguments . . . . . . . . . . . . . . . . .566

550 Appendix A Introduction to Graphing Utilities

Appendix A Introduction to Graphing Utilities

1. Keystrokes:

3

−10

−10

10

10

3. Keystrokes:

3 4 6

−10

−10

10

10

5. Keystrokes:

1 2

−10

−10

10

10

7. Keystrokes:

4 2

−10

−10

10

10

9. Keystrokes:

3

−10

−10

10

10

11. Keystrokes:

4

−10

−10

10

10

13. Keystrokes:

27 100

75

250

0 5

15. Keystrokes:

0.001 0.5

−100

100

−500 200

17. Keystrokes:

15 12

Xmin 5 4Xmax 5 20Xscl 5 1Ymin 5 14Ymax 5 22Yscl 5 1

x2c x 4 2

x 4 X,T,u

X,T,u

x 2 GRAPHx

5Y

GRAPHx

5YX,T,u GRAPH5Y

x X,T,u GRAPHx5Y

X,T,u X,T,ux 2

x 2

x 2

GRAPH5Y 2

2

1

X,T,u GRAPH5Y 1 X,T,u X,T,u GRAPH5Y 1

1

ABS

x X,T,u GRAPHx

5Y 2ABS

x X,T,u GRAPHx5Y 2ABS

Solutions to Odd-Numbered Exercises

Appendix A Introduction to Graphing Utilities 551

37. Keystrokes:

0.07 1.06 88.97

0.02 0.23 10.70y2

y1

19. Keystrokes:

15 12

Xmin 5 -20Xmax 5 -4Xscl 5 1Ymin 5 -16Ymax 5 -8Yscl 5 1

21. Keystrokes:

2 1

2 1


−5

5

−5 5

y2

y1

23. Keystrokes:

2 1 2

1y2

y1

25. Keystrokes:

9

Trace to x-intercepts:

Trace to y-intercept: s0, 9d

s23, 0d and s3, 0d

27. Keystrokes:

6 2



s28, 0d and s4, 0d

29. Keystrokes:

2 5

Trace to x-intercept:


s52, 0d

31. Keystrokes:

1.5 1



s22, 0d and s12, 0d

33. Keystrokes:

4

Triangle

−15

−10

15

10

y2

y1

35. Keystrokes:

8

8

Square

−15

−10

15

10

y2

y1

1 1x X,T,u GRAPHx

5Y ABS

ABS

x2c

1

1

1

x X,T,u GRAPHx

5Y ABS

1

1

1

1

xX,T,u X,T,u

x X,T,u X,T,u GRAPH

x

x

5Y

x2c

x2c

ENTER

x

GRAPH

x

5Y ENTER4

GRAPH

5Y ENTER

x 2X,T,u GRAPH5Y 2

x 2

1

1

1x 2

X,T,u X,T,u GRAPH5Y 2

2

X,T,u

X,T,u ABS

ABS

x2c GRAPH

5Y ENTER

X,T,u

X,T,u

GRAPH5Y 2

2

2


−4

6

−5 5

GRAPH

5Y ENTER

−5

115

0 7

First class

Periodicals

X,T,u

X,T,u

X,T,u

X,T,ux 2

552 Appendix B Further Concepts in Geometry

Appendix B Further Concepts in Geometry

Appendix B.1 Exploring Congruence and Similarity

1. Answers will vary. 3. 5. Two figures are similar if theyhave the same shape. Figures (a)and (b) are similar.

7. 9. The grid contains 7 congruenttriangles with 2-unit sides.

11. No. All the triangles in the grid areequilteral triangles, and all of thesetriangles have the same shape.Therefore all the triangles in thegrid are similar to each other.

13. False. For example, the twosquares shown below are similar,but they are not congruent.

15. True. Any two squares have thesame shape, so any two squaresare similar.

17. The ray from P through Q ismatched with notation (d).

19. The length of the segment between P and Q is matchedwith notation (b).

21. and are names for the same angle.

and are names for the same angle.

and are names for the same angle.

and are adjacent angles./YXW/ZXY

/WXY/YXW

/YXZ/ZXY

/WXZ/ZXW

23. (b) m/WXY < 308 25. (d) Equiangular

A triangle with angle measures of and is anequiangular triangle because all the angles are the samesize.

608608, 608,

27. (f) Right

A triangle with angle measures of and is aright triangle because it contains a right angle.

908308, 608,

29. (c) Obtuse

A triangle with angle measures of and is anobtuse triangle because it contains an obtuse angle.

158208, 1458,

31. The three points of congruent sidesare

andLP > OQ.LM > NO, MP > NQ,

33. If thenm/C 5 m/V.

DABC > DTUV, 35. If then LN > TV.DLMN > DTUV,

Appendix B.1 Exploring Congruence and Similarity553

37.

(Not possible)

(Not possible)

Scalene Isosceles Equilateral

Acute Yes Yes Yes

Obtuse Yes Yes No

Right Yes Yes No

39.

Therefore, all three sides of the triangle are of length 12.Yes, the triangle is equilateral.

AB 5 4x 5 4s3d 5 12.

AC 5 2s3d 5 6 1 6 5 12

BC 5 12

x 5 3

2x 5 6

2x 1 6 5 12

AC 5 BC 41.

Therefore, all three sides of the triangles are of length 6.Yes, the triangle is equilateral.

AB 5 x 1 3 5 3 1 3 5 6.

BC 5 4x 2 6 5 4s3d 2 6 5 12 2 6 5 6

AC 5 2x 5 2s3d 5 6

x 5 3

22x 5 26

2x 5 4x 2 6

AC 5 BC

43.

The scale drawing would be 12.5 feet by 12.5 feet. No,such a large drawing does not seem reasonable.

196

? 1200 5 12.5

1y8 inch

1 foot5

1ys8 ? 12d foot1 foot

51y96 foot

1 foot5

196

45. If V is located at either or thenDPQR > DTUV.

s3, 5d,s3, 1d

47. Form a tetrahedron, a three-dimensional figure with fourcongruent triangular faces.


Appendix B.2 Angles

1. Answers will vary.

12


21


1

2

7. and are adjacent,congruent, supplementary angles.

/COD/AOC 9. and are adjacent,supplementary angles.

/COE/BOC 11. and are adjacent,complementary angles.

/COF/BOC

13. False.

m/3 5 308 ⇒ m/1 5 1508 ⇒ m/4 5 308.

15. False.

For example,and

thus /2 À /3.m/3 5 308 ⇒ m/1 5 1508 ⇒ m/2 5 1508,

17. True. 19. because vertical angles are congruent.x 5 1108

21. because two angles that form a linear pair are supplementary.

x 5 558

x 51108

2

2x 5 1108

2x 1 708 5 1808

2x 2 58 1 758 5 1808

s2x 2 58d 1 758 5 1808

23. because vertical angles are congruent.

x 5 358

x 52708

22

22x 5 2708

22x 1 208 5 2508

3x 2 5x 1 208 5 2508

3x 1 208 5 5x 2 508

25. Answer (c)

because two angles that form a linear pair are supplementary.

because the sum of the measures of the interior angles of a triangle is

m/Q 5 708

1108 1 m/Q 5 1808

708 1 408 1 m/Q 5 1808

1808.m/PSQ 1 m/P 1 m/Q 5 1808

m/PSQ 5 708

m/PSQ 1 1108 5 1808

m/PSQ 1 m/QST 5 1808

Appendix B.2 Angles 555

27. and are alternate interior angles because they liebetween l and m and on opposite sides of t.

and are alternate interior angles because they liebetween l and m and on opposite sides of t.

/6/4

/5/3 29. and are corresponding interior angles because theylie between l and m and on the same side of t.

and are corresponding interior angles because theylie between l and m and on the same side of t.

/5/4

/6/3


by the Alternate Exterior Angle Theorem

m/2 5 1108

m/2 5 m/1

m/1 5 1108

m/1 1 708 5 1808


by the Alternate Interior Angles Theorem

Alternate approach for angle 1:

by Consecutive Interior Angles Theorem

m/1 5 708

m/1 1 1108 5 1808

m/1 5 708

m/1 5 m/2

m/2 5 708

m/2 1 1108 5 1808

35. because corresponding angles are congruent.

because corresponding angles are congruent.

208 5 b

608

35 b

608 5 3b

608 2 b 5 2b

a 5 308

a 52908

23

23a 5 908

a 5 4a 2 908

37. and are the interior angles of the triangle.

(These are the original three angles of the triangle.)

/7/2, /5,

39. Step 1: because two angles that form a linear pair are supplementary.

Step 2: because two angles that form a linear pair are supplementary.

Step 3: because vertical angles are congruent.



—CONTINUED—

m/8 5 1558

m/7 5 258

m/7 1 1558 5 1808

m/2 5 1108

m/3 5 708

m/3 1 1108 5 1808

m/1 5 708

m/1 1 1108 5 1808


41. because corresponding angles of congruenttriangles are congruent.m/B 5 358 43.

m/F 5 408

1408 1 m/F 5 1808

1058 1 358 1 m/F 5 1808

m/D 1 m/E 1 m/F 5 1808

45. True.

The sum of the measures of three angles of a triangle is

The sum of the measures of the two angles is

Therefore, the measure of the third angle is

Thus, the triangle has three angles, so the triangle is equiangular.608

1808 2 1208 5 608.

1208.608

1808.

47. Step 1: because the sum of the measures of three angles of a triangle is


Step 3: because the sum of the measurers of the three angles of a triangle is

Step 4: because the three angles combine to form a straight angle.



Step 7: because the sum of the measures of the three angles of a triangle is

Step 8: becaused two angles that form a linear pair are supplementary.

Step 9: because the sum of the measures of the three angles of a triangle is

m/9 5 358

m/9 1 1458 5 1808

1808.m/9 1 908 1 558 5 1808

m/5 5 908

m/5 1 908 5 1808

m/4 5 358

m/4 1 1458 5 1808

1808.m/4 1 558 1 908 5 1808

m/8 5 1258

m/8 1 558 5 1808

m/7 5 558

m/6 5 558

m/6 1 1258 5 1808

m/6 1 708 1 558 5 1808

m/3 5 508

m/3 1 1308 5 1808

1808.m/3 1 608 1 708 5 1808

m/2 5 608

m/1 5 308

m/1 1 1508 5 1808

1808.m/1 1 608 1 908 5 1808

39. —CONTINUED—

Step 6: because the sum of the measures of the interior angles of a triangle is


Step 8: because vertical angles are congruent.m/6 5 1358

m/4 5 1358

m/4 1 458 5 1808

m/4 1 m/5 5 1808

m/5 5 458

m/5 1 1358 5 1808

m/5 1 258 1 1108 5 1808

1808.m/5 1 m/7 1 m/2 5 1808

558 Appendix C Further Concepts in Statistics

Appendix C Further Concepts in Statistics

1. Organize scores by ordering the numbers.Let the leaves represent the units digits.Let the stems represent the tens digits.

Stems Leaves7 0 5 5 5 7 7 8 8 88 1 1 1 1 2 3 4 5 5 5 5 7 8 9 9 99 0 2 8

10 0 0

3. Organize scores by ordering the numbers.Let the leaves represent the units digits.Let the stems represent the tens digits.

Stems Leaves5 2 5 96 2 3 6 6 77 0 1 2 3 4 7 8 8 98 0 1 3 4 5 7 99 0 0 2 3 3 3 5 6 8 9

10 0 0

5. Frequency Distribution

Interval Tally

|||| ||f43, 50d||||f36, 43d||||f29, 36d|||| |f22, 29d|||| |||f15, 22d

Histogram

x

y

15 22 29 36 43 50

1

2

3

4

5

6

7

8

7.

Num

ber

of tr

avel

ers

(in m

illio

ns)

Place of origin

10

8

6

4

2

Canada

Mexico

Europe

FarEast

Other

12

14

169. 1985: 165 million tons

1995: 210 million tons

15.

Year

Enr

ollm

ent

2000

y

x

1900

1800

1700

1600

1994 1996 1998 2000

17.

Price of a share of stock (in dollars)

Stock Prices

Com

pany

Sears, Roebuck

Wal-Mart Stores

JC Penney

KMart Corp.

The Gap, Inc.

10 20 30 40 50 60 70 80 90 100

11. Total waste and recycled waste increased every year. 13. Total waste equals the sum of the other three quantities.

Appendix C Further Concepts in Statistics 559

19.

1991 1993 1995

2300

2200

2100

2000

1900

1800

Year

Cam

cord

er s

ales

(in m

illio

ns o

f dol

lars

)

21. x and y have a positive correlation because as x increases y also increases.

23. Yes, it appears that players withmore hits tend to have more runsbatted in.

25. Negative correlation, because as the age of the car increases the value of the car decreases.

27. Positive correlation, because as the age of a tree increases the height also increases.

29.

Altitude, A (in thousands of ft)

Pre

ssur

e,P(in

lb/ft

2 )

P

A10 20 30 40 50

2

4

6

8

10

12

14

16

31. The air pressure at 42,500 feet is approximately 2.45 pounds per square inch.

33.

Use graphing utility by entering data in 2 lists with

graph.

875 642Units of fertilizer

Yie

ld (

in b

ushe

ls)

3

75

60

65

70

1

55x

y 35. Use graphing utility to findregression line

(a)

(b) 71.8

875 642Units of fertilizer

Yie

ld (

in b

ushe

ls)

3

75

60

65

70

1

55x

y

y 5 57.49 1 1.43x

fLin Regsax 1 bdg.

37.


graph.

35302515Altitude (in 1000s of feet)

Spe

ed o

f sou

nd(in

ft/s

ec)

5 10 20

1050

1100

1150

950

1000

h

v

39. Use graphing utility to findregression line

(a)

(b) 1006.6

35302515Altitude (in 1000s of feet)

Spe

ed o

f sou

nd(in

ft/s

ec)

5 10 20

1050

1100

1150

950

1000

h

v

v 5 1117.3 2 4.1h

fLin Regsax 1 bdg.

41.


graph. Find regres-sion line with .

65432

20

10

15

1

5

x

y

y 5 22.179x 1 22.964 43.


graph. Find regres-sion line with .

4030

5060

844

10

8

y

x

y 5 2.378x 1 23.546

STAT PLOT

STAT PLOT

STAT PLOT

STAT CALC4

STAT CALC4

STAT PLOTSTAT CALC4

STAT CALC4

560 Appendix C Further Concepts in Statistics

45. (a) 12.78

(b)

t

y

Enr

ollm

ent (

in m

illio

ns)

Year (0 1990)↔1 2 3 4 5

11.0

11.4

11.8

12.6

12.2

(0, 11.21)

(1, 11.37)

(2, 11.54)

(3, 11.95)

(4, 12.33)

(5, 12.52)

y 5 11.1 1 0.28t

47.

↑Middle score

Mode: 7 occurs twice 5 7

Median: 14 12 9 8 7 7 5 5 8

Mean: 5 1 12 1 7 1 14 1 8 1 9 1 7

75 8.86 49.

↑Middle score

Mode: 7 occurs twice 5 7

Median: 24 12 9 8 7 7 5 5 8

Mean: 5 1 12 1 7 1 24 1 8 1 9 1 7

75 10.29

51. (a)

(b) Median: 87.8283.1881.7674.9867.9265.3565.3559.8457.9957.0052.5052.00

5 $67.14 s67.92 1 59.84 1 52 1 52.50 1 57.99 1 65.35 1 81.76 1 74.98 1 87.82 1 83.18 1 65.35 1 57d 4 12

average of 2 middle bills5 $65.35

53. (a)

(b) Median: List all the data. Find the average of the two 100th scores

(c) Mode: 3 occurs 54 times

5 3

Mean: 0 ? 1 1 1 ? 24 1 2 ? 45 1 3 ? 54 1 4 ? 50 1 5 ? 19 1 6 ? 7

200< 3.07

55. Answers vary. One possibility: H4, 4, 10J

57. (a)

(b) Median: list the scores from highest to lowest100, 99, 98, 92, 91, 90, 89, 88, 87, 84, 80, 79, 78, 77, 72, 64, 59, 42, 42, 20

Find the average of the two tenth scores: 84 and 80

(c) Mode: 42 occurs twiceThe median gives the most representative description since more of the test scores are in the 80’s.

5 82

42 1 20 1 89 1 42 1 100 1 98 1 84 1 78 1 91d 4 20 5 76.55s99 1 64 1 80 1 77 1 59 1 72 1 87 1 79 1 92 1 88 1 90 1

(c)

Use graphing utility by entering data in 2 listswith graph. Find regression linewith .

r < 0.987

STAT PLOTSTAT CALC4

Appendix D.1 Statements and Truth Tables561

Appendix D Introduction to Logic

Appendix D.1 Statements and Truth Tables

1. Statement, because only one truth value can be assigned.

3. Open statement, because a specific figure is needed to assign a truth value.

5. Open statement, because a value ofx is needed to assign a truth value.

7. Open statement, because values of x and y are needed to assign a truth value.

9. Nonstatement, because no truth value can be assigned.

11. Open statement, because a specific place is needed to assign a truth value.

13. (a)

(b)

20 Þ 0 False

4 1 10 1 6 5 0

s22d2 2 5s22d 1 6 5?

0

0 5 0 True

4 2 10 1 6 5 0

22 2 5s2d 1 6 5?

0 15. (a)

(b)

0 ≤ 4 True

02 ≤?

4

4 ≤ 4 True

s22d2 ≤?

4 17. (a)

(b)

3 Þ 2 False

4 2 |1| 5?

2

4 Þ 2 False

4 2 |0| 5?

2

19. (a)

(b)

Undefined Þ 1 False

00

5?

1

1 5 1 True

2424

5?

1 21. (a) ~p: The sun is not shining.

(b) ~q: It is not hot.

(c) The sun is shining and it is hot.

(d) The sun is shining or it is hot.p ∨ q:

p ` q:

23. (a) ~p: Lions are not mammals.

(b) ~q: Lions are not carnivorous.

(c) Lions are mammals and lions are carnivores.

(d) Lions are mammals or lions are carnivorous.p ∨ q:

p ` q:

25. (a) The sun is not shining and it is hot.

(b) The sun is not shining or it is hot.

(c) The sun is shining and it is not hot.

(d) The sun is shining or it is not hot.p ∨ ~q:

p ` ~q:

~p ∨ q:

~p ` q:

27. (a) Lions are not mammals and lions are carnivorous.

(b) Lions are not mammals or lions are carnivorous.

(c) Lions are mammals and lions are not carnivorous.

(d) Lions are mammals or lions are not carnivorous.

p ∨ ~q:

p ` ~q:

~p ∨ q:

~p ` q: 29. It is four o’clock.

It is time to go home.

p ` ~q

q:

p :

31. It is four o’clock.

It is time to go home.

~p ∨ q

q:

p : 33. The dog has fleas.

The dog is scratching.

~p ∨ ~q

q:

p : 35. The dog has fleas.

The dog is scratching.

~p ` q

q:

p :

37. The bus is blue. 39. x is not equal to 4. 41. The earth is flat.

562 Appendix D Introduction to Logic

49. p q ~p ~q

T T F F F T

T F F T F T

F T T F T F

F F T T F T

p ∨ ~q~p ` q

not identicalnot logically equivalent

55.

(a)

(b) p ∨ ~q

5 p ` ~q

q 5 It is made of wood.

Let p 5 The house is red.

p q ~q

T T F F T

T F T T T

F T F F F

F F T T T

p ∨ ~qp ` ~q

not identicalnot logically equivalent

43. 45. 47.p q ~p ~q

T T F F F

T F F T T

F T T F T

F F T T T

~p ∨ ~qp q ~p

T T F F

T F F F

F T T T

F F T F

~p ` q p q ~q

T T F T

T F T T

F T F F

F F T T

p ∨ ~q

53. p q ~p ~q

T T F F F T F

T F F T T F T

F T T F F T F

F F T T F T F

~s~p ∨ qd~p ∨ qp ` ~q

identicallogically equivalent

57.

(a)

(b) ~p ` ~q

~sp ∨ qd

q 5 It is blue.

Let p 5 The house is white.

p q ~p ~q

T T T F F F F

T F T F F T F

F T T F T F F

F F F T T T T

~p ` ~q~sp ∨ qdp ∨ q


p ~p

T F F

T F F

F T F

F T F

~p ` p59.

not a tautology

51. p q ~p ~q

T T F F T F F

T F F T T F F

F T T F F T T

F F T T T F F

~p ` q~sp ∨ ~qdp ∨ ~q


Appendix D.2 Implications, Quantifiers, and Venn Diagrams563

63. p q ~p ~q

T T F F T F F

T F F T F T T

F T T F F T T

F F T T F T T

~p ∨ ~q~sp ` qdp ` q


Appendix D.2 Implications, Quantifiers, and Venn Diagrams

1. (a) If the engine is running, then the engine is wasting gasoline.

(b) If the engine is wasting gasoline, then the engine is running.

(c) If the engine is not wasting gasoline, then the engine is not running.

(d) If the engine is running, then the engine is not wasting gasoline.

p → ~q :

~q → ~p :

q → p :

p → q : 3. (a) If the integer is even, then it is divisible by 2.

(b) If it is divisible by 2, then the integer is even.

(c) If if it is not divisible by 2, then the integer is not even.

(d) If the integer is even, then it is not divisible by 2.

p → ~q :

~q → ~p :

q → p :

p → q :

5. The economy is expanding.

Interest rates are low.

q → p

q 5

Let p 5 7. The economy is expanding.


p → q

q 5

Let p 5

9. The economy is expanding.


p → q

q 5

Let p 5 11. Hypothesis Conclusion Implication

T T T

13. Hypothesis Conclusion Implication

F T T


T F F


F T T


T T T

21. Converse:If you can see the eclipse, then the sky is clear.

Inverse:If the sky is not clear, then you cannot see the eclipse.

Contrapositive:If you cannot see the eclipse, then the sky is not clear.

23. Converse:If the deficit increases, then taxes were raised.

Inverse:If taxes are not raised, then the deficit will not increase.

Contrapositive:If the deficit does not increase, then taxes were not raised.

61.

a tautology

p ~p

T F T T

T F T T

F T F T

F T F T

~s~pd ∨ ~p~s~pd


33. Negation: No students are in extracurricular activities. 35. Negation: Some contact sports are not dangerous.

37. Negation: Some children are allowed at the concert. 39. Negation: None of the $20 bills are counterfeit.

25. Converse:It is necessary to apply for the visa to have a birth certificate.

Inverse:It is not necessary to have a birthcertificate to not apply for the visa.

Contrapositive:It is not necessary to apply for thevisa to not have a birth certificate.

27. Negation:Paul is not a junior and not a senior.

29. Negation:If the temperature increases, then the metal rod will not expand.

31. Negation:We will go to the ocean and theweather forecast is not good.

p q ~q

T T F F T

T F T T F

F T F T F

F F T T F

~sp → ~qdp → ~q 43.41. p q

T T T F F

T F T F F

F T F T T

F F T F F

~sq → pd ` q~sq → pdq → p

p q ~p

T T T F F T

T F T F F T

F T T T T T

F F F T F T

fsp ∨ qd ` s~pdg → qsp ∨ qd ` s~pdsp ∨ qd45.

p q ~p ~q

T T F F F T F T

T F F T T T T F

F T T F T T T T

F F T T T F F T

sp ↔ ~qd → ~pp ↔ ~q~q → pp → ~q47.

p q ~p ~q

T T T F F T

T F T F T T

F T F T F F

F F T T T T

~p → ~qq → p49. p q ~q

T T T F F F

T F F T T T

F T T F F F

F F T F T F

p ` ~q~sp → qdp → q51.

identical identical

Appendix D.2 Implications, Quantifiers, and Venn Diagrams565

53. p q ~q ~p

T T T F T F T

T F F T T F T

F T T F T T T

F F T T T T T

p ∨ ~psp → qd ∨ ~qp → q

identical

p q ~p

T T F F F

T F F F F

F T T T T

F F T F T

p → s~p ` qd~p ` q55.

identical

57.

Statement is

(c) If a number is not divisible by 2, then it is not divisible by 6.

p → q ; ~q → ~p

~q → ~p

p → q

q 5 It is divisible by 2.

Let p 5 A number is divisible by 6. 59. (a) Some citizens over the age of 18 have the right to voteis not logically equivalent to above statement.

p q ~p ~q

T T T F F T

T F F F T F

F T T T F T

F F T T T T

~q → ~pp → q

identical

61.

A

B

B 5 college students

Let A 5 people who are happy 63.

A B



A B


Let A 5 people who are happy

67.

A B



A B


Let A 5 people who are happy 71. (a) Statement does not follow.

(b) Statement follows.

Greenthings

Toads


Appendix D.3 Logical Arguments

73. (a) Statement does not follow.

(b) Statement does not follow.Blue cars Old cars

p q ~p ~q

T T F F F F

T F F T T F

F T T F T T

F F T T T F

sp → ~qd ` qp → ~q1.

T

T

T

T

fsp → ~qd ` qg → ~p

3. p q ~p

T T F T F

T F F T F

F T T T T

F F T F F

sp ∨ qd ` ~pp ∨ q

T

T

T

T

fsp ∨ qd ` ~pg → q

p q ~p ~q

T T F F T T

T F F T T T

F T T F T F

F F T T F F

s~p → qd ` p~p → q5.

F

T

T

T

fs~p → qd ` pg → ~q

7. p q

T T T T T

T F T F T

F T T T F

F F F F T

fsp ∨ qd ` qg → psp ∨ qd ` qp ∨ q

9.

Premise #1:

Premise #2:p

Conclusion: q

p → q

q 5 businesses will leave the state

Let: p 5 taxes are increased p q

T T T T T

T F F F T

F T T F T

F F T F T

sp → qd ` p → qsp → qd ` pp → q

Argument is valid.

Appendix D.3 Logical Arguments 567

11.

Premise #1:

Premise #2:q

Conclusion: p

p → q

q 5 businesses will leave the state

Let: p 5 taxes are increased p q

T T T T T

T F F F T

F T T T F

F F T F T

sp → qd ` q → psp → qd ` qp → q

Argument is invalid.

13.

Argument is valid.

q 5 car was not stolen

Let: p 5 doors are locked

p q ~p ~q

T T T F F F T

T F F F T F T

F T T T F F T

F F T T T T T

sp → qd ` ~q → ~psp → qd ` ~qp → q

15.

Argument is valid.

Reliable cars

Fords

Lincolns

17.


PaperworkReduction

Act

FederalIncome

Tax forms

Schedule A

19.

Argument is valid.

q 5 He is at the handball court.

Let: p 5 Eric is at the store.

p q ~p

T T F T F T

T F F T F T

F T T T T T

F F T F F T

sp ∨ qd ` ~p → qsp ∨ qd ` ~pp ∨ q

Premise #1:

Premise #2: ~q

Conclusion: ~p

p → q

Premise #1:

Premise #2: ~p

Conclusion: q

p ∨ q


21.


q 5 It sparkles in the sunlight.

Let: p 5 It is a diamond.

p q

T T T F F T

T F F T F T

F T F T T F

F F F T F T

~sp ` qd ` q → p~sp ` qd ` q~sp ` qdp ` q

Premise #1:

Premise #2:q

Conclusion: p

~sp ` qd

23.

Premise #1:

Premise #2:

So conclusion must be ~p or 7 is not a prime numberwhich is (b).

~q

p → q

q 5 7 does not divide evenly into 21

Let: p 5 7 is a prime number 25.

Premise #1:

Premise #2:

So conclusion must be ~p or the economy does notimprove which is (c).

~q

p → q

q 5 Interest rates lowered

Let: p 5 Economy improves

27.

Premise #1:

Premise #2:

So conclusion must be q or acid rain will continue as anenvironmental problem which is (b).

~p

p ∨ q

an environmental problem q 5 Acid rain will continue as

Let: p 5 Smokestack emissions must be reduced 29.

Premise #1:

Premise #2:

Conclusion:

If Rodney doesn’t get a good job

He didn’t study

So by the Law of Contraposition the answer is (c).

~p :

~r :

p → r Law of Transitivity

q → r

p → q

r 5 He will get a good job

q 5 He will make good grades

Let: p 5 Rodney studies

31.

Argument is valid.

A

B

50

B 5 All numbers divisible by 10

Let A 5 All numbers divisible by 5 33.


A

B

C

C 5 College students

B 5 People under the age of 18

Let A 5 People eligible to vote

Appendix D.3 Logical Arguments 569

35. Let p represent the statement “Sue drives to work,” let q represent “Sue will stop at the grocery store,” and let r represent “Sue will buy milk.”

First write:

Premise #1:

Premise #2:

Premise #3:p

Reorder the premises:

Premise #3:p

Premise #1:

Premise #2:

Conclusion: r

Then we can conclude r. That is, “Sue will buy milk.”

q → r

p → q

q → r

p → q

37. Let p represent “This is a good product,” let q represent“We will buy it,” and let r represent “the product wasmade by XYZ Corporation.”

First write:

Premise #1:

Premise #2:

Premise #3: ~r

Note that and reorder the premises:

Premise #2:

Premise #3: ~r

(Conclusion from Premise #2, Premise #3: )

Premise #1:

Conclusion:

Then we can conclude . That is, “It is not a goodproduct.”

~p

~p

~q → ~p

~q

r ∨ ∼q

p → q ; q → ~p,

r ∨ ∼q

p → ~q

intermediate algebra, third edition

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