chapter 1 real numbers and variable...

Chapter 1 Real Numbers and

Variable Expressions

Prep Test 1. 127.16 2. 55.107 3. 4517 4. 11,396 5. 24 6. 24 7. 4 8. 3 7⋅ 9. 4 2

10 5=

Go Figure

Multiplying by numbers that end in zero will produce a zero at the end of the final answer. Since 10 and 20 are included in the list, they are two numbers that would produce a zero at the end of the number. Also, multiplying 5 and 6 produces 30, and multiplying 2 and 15 also produces 30. So there are 4 zeros at the end of the product of the first 20 natural numbers.

Section 1.1

....................................................... Objective A Exercises 1. 8 > –6 3. –12 < 1 5. 42 > 19 7. 0 > –31 9. 53 > –46 11. False 13. True 15. False 17. True 19. False 21. x < –8 is true for

–23 and –18. 23. a > –10 is true for

21 and 37. 25. –6 > n is true for –23.

....................................................... Objective B Exercises 27. –4 29. 9 31. 28 33. 14 35. –77 37. 0 39. 74 41. –82 43. –81 45. − >83 58 47. 43 52< − 49. − > −68 42 51. − < −45 61 53. − −( ) =

−( ) =−( ) = −

19 190 0

28 28

55. ( )( )( )

45 45 450 0 0

17 17 17

− − = − = −− = − =− = − = −

....................................................... Objective C Exercises 59. –3 + (–8) = –11 61. –8 + 3 = –5 63. − + −( ) = −3 80 83 65. − + −( ) = −23 23 46 67. 16 16 0+ −( ) = 69. 48 53 5+ −( ) = − 71. − + −( ) + = − + =17 3 29 20 29 9 73. − + −( ) + = − + =3 8 12 11 12 1 75. 16 8 16 8 8− = + −( ) = 77. 7 14 7 14 7− = + −( ) = − 79. − − = − + −( ) = −7 2 7 2 9 81. 7 2 7 2− −( ) = 9+ = 83. − − −( ) = − + = −6 3 6 3 3 85. 6 12 6 12 18− −( ) = + =

Copyright © Houghton Mifflin Company. All rights reserved.

2 Chapter 1: Real Numbers and Variable Expressions

87. 13 22 4 5 9 4 5 5 5 1+ −( ) + + −( ) = − + + −( ) = − + −( ) 0= − 89. − + −( ) + −( ) + = − + −( ) + = − += −

16 17 18 10 33 18 10 51 1041

91. 26 15 11 12 11 11 12 0 12

12+ −( ) + −( ) + −( ) = + −( ) + −( ) = + −( )

= − 93. − + −( ) + −( ) + = − + −( ) + = − + =14 15 11 40 29 11 40 40 40 0

95. − − − = − + −( ) + −( ) = − + −( ) = −4 3 2 4 3 2 7 2 9 97. 12 7 8 12 7 8 19 8 11− −( ) − = + + −( ) = + −( ) = 99. − − −( ) − = − + + −( ) = + −( ) = −19 19 18 19 19 18 0 18 18 101. − − −( ) − −( ) = − + + = − + =17 8 9 17 8 9 9 9 0 103. − − −( ) − − = − + + −( ) + −( )

= + −( ) + −( ) = + −( ) =30 65 29 4 30 65 29 4

35 29 4 6 4 2

105. − − − − = − + −( ) + −( ) + −( )= − + −( ) + −( ) = − + −( )= −

16 47 63 12 16 47 63 1263 63 12 126 12138

107. − − −( ) − − = − + + −( ) + −( )

= + −( ) + −( ) = + −( ) = −47 67 13 15 47 67 13 15

20 13 15 7 15 8

109. − − − −( ) − = − + −( ) + + −( )= − + + −( ) = + −( ) = −

19 17 36 12 19 17 36 1236 36 12 0 12 12

111. 21 14 43 12 21 14 43 12

35 43 12 8 12 20− −( ) − − = + + −( ) + −( )

= + −( ) + −( ) = − + −( ) = −

....................................................... Objective D Exercises 115. 42 117. –28 119. 60 121. –253 123. –238 125. –114 127. –2 129. 8 131. –7 133. –12 135. –6 137. –7 139. 11 141. –14 143. 15 145. –16 147. 0 149. –29 151. undefined 153. –11 155. undefined 157. 7 5 3 35 3 105( ) −( ) = −( ) = − 159. 9 7 4 63 4

252−( ) −( ) = − −( )

=

161. 16 3 5 48 5 240−( ) = −( ) = − 163. − −( ) = ( ) =4 3 8 12 8 96 165. − −( ) −( ) = −( )= −

3 8 9 24 9216

167. −( ) ( ) = − ( ) = −9 7 5 63 5 315 169. 7 2 5 6 14 5 6 70 6

420−( )( ) −( ) = − ( ) −( ) = − −( )

= 171. − −( ) −( ) −( ) = −( ) −( )

= − −( ) =9 4 8 10 36 8 10

288 10 2880

173. 7 9 11 4 63 11 4 693 4

2772( ) −( ) = −( ) = −( )

= − 175. −( ) −( ) = − −( ) = ( ) =14 9 11 0 126 11 0 1386 0 0

....................................................... Objective E Application Problems 177. Strategy To find the difference, subtract the elevation of Death Valley (–86) from the elevation of Mt. Aconcagua (6960). Solution 6 960 86 6 960 86 7046, ,− −( ) = + =

The difference in elevation is 7046 m. 179. Strategy To find the greatest difference, subtract the lowest elevation from the highest elevation for each continent. Solution Afr ica 5895 133 5895 133 6028 m− −( ) = + =

Asia 8848 400 8848 400 9248 m− −( ) = + = Europe 5634 28 5634 28 5662 m− −( ) = + = Am erica 6960 86 6960 86 7046 m− −( ) = + = The greatest difference in elevation occurs in Asia (9248 m).


Section 1.2 3

181. Strategy To find the difference, subtract the melting point of xenon − °( )112 C from its boiling point . − °( )107 C Solution − − −( ) = − + = °107 112 107 112 5 C

The difference in temperature is . 5°C 183. Strategy To find the score: • Multiply the number of correct answers by 7. • Multiply the number of incorrect answers by –3. • Multiply the number of blank answers by –1. • Add the results. Solution 17 7 119

8 3 242 1 2

119 24 2 93

( )( ) =( ) −( ) = −( ) −( ) = −

+ −( ) + −( ) =

The student's score was 93. 185. Strategy To find the difference, subtract the temperature at 40,000 ft from the temperature at 50,000 ft. Solution –70 – (–70) = 0

The difference is 0°. 187. Strategy To find the difference, subtract the 25 mph wind-chill factor ( )31 F− ° from the 10 mph wind-chill factor ( )22 F− ° . Solution ( )22 31 22 31 9− − − = − + =

The wind-chill factor difference is . 9 F°

....................................................... Applying the Concepts 189.

The product of two negative integers is a positive number. Because –4 is a negative number, it must be multiplied by a negative integer in order for the product to be positive. Therefore, if −4x equals a positive integer, then x must be a negative integer.

Section 1.2

....................................................... Objective A Exercises

1. 8 1 0000125

8201640400

..

−

−

−

18

0125= .

3. 9 2 0000 222

1820182018

2

.

.

−

−

−

29

0 2= .

5. 6 1 0000166

6403640364

..

−

−

−

16

0 16= .

7. 16 9 00000 5625

80100

96403280800

.

.

−

−

−

−

916

0 5625= .

9. 12 7 00000 5833

6010096403640364

.

.

−

−

−

−

712

0 583= .

11. 25 6 000 24

50100100

0

.

.

−

−

625

0 24= .



13. 40 9 0000 225

80100

80200200

0

.

.

−

−

−

940

0 225= .

15. 11 5 00000 4545

446055

504460555

.

.

−

−

−

−

511

0 45= .

....................................................... Objective B Exercises

19. 40% 40 1100

40100

25

40% 40 0 01 0 40

= FHIK = =

= ( ) =. .

21. 88% 88 1100

88100

2225

88% 88 0 01 0 88

= FHIK = =

= ( ) =. .

23. 160% 160 1100

160100

85

160% 160 0 01 1 6

= FHIK = =

= ( ) =. .

25. 87% 87 1100

87100

87% 87 0 01 0 87

= FHIK =

= ( ) =. .

27. 450% 450 1100

450100

92

450% 450 0 01 4 50

= FHIK = =

= ( ) =. .

29. 4 27

4 27

1100

307

1100

370

% = FHIK =FHIK =

31. 37 12

37 12

1100

752

1100

38

% = FHIK =FHIK

=

33. 14

14

1100

1400

% = FHIK = 35. 6 1

46 1

41

100254

1100

116

% = FHIK =FHIK =

37. 5 34

5 34

1100

234

1100

23400

% = FHIK =FHIK

=

39. 9 1% 9 1 0 01 0 091. . . .= ( ) = 41. 16 7% 16 7 0 01 0 167. . . .= ( ) =

43. 0 9% 0 9 0 01 0 009. . . .= ( ) = 45. 9 15% 9 15 0 01 0 0915. . . .= ( ) = 47. 18 23% 18 23 0 01 0 1823. . . .= ( ) = 49. 0 37 0 37 100% 37%. .= ( ) = 51. 0 02 0 02 100% 2%. .= ( ) = 53. 0 125 0 125 100% 12 5%. . .= ( ) = 55. 1 36 1 36 100% 136%. .= ( ) = 57. 0 004 0 004 100% 0 4%. . .= ( ) = 59.

83100

83100

100% 8300100

83%= ( ) = =%

61. 38

38

100% 3008

37 12

= ( ) = =% % 63. 49

49

100% 4009

44 49

= ( ) = =% % 65. 9

209

20100% 900

2045%= ( ) = =%

67. 2 12

2 12

100% 52

100% 5002

250%= ( ) = ( ) = =%

....................................................... Objective C Exercises

69. −

+ =−

+ =− +

=6

131726

1226

1726

12 1726

526

71. 58

34

58

68

58

68

5 68

118

− −FHIK = − −FH

IK = + =

+=


Section 1.2 5

73. 1112

56

1112

1012

1112

1012

11 1012

112

− = − = +−

=+ −( )

= 75. −

−−FHIK =

−−

−( )=−

+ =− +

=5

811

1215

2422

2415

242224

15 2224

724

77. 12

23

16

36

46

16

36

46

16

3 4 16

06

0− + = − + = +−

+ =+ −( ) +

= = 79. 12

38

14

48

38

28

48

38

28

4 38

38

− − −2F

HIK = − − −FH

IK = +

−+ =

+ −( ) +

=

81. 13

14

15

2060

1560

1260

2060

1560

1260

20 15 1260

760

− − = − − = +−

+−

=+ −( ) + −( )

= −

83. 5

1618

12

516

216

816

516

216

816

5 2 816

116

+ − = + − = + +−

=+ + −( )

= −

85. 3.60

2.541.06

−

2 54 3 6 1 06. . .− = −

87. 16 9206 925

23 845

.

.

.

+

− − = −16 92 6 925 23 845. . .

89. 17.69206.9027

10.7893−

6 9027 17 692 10 7893. . .− = −

91. − + − = − − = −18 39 4 9 23 7 13 49 23 7 37 19. . . . . . 93. − − −( ) − = − + − = − −

= −3 07 2 97 17 4 3 07 2 97 17 4 010 17 4

17 5. . . . . . .

..

.

95. − − − −( ) = − − + = − +

=3 09 4 6 27 3 3 09 4 6 27 3 7 69 27 3

19 61. . . . . . .

.

....................................................... Objective D Exercises

97. 12

34

12

34

1 32 4

38

−FHIK = − ⋅FH

IK = −

⋅⋅

= − 99. −FH IK −FH IK = FH IKFH IK =⋅⋅

38

415

38

415

3 48 15

=⋅ ⋅

⋅ ⋅ ⋅ ⋅=

3 2 22 2 2 3 5

110

1 1 1

1 1 1

101. − FH IK = − ⋅FH IK = − ⋅FH IK = −⋅⋅FH IK = −

⋅ ⋅⋅ ⋅

= −12

89

12

89

12

89

1 82 9

2 2 22 3 3

49

1

1

103. 58

712

1625

58

712

1625

5 7 168 12 25

5 7 2 2 2 22 2 2 2 2 3 5 5

730

−FH IK = − ⋅ ⋅FH IK = −⋅ ⋅⋅ ⋅

= −⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅= −

11 1 1 1

1 1 1 1 1

105. 12

34

58

12

34

58

1 3 52 4 8

3 52 2 2 2 2 2

1564

FHIK −FHIK −FHIK = ⋅ ⋅ =

⋅ ⋅⋅ ⋅

=⋅

⋅ ⋅ ⋅ ⋅ ⋅= 107.

56

34

56

34

56

43

5 46 3

5 2 22 3 3

109

÷ −FH IK = − ÷FH IK = − ⋅FH IK = −⋅⋅FH IK

= −⋅ ⋅⋅ ⋅FH

IK = −

1

1

109. − ÷ = − ÷FHIK = − ⋅FH

IK = −

⋅⋅FHIK

= −⋅ ⋅

⋅ ⋅ ⋅ ⋅FH

IK = −

78

421

78

421

78

214

7 218 4

7 3 72 2 2 2 2

14732

111. − ÷ −FH IK = ÷ = ⋅ =⋅⋅

=⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅

=

1564

340

1564

340

1564

403

15 4064 3

3 5 2 2 2 52 2 2 2 2 2 3

258

1

1

1

1

1

1

1

1

113. − ÷ −FH IK = ÷ = ⋅ =⋅⋅

=⋅ ⋅⋅ ⋅

=49

23

49

23

49

32

4 39 2

2 2 33 3 2

23

1 1

11

115. 3 471 20

6943474 167

.

.

.

×

1 2 3 47 4 164. . .× =



117. 1 892 30

567378

4 347

.

.

.

×

−( ) −( ) =1 89 2 3 4 347. . .

119. 1 063 80

848318

4 028

.

.

.

×

1 06 3 8 4 028. . .× −( ) = −

121. 1 2 0 5 3 7 0 6 3 72 22

. . . . ..

−( )( ) = − ( )= −

123. ( )( )

( )2.3 0.6 0.8

1.38 0.81.104

−= −= −

125. 3 5 9 07002 591

70207175

320315

503515

. ..

−

−

−

−

9 07 3 5 2 59. . .÷ −( ) ≈ −

127. 1 35 6 904005 114

6 75154135

190135

550540

10

. ..

.−

−

−

−

− ÷ ≈ −6 904 1 35 5 11. . .

129. 0 1719 354 2086002060 550

34381040

01040810314

9460

94608650

86508595

5500

550

. ..

−

−

−

−

−

−

−

− ÷ ≈ −354 2086 01719 2060 55. . .

....................................................... Objective E Exercises 131. 7 7 7 7 7 24014 = ⋅ ⋅ ⋅ = 133. − = − ⋅ ⋅( ) = −4 4 4 4 63 4 135. −( ) = −( ) −( ) −( )

= −2 2 2 2

8

3 137. −( ) = −( ) −( ) −( )

= −5 5 5 5

125

3

139. 34

34

34

34

3 3 34 4 4

2764

3FHIK = −FH

IK −FHIK −FHIK

= −⋅ ⋅⋅ ⋅

= −

141. 1 5 1 5 1 5 1 5

3 375

3. . . ..

( ) = ( )( )( )=

143. −FH IK ⋅ = −FH IK −FH IK −FH IK ⋅ = −FH IKFH IKFH IK ⋅

= −⋅ ⋅ ⋅⋅ ⋅

= −⋅ ⋅⋅ ⋅

= −

12

8 12

12

12

8 12

12

12

8

1 1 1 82 2 2

2 2 22 2 2

1

3

1 1 1

1 1 1

145. −( ) ⋅ −( ) = −( ) −( ) −( ) = −2 2 2 2 2 82

147. −( ) ⋅ ⋅ = −( ) −( ) −( ) ⋅ ⋅ ⋅ = −( ) ⋅ ⋅

= − ⋅ = −3 5 10 3 3 3 5 5 10 27 25 10

675 10 6750

3 2 149. −( ) ⋅ ⋅ −( ) = −( ) ⋅ ⋅ ⋅ ⋅ −( ) −( ) = −( ) ⋅ ⋅ = − ⋅

= −2 2 3 2 2 2 2 3 3 2 8 9 16 9

144

3 2

151. 34

4 2 34

34

4 2 2 2 3 3 4 2 2 24 4

3 3 2 2 2 2 22 2 2 2

18

23FH IK ⋅ −( ) ⋅ = ⋅ ⋅ −( ) ⋅ ⋅ ⋅ = −

⋅ ⋅ ⋅ ⋅ ⋅⋅

= −⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅= −

1 1 1 1

1 1 1 1


Section 1.2 7

....................................................... Objective F Exercises 153. 16 2 2 44 2= = = 155. 49 7 72= = 157. 32 2 2 2 2 2

2 2 4 2

5 4 4

2

= = ⋅ =

= =

159. 8 2 2 2 2 2 23 2 2= = ⋅ = = 2 161. 6 18 6 2 3 6 3 2

6 3 2 18 2

2 2= ⋅ =

= ⋅ = 163. ( )3 2

2

5 40 5 2 5 5 2 2 5

5 2 2 55 2 10 10 10

= ⋅ = ⋅

= ⋅= ⋅ =

165. 15 3 5 15= ⋅ = 167. 29 169. 3 2 2 2

2 29 72 9 2 3 9 2 3 2

9 2 3 29 2 3 2 54 2

− = − ⋅ = − ⋅ ⋅

= − ⋅= − ⋅ ⋅ = −

171. 45 3 5 3 5

3 5

2 2= ⋅ =

=

173. 0 0= 175. 7 6 6

36 128 6 2 6 2 2 6 2 2

6 2 2 48 2= = ⋅ == ⋅ =

177. 240 15 492≈ . 179. 288 16 971≈ . 181. 256 2 2 168 4= = = 183. 275 16 583≈ . 185. 245 15 652≈ . 187. 352 18 762≈ .

....................................................... Objective G Application Problems 189. Strategy a. To find the average monthly net income, divide TiVo’s annual net income by the number of months per year. b. To find difference, subtract Warnaco’s annual income from 2004 from annual income from 2003.

Solution a. 32.018 $2.66812

−= − million

b. million 2,360.423 ( 964.863) $3325.286− − = − 191. Strategy To find the percent, divide the amount spent on decorations by the total amount spent on Halloween, $44.

Solution 11 100 25%44

⎛ ⎞ ⋅ =⎜ ⎟⎝ ⎠

....................................................... Applying the Concepts

193. 8 2 82≈ . and 90 9 49≈ . The whole numbers between 8 and 90 are 3, 4, 5, 6, 7, 8, and 9.



Section 1.3

....................................................... Objective A Exercises 3. 4 8 2 4 4

0− ÷ = −

= 5. 2 3 4 3 2 1 3

2 1 92 911

2 2−( ) − −( ) = −( ) − −( )= −( ) −= − −= −

7. 24 18 3 2 24 6 218 220

− ÷ + = − += +=

9. 8 2 3 8 2 9

8 1810

2− ( ) = − ( )= −= −

11. 12 16 4 2 12 4 212 820

+ ÷ ⋅ = + ⋅= +=

13. 27 18 3 27 18 9

27 227 229

2− ÷ −( ) = − ÷ −( )= − −( )= +=

15. 16 15 5 2 16 3 2

16 3 213 211

+ ÷ −( ) − = + −( ) −= − −= −=

17. 14 2 4 7 14 2 3

14 2 314 4 310 37

2

2

− − − = − − −

= − −= − −= −

2

=

19. 3 2 8 3 2 3 2 8 13 2 73 14

11

− − −( ) = − −= −= −= −

21. 6 16 42 2

2 6 124 2

2

6 126

2

6 2 28 26

2+−+

− = ++

−

= + −

= + −= −=

23. ( ) ( )( )

( )( )

318 9 2 3 18 9 8 318 1 318 1 318 318 315

÷ − + − = ÷ − + −

= ÷ + −= ÷ + −= + −= −=

25. 4 16 7 1 10 4 16 6 10

4 10 1040 104

− −( ) ÷ = − ÷= ÷= ÷=

27. 20 10 2 5 20 10 8 5

20 2 510 510 55

3÷ − + −( ) = ÷ −( ) + −( )

= ÷ ( ) + −( )= + −( )= −=

c h

29. 4 8 2 7 3 4 8 2 4

4 8 2 164 8 32

32 321

2 2−( ) ÷ −( ) = −( ) ÷ ( )

= −( ) ÷ ( )= −( ) ÷= − ÷= −

31. 16 4 3 72 2

2 16 4 27 78 2

4

16 4 2010

4

16 4 2 416 8 48 44

3

32− ⋅

−+

− −( ) = − ⋅−+

−

= − ⋅ −

= − ⋅ −= − −= −=

33. 0 3 1 7 4 8 1 2 0 3 3 1 1 2

0 3 3 1 1 440 93 1 44

0 51

2 2. . . . . . .. . .

. ..

−( ) + ( ) = −( ) + ( )= −( ) += − +=

35. 1 65 1 05 0 4 0 8 0 60 0 4 0 80 36 0 4 0 80 9 0 81 7

2 2. . . . . .. . .. ..

−( ) ÷ + = ( ) ÷ += ÷ += +

.

=


Section 1.4 9

....................................................... Applying the Concepts

37. 23

16

0

12

16

56

13

12

13

−

−

−

39. Strategy To find the number of months: • Divide the miles per year by the current miles per gallon to find the gallons used per year. • Divide the miles per year by the EPA rating of the new car to find the gallons that would be used per year. • Find the difference in the number of gallons used annually. • Multiply the difference in gallons per year by the cost per gallon to find the cost. • Divide the increased cost of the new car by the money saved per year to find the time in years before the savings

equal the added cost. • Multiply the number of years by 12 to convert years to months. Solution

( )

( )

10,000 28 357.1410,000 80 125

357.14 125 232.14232.14 2.00 464.28

1500 464.28 3.233.23 12 39

÷ ≈÷ =− =

≈÷ ≈

≈

It would take 39 months.

Section 1.4

....................................................... Objective A Exercises 1. 6b a÷ −( )

6 3 2 18 2 9( ) ÷ −( ) = ÷ −( ) = − 3. b a2 4 c−

3 4 2 4 9 4 2 49 32 9 341

2( ) − ( ) −( ) = − ( ) −( )= − −( ) = + 2=

5. b c2 2− 3 4 9 162 2− −( ) = − = −7

3

5

7. a b2 2+

2 3 4 9 12 2+ = + = 9. b a c−( ) +2 4

3 2 4 4 1 4 41 16 1

2 2−( ) + −( ) = + −( )= + −( ) = −

11. 5

63ab cb−

5 2 36

3 4 3 306

36

5 36 41

( )( )− −( )( ) = − −( )

= − −( ) =

13. b c

d+

4 13

33

1+ −( )= =

15. 2d b

a+

−

2 3 42

6 42

102

5( ) +− −( )

=+

= =

17. b dc a−−

4 31 2

11

1−− − −( )

= =

19. b d a+( ) −2 4 4 3 4 2 7 4 2

49 8 57

2 2+( ) − −( ) = − −( )= − −( ) =

21. d a−( ) ÷2 5 3 2 5 5 5 25 5 52 2− −( ) ÷ = ÷ = ÷ =

23. b a

bc d−−2

2

4 2 24 1 3

4 44 1 3

84 3

81

8

2− −( )−( ) −

=− −( )( ) −

=−

=

=



25. 13

38

2 2d b−

13

3 38

4 13

9 38

16 3 6

3

2 2( ) − ( ) = ( ) − ( ) = −

= −

27. −−

42

bca b

− ( ) −( )−( ) −

=− −

=−

= −4 4 1

2 2 4164 4

168

2

29. − − −( )23

15

d bd ac

− ( ) − ( ) − −( ) −( ) = − ( ) − −

= − ( ) − ( ) = − − = −

23

3 15

4 3 2 1 23

3 15

12 2

23

3 15

10 2 2 4

31. b c a d+( ) + +( )2 2 4 1 2 3 3 1 9 1 12 2 2 2+ −( ) + −( ) + = + = + = 0

33. 3 4 2dc c− ( )

3 3 1 4 1 3 3 1 4 9 16 252 2( ) −( ) − −( ) = ( ) −( ) − −( ) = − − = −

....................................................... Objective B Exercises 35. 12 13 25x x x+ = 37. 12 3 9a a a− = 39. 8 6 2y y y+ −( ) = 41. − − = − −12 3 12 3y y 43. − + =3 12 9a a a 45. 9 3 6ab ab ab− = 47. − + = −15 3 12xy xy xy 49. − + =7 7ab ab 0

51. − + = − + = −25

310

410

310

110

y y y y y 53. 23

49

69

49

29

2 2 2 2y y y y− = − = 2y

55. 8 5 7 20x x x x+ + = 57. 10 17 3 4a a a a− + = − 59. − − + = −y y y y2 2 28 7 2 2

y

61. 8 10 8 2 8y x x x+ −( ) + = − + 63. 8 8 8 8y x y x+ − = 65. − + − + = −5 7 7 12 19 12b a b a a b 67. ( )3 12 7 2 12 2y x y y x+ − − + = − − y x 69. 3 5 10 10 7 52 2 2x x x x x+ − − = − −

....................................................... Objective C Exercises 71. 60x 73. –10a 75. 30y 77. 72x 79. –28a 81. 108b 83. −56 2x 85. x 2 87. x 89. a 91. b 93. x 95. n 97. 2x 99. –2x 101. −15 2a 103. 6y 105. 3y 107. –2x 109. –9y

....................................................... Objective D Exercises 111. − +( ) = − −x x7 7 113. 5 2 7 10 35x x−( ) = − 115. − +( ) = − −5 16 5 8a a 0 117. − −( ) = − +5 3 7 15 35y y

119. 10 7 2 20 14−( ) = −b b 121. 12

8 4 4 2− +( ) = − +x y x y 123. 6 3 2 18 122 2x x x+ = +c h x 125. − − +( ) = −5 2 7 10 35x x

127. − +( ) = − +2 7 7 14 49x x 129. 5 6 3 30 152 2− − = − −x xc h 131. − − = − +8 3 12 24 962 2y yc h

133. 5 52 2 2 5x y x y+ = +c h 2 135. − −( ) = − +12

4 12

2x y x y 137. 3 2 6 3 6 12 2x x x x+ − = + −c h 8

8

139. − − + = − + −2 2 4 2 42 2y y y yc h 141. − − +( ) = − + −13

6 9 1 2 3 13

x y x y 143. − − − + = + −5 2 3 7 10 15 352 2x x x xc h


Section 1.4 11

145. 3 2 3 6 3 92 2 2x xy y x xy y+ − = + −c h 2 4

6

147. − + − = − − +3 5 4 3 52 2a a a ac h 149. 4 2 3 8 4 6 16 2 1x x x x x− +( ) = − − = − − 151. 9 3 4 6 9 12 18 12 9− +( ) = − − = − −y y y

7

153. 5 7 2 5 7 2 7n n n n n− −( ) = − + = − 155. 3 2 5 7 3 6 5 35 2 41x x x x x+( ) − −( ) = + − + = − + 157. 12 2 3 7 3 12 24 21 9 3 3y y y y−( ) + −( ) = y− + − = − 159. 3 3 3a b a b a b a b a b2 4−( ) − +( ) = − − − = − 161. 4 2 3 4 2 6 4 6 4 24x x x x x x− −( ) = − + = − + = − + 163. − + −( ) = − + − = − + = − −2 3 2 4 2 3 8 2 2 8 2 16x x x x x x 165. − − +( ) = − − − = − − = − +3 2 7 3 2 7 3 7 3 21x x x x x x 167. 2 3 4 2 3 4 2 3 2 4

2 6 12 4 12x x x x x x x x

x x x− − −( ) = − − + = − −

= − + = − +

169. ( ) [ ]

[ ]5 2 2 4 7 6 5 2 2 4 28 6

5 2 2 28 65 4 56 6 50

x x x x x xx xx x x

− − − + − = − − − − −⎡ ⎤⎣ ⎦= − − − − −= − + + − = − +

....................................................... Objective E Exercises 171. the unknown number: x

x18

173. the unknown number: x x + 20

175. the unknown number: x the difference between the number and 50: x − 50 10 50 10 500x x−( ) = −

177. the unknown number: x

five-eighths of the number: 58

x

58

6x +

179. the unknown number: x three more than the number: x + 3 x x x x− +( ) = − − = −3 3 3

181. the unknown number: x the sum of the number and nineteen: x +19 4 19 4 76x x+( ) = +


the sum of the number and twelve: x +12

1512x +

185. the unknown number: x the sum of the number and seven: x + 7 23

7 23

143

x x+( ) = +

187. the unknown number: x the quotient of the number and

twenty: 20x

4020

−x


the square of the number: x 2 twice the number: 2x x x2 2+

191. the unknown number: x eight more than the number: x + 8 one-third of the number: 1 3a fx

x x x

x

+( ) + = + +

= +

8 13

8 13

43

8

x

193. the unknown number: x the total of the number and 9: x + 9 x x x+ +( ) = +9 2 9


the difference between eight and the number: 8 − x x x x x x− −( ) = − + = −8 8 2 8


one-third of the number: 13

x

five-eighths of the number: 58

x

1 5 8 15 73 8 24 24 24

x x x x− = − = − x

199. the unknown number: x the total of the number and 5: x + 5 x x+( ) + = +5 2 7



201. the unknown number: x six times the number: 6x the sum of six times the number and seven: 6 7x + 2 6 7 12 14x x+( ) = +

203. number of nations participating in 2004: n number of nations participating in 1896: n + 1990

205. amount in larger container: g amount in smaller container: 20 − g

207. pounds of pecans in Texas: p

pounds of pecans in Alabama: 12

p

209. diameter of a baseball: d diameter of a basketball: 4d

....................................................... Applying the Concepts 211. Yes. 213. number of oxygen atoms: x

number of hydrogen atoms: 2x

215. length of wire: x

length of side of square: 14

x

217. distance rope was pulled: x

distance weight will move: 35

x

Section 1.5

....................................................... Objective A Exercises 1. A = { }16 17 18 19 20 21, , , , , 3. A = { }9 11 13 15 17, , , , 5. A b c= { },

9. A B∪ = { }3 4 5 6, , , 11. A B∪ = − − −{ }10 9 8 8 9 10, , , , , 13. A B a b c d e f∪ = { }, , , , , 15. A B∪ = { }1 3 7 9 11 13, , , , , 17. A B∩ = { }4 5, 19. A B∩ = ∅ 21. A B c d e∩ = { }, ,

....................................................... Objective B Exercises 23. x x x> − ∈{ }5, negat ive in teger 25. x x x> ∈{ }30, in teger 27. x x x> ∈{ }5, even in teger 29. x x x> ∈{ }8, real num bers

....................................................... Objective C Exercises 31. x > 2

0 2 41 3 5–1–2–3–4–5(

33. 0 0≥ ≤x xor

0 2 41 3 5–1–2–3–4–5]

35. x x x x> −{ }∪ < −{ }2 4

0 2 41 3 5–1–2–3–4–5()

37. x x x x> −{ }∩ <{ }2 4

0 2 41 3 5–1–2–3–4–5( )


Chapter Review Exercises 13

39. x x x x≥ −{ }∪ <{ }2 4

0 2 41 3 5–1–2–3–4–5

....................................................... Applying the Concepts 41. a. Never true. A positive number times a negative number is always negative. b. Always true. The square of a non-zero number is always positive. c. Always true. The square of a non-zero number is always positive and is therefore greater than a negative number. 43. Let A = , { }1 2 3 4, , , B = { }1 3, , 5 , C = { }2 4 6, , a. A B∩ = { }1 3, , B C∩ = ∅ , B A∩ = { }1 3, , C B∩ = ∅ The intersection of two sets is commutative. b. A B C∩( )∩ = { }∩{ } = ∅1 3 2 4 6, , , , A B C∩ ∩( ) = { }∩∅ = ∅1 2 3 4, , , The intersection of two sets is associative.

....................................................... Chapter Review Exercises 1. Replace x with each element

in the set and determine whether the inequality is true.

x <− <

<<

14 10 1

11 1

TrueTrueFalse

x < 1 is true for the values –4 and 0.

2. 4 3. − − = −( ) = −5 5 5

4. − + −( ) + + −( ) = − + + −( ) = − + −( ) = −3 12 6 4 15 6 4 9 4 13 5. 16 3 18 16 3 18 19 18 1− −( ) − = + − = − =

6. − ( ) = −6 7 42 7. –20 8. 25 7 000 28

50200200

.

.

−

−

= 0 28.

725

0 28= .

9. 6 2% 6 2 0 01 0 062. . . .= ( ) =

10. 58

58

100% 5008

62 5%= ( ) = =% . 11. 13

16

512

412

212

512

4 2 512

712

− + = − +

=− +

=

12. 517 6 238 1 068. . .− = −

13. − ÷ = − ⋅ = − ⋅⋅

= − ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅

= −

1835

1728

1835

2817

18 2835 27

2 3 3 2 2 75 7 17

7285

1

1

14. 4 32 1 07 4 6224. . .−( ) = −

15. −FHIK = −FH

IK −FHIK −FHIK −FHIK =

23

23

23

23

23

1681

4

16. 2 36 2 6 2 6 122= = ⋅ =

17. − = − ⋅ ⋅ = − ⋅ ⋅( ) = − ⋅ ⋅

= − ⋅ ⋅ ⋅ = −

3 120 3 2 3 5 3 2 2 3 5 3 2 2 3 5

3 2 2 3 5 6 30

3 2 2c h 18. − + + −( ) = − + + −( ) = − +

= − + =3 4 18 12 20 3 4 18 8 9 4 10

9 40 31

2 2



19. b a c−( ) +2

3 2 4 3 2 4 525 4 29

2 2 2− −( ) + = + + = += + =

4

20. 6 4 2 6 2 46 2 4

8 4

a b a a a ba b

a b

− + = + −= +( ) −= −

21. − −( ) = − −( ) =3 12 3 12 36y y y

22. 5 2 7 5 2 5 7 10 35x x x−( ) = ( ) + −( ) = − 23. − −( ) + +( ) = − ( ) − −( ) + ( ) + ( )

= − + + += − + + = +

4 2 9 5 3 2 4 2 4 9 5 3 5 28 36 15 10

15 8 36 10 7 46

x x x xx xx x x

24. 5 2 3 6 1 5 2 18 3

5 5 1890 25

− −( ) = − += −= − +

xxx

x

25. 1 3 5 7, , ,{ } 26. A B∩ = { }1 5 9, ,

27.

0 2 41 3 5–1–2–3–4–5(

28.

0 2 41 3 5–1–2–3–4–5]

29. Strategy To find the score: • Multiply the number of correct answers by 6. • Multiply the number of incorrect answers by –4. • Multiply the number of blanks by –2. • Add the results.

Solution 21 6 1265 4 204 2 8

126 20 8 98

( )( ) =( ) −( ) = −( ) −( ) = −

+ −( ) + −( ) =

The student's score is 98. 30. 3.3 100% 50.8%

0.5 3.3 2.7⎛ ⎞ =⎜ ⎟+ +⎝ ⎠

31. unknown number: x

2 12

42

12

32

x x x x− = − = x

32. number of American League cards: A

number of National League cards: 5A 33. number of ten-dollar bills: T

number of five-dollar bills: 35 −T

....................................................... Chapter Test 1. − > −2 40 2. 4 3. –4 4. –14 5. − + + −( ) = − − = −22 14 8 8 8 16 6. 16 30 42 16 30 42 4− −( ) − = + − = 7. 17

8. 0 7. 9. 45

1009

20= , 0.45 10. − + = − +

=

25

715

615

715

115

11. –5.3578

12. 512

56

512

65

5 2 32 2 3 5

12

÷ −FH IK = − ⋅ = − ⋅ ⋅⋅ ⋅ ⋅

= −1

1

1

1

1

1 13.

34

4 34

16 122⋅ ( ) = ⋅ =

14. − = − ⋅ = −2 45 2 3 5 6 52 15. 16 2 8 3 4 2 1 16 2 8 3 2 1 16 2 8 6 116 2 2 1 8 2 1 17

÷ − −( ) + = ÷ − ⋅ + = ÷ − += ÷ ⋅ + = ⋅ + =

16. b ab2 23 2 3 3 24 18 4 18 2

− = −( ) − ( ) −( )= − −( ) = + = 2

17. 3 5 7 2 7 5x x x x x x− + = − + = 18.

15

10 2x x( ) =

19. − − = − +3 2 7 6 212 2 2 2 x y x yc h 620. 2 3 2 2 3 6x x x x x− −( ) = − + = − +


Chapter Test 15

21. 2 3 4 3 7 2 3 4 3 72 12 9 21 7 3

x x x x3x x x

+ − −( ) = + − += + − + = − +

22. − −{ }2 1 0 1 2 3, , , , ,

23. { }3, real numbersx x x< − ∈ 24. A B∪ = { }1 2 3 4 5 6 7 8, , , , , , ,

25.

0 2 41 3 5–1–2–3–4–5)

26.

0 2 41 3 5–1–2–3–4–5(]

27. 10 3 10 30x x−( ) = − 28. speed of return: s

speed of fastball: 2s

29. a. ’81, ’88, ’89, ’90, ’91, ‘95 b. 81.1 (369.7) = $288.6 billion

c. The difference in trade was greatest from 1999 to 2000. d. 81.1 4.18 4

19.4−

= ≈−

times greater

e. 369.7 $92.4254

−= − billion 30. 134.0 ( 81.4) 215.4 F− − =


chapter 1 real numbers and variable...

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