integrated algebra 1 · 2017-01-17 · integrated algebra 1 ann xavier gantert amsco school...
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INTEGRATEDALGEBRA 1Ann Xavier Gantert
AMSCO SCHOOL PUBLICATIONS, INC.315 HUDSON STREET, NEW YORK, N.Y. 10013
Teacher’s Manualwith Answer Key
A M S C O
14271FM.pgs 9/25/06 10:54 AM Page i
AMSCO SCHOOL PUBLICATIONS, INC.,a division of Perfection Learning®
Contents
Answer Keys
For Enrichment Activities 264
For Suggested Test Items 280
For SAT Preparation Exercises 288
For Textbook Exercises
Chapter 1 290
Chapter 2 293
Chapter 3 299
Chapter 4 303
Chapter 5 308
Chapter 6 312
Chapter 7 315
Chapter 8 321
Chapter 9 324
Chapter 10 345
Chapter 11 355
Chapter 12 358
Chapter 13 362
Chapter 14 371
Chapter 15 375
Chapter 16 384
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264
Enrichment Activity 1-1: Guessing aNumber by Bisection
a. 8 b. 6c. A maximum of 11 guesses would be needed to
locate a number between 1 and 2,000. The firstguess would be the average of 0 and 2,000, or1,000. Assuming that the number itself was not1,000, this would divide the possible numbersinto two equal sets: numbers between 0 and 999and numbers between 1,001 and 2,000. Sets ofthis size require a maximum of 10 guesses tofind the number.
Use a simpler related problem to find a generalsolution.
Range Max. Numberof Numbers of Guesses
From 1 to 3 2 First guess would be 2.If 2 is not the number,the next guess wouldbe correct.
From 1 to 7 3 First guess would be 4.If 4 is not the number,the number must beone of a set of threepossible numbers, andat most two moreguesses are needed.
From 1 to 15 4 First guess would be 8.If 8 is not the number,the number must beone of a set of sevenpossible numbers, andat most three moreguesses are needed.
Range Max. Numberof Numbers of Guesses Powers of 2
From 1 to 3 2 22 � 4From 1 to 7 3 23 � 8From 1 to 15 4 24 � 16From 1 to 31 5 25 � 32From 1 to 63 6 26 � 64From 1 to 127 7 27 � 128From 1 to 255 8 28 � 256
If 2n � 1 � the number of integers in the setfrom which the chosen number is selected,n is the maximum number of guesses needed toidentify the number.
To find a word in the dictionary, choose a middleword alphabetically and ask if the chosen wordcomes before or after the word for which you aresearching. Continue the process. To find a nameon a list, choose a middle name and continue theprocess.
Enrichment Activity 1-2: RepeatingDecimals
1. 2.
3. 4.
5. 6.
7. 1 8.9. 15 10. 120
Enrichment Activity 1-3: A Piece of Pi1. 0 1 2 3 4
5 6 7 8 9 2. No. The digit 9 appears most often while the digit
1 appears least often.3. 0 1 2 3 4
5 6 7 8 9 4. Answers will vary. Example: The digits 0, 1, 4, 5,
and 8 appear more often than in the first group,the digits 3, 6, 7, and 9 appear less often than inthe first group, and the digit 8 appears with samefrequency as in the first group.
5. 0 1 2 3 4 5 6 7 8 9
6. Answers will vary. Examples: The digit 7 appearssignificantly less frequently than the other digits.Appearances of the digits 0, 2, 3, 4, 5, 8, and 9 areevening out. As more digits in the expansion arechecked, the digits may seem to appear the samenumber of times.
Enrichment Activity 1-4: Making 1001. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. Answers will vary. Examples:
40 5 275,14839616 5 123,576
894 ,
13 5 95,4721,368,
369,258714815,643
297
817,524396823,546
197
915,823647917,524
836
941,578263961,428
357
961,752438962,148
537
23251216202219241623
9134712128121112
1412898101112511
155198
32299 or 325
9947111
1033
712
4199
115
Answers for Enrichment Activity ExercisesAnswers for Enrichment Activity Exercises
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Enrichment Activity 2-1: Special Ops1. 7 2. 5 3.4. 6.2 5. 20 6.7. 0.4 8. 9. 23
10. 8 11.12. a. a � b � a � 2b
b. 413. Answers will vary.
Enrichment Activity 2-2A: Clock Arithmetic
a. � 1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 1
2 3 4 5 6 7 8 9 10 11 12 1 2
3 4 5 6 7 8 9 10 11 12 1 2 3
4 5 6 7 8 9 10 11 12 1 2 3 4
5 6 7 8 9 10 11 12 1 2 3 4 5
6 7 8 9 10 11 12 1 2 3 4 5 6
7 8 9 10 11 12 1 2 3 4 5 6 7
8 9 10 11 12 1 2 3 4 5 6 7 8
9 10 11 12 1 2 3 4 5 6 7 8 9
10 11 12 1 2 3 4 5 6 7 8 9 10
11 12 1 2 3 4 5 6 7 8 9 10 11
12 1 2 3 4 5 6 7 8 9 10 11 12
b. 1. Yes2. Yes3. Yes4. 125. Number 1 2 3 4 5 6 7 8 9 10 11 12
Inverse 11 10 9 8 7 6 5 4 3 2 1 12
Enrichment Activity 2-2B: Digital Addition;Digital Multiplication
1. { 0 2 4 6 8
0 0 2 4 6 8
2 2 4 6 8 0
4 4 6 8 0 2
6 6 8 0 2 4
8 8 0 2 4 6
2. a. Yes b. Yesc. Yes d. 0e. Number { Inverse � Identity
0 { 0 � 02 { 8 � 04 { 6 � 06 { 4 � 08 { 2 � 0
14
75
58
58
3. z 0 2 4 6 8
0 0 0 0 0 0
2 0 4 8 2 6
4 0 8 6 4 2
6 0 2 4 6 8
8 0 6 2 8 4
4. a. Yes b. Yesc. Yes d. 6e. Number z Inverse � Identity
2 z 8 � 64 z 6 � 66 z 4 � 68 z 2 � 6
5. YesConclusion: This is a field; all 11 properties hold.
Enrichment Activity 2-5: Paying the Toll1(7) � 2(�3) � 1 1(11) � 11 1(11) � 1(7) � 1(3) � 21
1(11) � 3(�3) � 2 4(3) � 12 2(11) � 22
1(3) � 3 1(7) � 2(3) � 13 2(7) � 3(3) � 23
1(7) � 1(�3) � 4 1(11) � 1(3) � 14 8(3) � 24
1(11) � 2(�3) � 5 5(3) � 15 2(11) � 1(3) � 25
2(3) � 6 1(7) � 3(3) � 16 3(11) � 1(�7) � 26
1(7) � 7 1(11) � 2(3) � 17 9(3) � 27
1(11) � 1(�3) � 8 1(11) � 1(7) � 18 4(7) � 28
3(3) � 9 1(7) � 4(3) � 19 2(11) � 1(7) � 29
1(7) � 1(3) � 10 2(7) � 2(3) � 20 3(7) � 3(3) � 30
All purchases can be made since any wholenumber can be formed using multiples of thenumbers above.
Enrichment Activity 2-8: GraphingOrdered Pairs of Numbers
a.
b. M is two blocks west and one block south ofO(�2, �1).
c. The first number gives the number of blocks eastor west of O, the center of town. If the number ispositive, the position is east of O; if the numberis negative, the position is west of O. The secondnumber gives the number of blocks north or
Stat
e St
reet
Main Street
S
P F
C
H
A LV
M
D
GO
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south of O. If the number is positive, the positionis north of O; if the number is negative, theposition is south of O.
Enrichment Activity 3-1: Number PlayColumns 2 and 3 will vary. The answers for columns 1and 4 are shown below.
Col. 1 Col. 4
1. 15 n2. 3 2n3. 38 2n � 84. 3,800 200n � 8005. 3,600 200n � 6006. 360 20n � 607. 400 20n � 1008. 20 n � 59. 15 n
Conclusion: The result on line 9 will always equal thestarting number on line 1.
Enrichment Activity 3-2: Variable Codes1. a. PAOIK b. ZUSUXXUC
c. SGZNKSGZOIY d. KDZXGUXJOTGXE2. Answers will vary.3. I MUST STUDY FOR AN EXAM4. a. XIVRK b. GLQQCV
c. RJJZXEDVEK d. VOGCFIRKZFE5. x � 96. NZIXS, MLEVNYVI; the correspondence is
symmetric. If a line is drawn between M and N inthe alphabet list, the corresponding letters areequal distances from that line. A corresponds toZ, B corresponds to Y, and so on.
7. a. MLBNCFHb. TUESDAY; the result is the original word.
7 � x is the encoding and decoding expression.8. a. YKBWTR b. RDUPMK; no
Enrichment Activity 3-7: Formulas forHealth
1. 22 � 2. 24 � 3. 17 �4. 31 � 5. 26 � 6. 22 �7. 24 � 8. 34 � 9.
10. 114 bpm 11. 19 bpm 12. 23 bpm13. 19 bpm 14. 84%
Enrichment Activity 4-2: Book Value1. Let x represent the middle book because the
values of the other books can be expressed inrelation to it.
2. 16 books 3. x � 100; x � 200; x � 300
BMI 5 703WH2
4. 16x � 13,600 5. x � 150; x � 300; x � 4506. 16x � 20,400 7. 33x � 34,0008. 33x � 34,000 � 65,0009. $3,000 10. $3,500 11. $45,200
Enrichment Activity 4-3: ConsecutiveIntegers
1. Number First Even of Integers Integer Sum or Odd
3 1 6 even
3 4 15 odd
3 7 24 even
3 x 3x � 3 even or odd
4 1 10 even
4 4 22 even
4 7 34 even
4 x 4x � 6 even
5 1 15 odd
5 4 30 even
5 7 45 odd
5 x 5x � 10 even or odd
6 1 21 odd
6 4 39 odd
6 7 57 odd
6 x 6x � 15 odd
7 1 28 even
7 4 49 odd
7 7 70 even
7 x 7x � 21 even or odd
2. Let x be the first consecutive integer.a. Odd; 2x � 1 is the sum of an even number and
an odd number.b. Even when x is odd; 3x � 3 is the sum of two
odd numbers.Odd when x is even; 3x � 3 is the sum of aneven number and an odd number.
c. Even; 4x � 6 is the sum of two even numbers.d. Even when x is even; 5x � 10 is the sum of
two even numbers.Odd when x is odd; 5x � 10 is the sum of anodd number and an even number.
e. Odd; 6x � 15 is the sum of an even numberand an odd number.
f. Even when x is odd; 7x � 21 is the sum of twoodd numbers.Odd when x is even; 7x � 21 is the sum of aneven number and an odd number.
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g. Even when the number of integers divided by2 is even.Odd when the number of integers divided by2 is odd.
h. Even when the first integer is even and thenumber of integers after the first, divided by 2,is even.Odd when the first integer is odd and thenumber of integers after the first, divided by 2,is even.Even when the first integer is odd and thenumber of integers after the first, divided by 2,is odd.Odd when the first integer is even and thenumber of integers after the first, divided by 2,is odd.
3. a. 3 b. 5 c. 7d. The sum always has the number of consecutive
integers as a factor.The sum is always thenumber of integers times the middle number.
4. a. Nob. The sum is 6 more than 4 times the first integer.
The sum is twice the sum of the second andthird integers.The sum is 6 less than 4 times the last integer.
5. a. Student tablesb. The sum of even integers is always even.
6. a. Student tablesb. The sum of odd integers is even if there is an
even number of integers and odd if there is anodd number of integers.
Enrichment Activity 4-4: Law of the Lever1. 2.4 ft2. 15 lb, 25 lb3. Heavier carton, 9 ft; lighter carton, 12 ft4. Kelly, 49 lb; Laurie, 70 lb5. a. The side with the 32-lb weight
b. 3 ft
Enrichment Activity 4-7: Graphing anInequality
1. x � 4
2. x � 4
3. x � �3
4. All real numbersbetween �4 and4; [�4, 4]
5. All real numbersbetween �1 and1; (�1, 1)
6. All real numbersgreater than 1;(1, �). Since thereis no real numberthat is the squareroot of a nega-tive number, theinequality is mean-ingless for x � 0.
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7. All real numbers;(��, �).
8. All positive realnumbers; (0, �).
9. Since �x� is alwayspositive, there isno real numberthat makes thisinequality true; �.
Enrichment Activity 5-1: Sums and Squares1. 42 � 4 � 5 � 25 � 52
2.
3. 202 � 20 � 21 � 400 � 20 � 21 � 4414. 102 � 1005. (n � 1)2
n
n2 n
n + 1
n
+1
+ 1
Enrichment Activity 5-2: An Odd Triangle1. Row 6: 31, 33, 35, 37, 39, 41
Row 7: 43, 45, 47, 49, 51, 53, 552. 1, 9, 25, 49; the middle number in odd-numbered
Row n is n2.3. 4, 16, 36; the average of the middle numbers in
even-numbered Row n is n2.4. a. 169 b. 6
c. 157, 159, 161, 163, 165, 167, 169, 171, 173, 175,179, 181, 183
5. Row Number Sum of Numbers in Row
1 1
2 8
3 27
4 64
5 125
6 216
7 343
6. Sum of the numbers in Row n � n3
7. a. 1,728 b. Row 198. a. 3 � 22 � 1, 7 � 32 � 2, 13 � 42 � 3,
21 � 52 � 4, 31 � 62 � 5, 43 � 72 � 6b. n2 � (n � 1) � n2 � n � 1
9. a. 5 � 22 � 1, 11 � 32 � 2, 19 � 42 � 3,29 � 52 � 4, 41 � 62 � 5, 55 � 72 � 6
b. n2 � (n �1) � n2 � n � 1
10. a. Row Sum of All Sum WrittenNumber Numbers as a Square
1 1 12
2 9 32
3 36 62
4 100 102
5 225 152
6 441 212
7 784 282
b.
Enrichment Activity 5-4: Products, Sums,and Cubes
1. 3(4)(5) � 4 � 60 � 4 � 64 � 43
4(5)(6) � 5 � 120 � 5 � 125 � 53
5(6)(7) � 6 � 210 � 6 � 216 � 63
2. The product of three consecutive integers plusthe middle integer is equal to the cube of themiddle integer.
Cn(n 1 1)2 D2
268
n2 � n � (n � 1) � n2 � 2n � 1
(n2 � 2n � 1) (n � 1)2
n2 � 2n � 1 (n � 1)(n � 1)n2 � 2n � 1 � n2 � 2n � 1n2 � 2n � 1 � (n � 1)2
5?
5?
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3. 6(7)(8) � 7 � 336 � 7 � 343 � 73
4. 7(8)(9) � 8 � 504 � 8 � 512 � 83
5. (n � 1)(n)(n � 1) � n � n(n � 1)(n � 1) � n� (n2 � n)(n � 1) � n� n3 � n2 � n2 � n � n� n3
6. (n)(n � 1)(n � 2) � (n � 1) (n � 1)3
(n)(n2 � 3n � 2) � (n � 1) (n � 1)(n � 1)(n � 1)
(n3 � 3n2 � 2n) � (n � 1) (n � 1)(n2 � 2n � 1)n3 � 3n2 � 3n � 1 � n3 � 3n2 � 3n � 1
Enrichment Activity 6-1A: FibonacciSequence and the Golden RatioPart 1. For clarity, the first 26 terms of the Fibonacci
sequence are written below from smallest tolargest in three-column format.
1 55 4,181
1 89 6,765
2 144 10,946
3 233 17,711
5 377 28,657
8 610 46,368
13 987 75,025
21 1,597 121,393
34 2,584
Part 2.
Part 3.As the terms of the Fibonacci sequenceincrease, the ratio comparing the greaterof two consecutive Fibonacci numbers tothe smaller approaches the value of thegolden ratio.
1 1 "52 < 1.618033989
121,39375,025 < 1.61803398975,025
46,368 < 1.618033989
46,36828,657 < 1.61803398828,657
17,711 < 1.61803399
17,71110,946 < 1.61803398510,946
6,765 < 1.618033999
6,7654,181 < 1.6180339634,181
2,584 < 1.618034056
2,5841,597 < 1.6180338131,597
987 < 1.618034448
987610 < 1.618032787610
377 < 1.618037135
377233 < 1.618025751233
144 5 1.61805
14489 < 1.61797752889
55 5 1.618
5534 < 1.61764705934
21 5 1.619047
2113 5 1.61538413
8 5 1.625
5?
5?
5?
Enrichment Activity 6-1B: Ratio: Estimatesand ComparisonsColumns 1 and 2 will vary according to studentestimates.
Column 3
Actual Ratio of Size
Australia 0.80
Brazil 0.88
Canada 1.04
China 1.00
India 0.34
Japan 0.04
Mexico 0.20
Russia 1.77
Spain 0.05
United States 1.00
Bonus: Actual Ratio of Size
Australia 3.90
Brazil 4.32
Canada 5.06
China 4.87
India 1.67
Japan 0.19
Mexico 1.00
Russia 8.66
Spain 0.26
United States 4.88
In a and b, student responses will vary.a. With Mexico’s area used as the base of compari-
son, its ratio changes from 0.20 to 1.00. In turn,the ratio for every nation becomes about 5 timesas large as the ratio using the United States asthe base of comparison.
b. The ratios in both lists show the sizes of thecountries in relation to one another: Japan hasthe smallest area, followed by Spain, up to Russiawith the largest area.
269
Source for Teacher:Area (sq mi)
2,967,908
3,286,487
3,855,101
3,705,405
1,269,345
145,883
761,606
6,592,769
194,897
3,718,709
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Enrichment Activity 6-2: Population DensityPart 1.
State Land Area PopulationPopulation (sq mi) Density
Alaska 648,818 571,951 1
California 35,484,453 155,959 228
Florida 17,019,068 53,927 316
Montana 917,621 145,552 6
New Jersey 8,638,396 7,414 1,165
New York 19,190,115 47,214 406
North Carolina 8,407,248 48,711 173
Texas 22,118,509 261,797 84
Part 2. a. Student estimates will vary.b. Canada, 9; China, 361; India, 943;
Japan, 880; Mexico, 141; Russia, 22
Enrichment Activity 6-3: Catching Up1. a. 35 steps b. 5 steps c.2. a. 43 steps b. 5 steps; 10 steps
c.d. ; Evan is gaining on Marisa.
3.4. Evan continues to gain on Marisa until he
catches her after he has taken a total of 30 stepsand she has taken a total of 75 steps.
Enrichment Activity 6-4: Ratios andInequalities
1. Approach 1 proof:
Approach 2 proof:
2. a. ad bc b. ad � bcc. d � b d. d bc
3. must be true because 1(b � 1) b(1)or b � 1 b.
4. is true because a2 b2, so and
a b.5. a b is true because a(b � c) b(a � c), so
ab � ac ba � bc and ac bc.6. a. False b. Leads to contradiction (x � 0)7. a. True b. Consistent with x 0, y 08. a. False b. Leads to contradiction (y � 0)
"a . "bab . 1
1b . 1
b 1 1
ab , cd
ab A d
d B , A bb B c
d
adbd , bc
bd ad , bc
ab , cd
ab(b)(d) , c
d(b)(d) ad , cb ad , bc
37.551 or 75
102, 5059, 62.5
67 or 125134, 75
75
2543 . 25
70
2543
12.535 or 25
70
9. a. False b. Leads to the contradiction (y�0)10.
Enrichment Activity 7-4: Hero’s Formula1. a. 24 sq in. b. 24 sq in.2. a. 431.6 m2 b. 11,161.36 sq ft3. a. 388.8 cm2
b. Since the area and base are known, substitutethe values into the formula ; b � 18 cm
c. Right triangle
Enrichment Activity 7-5: RectangleCover-Up
1. 5 ways
2. 8 ways
3. Rectangles of Width 2
Number of Ways to Length of Rectangle Cover with Dominoes
0 1
1 1
2 2
3 3
4 5
5 8
6 13
7 21
8 34
9 55
10 89
4. Each number is the sum of the two previousnumbers. This is the Fibonacci sequence.
A 5 12bh
ab , a 1 c
b 1 d , cd
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Enrichment Activity 7-6: Area of Polygons1. a. 60 cm2 2. a. 480 cm2 3. a. 88 sq in.
b. b. 4r(ED) b.4.
5. a. sq in. b. cm2 c. 8 m2
d. 28,500 cm2 e. 800 mm2
Enrichment Activity 8-1:Pythagorean Triples
1. a. U V U2 � V2 2UV U2 � V2
2 1 3 4 5
3 1 8 6 10
4 1 15 8 17
5 1 24 10 26
6 1 35 12 37
b. The difference between U2 � V2 and U2 � V2
is 2.The sum of U2 � V2 and U2 � V2 is U times2U [5 � 3 � 2(4)].The sum of the three num-bers is twice the product of U and the next con-secutive integer [3 � 4 � 5 � 2(2 3)]. Otherrelationships are possible.
2. a. U V U2 � V2 2UV U2 � V2
3 2 5 12 13
4 3 7 24 25
5 4 9 40 41
6 5 11 60 61
7 6 13 84 85
b. The difference between U2 � V2 and 2UV is 1.The sum of U2 � V2 and 2UV is the square ofU2 � V2. The sum of the three numbers is theproduct of U2 � V2 and U2 � V2 � 1 [5 � 12 �13 � 5 6]. The smallest number, U2 � V2, isthe sum of U and V. Other relationships arepossible.
3. a. U V U2 � V2 2UV U2 � V2
4 2 12 16 20
5 3 16 30 34
6 4 20 48 52
7 5 24 70 74
8 6 28 96 100
b. The difference between U2 � V2 and 2UV is 4.The sum of U2 � V2 and U2 � V2 is twice U2
[20 � 12 � 2(42)]. The sum of U2 � V2 and2UV is the square of half the smallest number,U2 � V2 [20 � 16 � (12 � 2)2]. Other relation-ships are possible.
4. 10 and 75. 33, 56, 65 or 42, 56, 70 (Other answers are possible.)
96"39"3
12nsr
52r(YZ)1
2r(ED)
6. 69,260,269 or 69,92,115 (Other answers are possible.)7. a. Yes; 6, 8, 10
b. No. If a number is odd, its square is odd.Thesum of the squares of two odd numbers is even.Therefore, the third number must be even.
c. Yes; 3, 4, 5d. No. If either a or b is odd and the other even,
then c must be odd. If a and b are both even,then c must be even. Therefore, it is notpossible to have exactly one odd number in aPythagorean triple.
Enrichment Activity 8-3: Polytans1.
2.3.
4.
Enrichment Activity 8-6: TrigonometricIdentities
1. a. 1 b. 1 c. 1d. 1 e. 1 f. 1
2. tan A tan (90° � A) � 1, where 0° � A � 90°3.
B � 90° � Atan A tan (90° � A) � 1
4. a. 1 b. 1 c. 1d. 1 e. 1 f. 1
5. (sin A)2 � (cos A)2 � 16.
(Note the use of the Pythagorean Theorem.)
sin A 5 ac, (sin A)2 5 A a
c B 2 5 a2
c2
cos A 5 bc, (cos A)2 5 A b
c B 2 5 b2
c2
(sin A)2 1 (cos A)2 5 a2
c2 1 b2
c2 5 a2 1 b2
c2 5 c2
c2 5 1
tan A 3 tan B 5 ab 3 b
a 5 1
tan A 5 ab
tan B 5 ba
4"2, 6, 4 1 2"2, 2 1 4"2
4√2 6 4 + 2√2 4 + 2√2 4 + 2√2
4 + 2√2 4 + 2√2 4 + 2√2 4 + 2√2
4 + 4√2 4 + 4√2 4 + 4√2 4 + 4√2 4 + 4√2
4 1 "2, 2 1 3"2
4 + √2 2 + 3√2 2 + 3√2 2 + 3√2
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7. a. b.
c. d.
e. f.
8. , where 0° � A � 90°
9.
Enrichment Activity 9-3: Graphing withThree Variables1–4.
5. Octant 6 6. Octant 5 7. Octant 48. Octant 8 9. Answers will vary; (0, y, 0)
10.
11
1
x
y
z
(6, 0, 0)
(0, 0, 4)
(0, 3, 0)
11
1
x
y
z
(5, –4, 1)
(2, –5, 7)
(3, 2, 6)
(4, 2, –3)
sin A 5 ac
cos A 5 bc
tan A 5 ab
sin A cos A 5 a
c 4 bc 5 a
c 3 cb 5 a
b 5 tan A
sin A cos A 5 tan A
sin 78 cos 78 5 tan 78 sin 158
cos 158 5 tan 158
sin 348 cos 348 5 tan 348 sin 788
cos 788 5 tan 788
sin 428 cos 428 5 tan 428 sin 208
cos 208 5 tan 208 11.
Enrichment Activity 9-7: Iterating LinearFunctions
1. y � 0.4x � 6
Starting Value 5 Starting Value 15
8 12
9.2 10.8
9.68 10.32
9.872 10.128
9.9488 10.0512
9.97952 10.02048
2. The iterates are approaching 10; for the startingvalue 5, the iterates approach 10 from below;for the starting value 15, the iterates approach10 from above.
3. All the y-values (outputs) are exactly 10.4. a. 3 b. �4
c. 0 d. 25. y � x6. a.
b. Undefined for m � 1 since the denominatorwould be 0. Functions of the form y � x � b(b � 0) cannot have a fixed point because they-value can never be the same as the x-valueif a nonzero number is being added to thex-value.
7. a. Starting point 0: �2, �6, �14, �30, �62, �126Starting point 4: 6, 10, 18, 34, 66, 130
b. Noc. Here, m � 0; in question 1, 0 � m � 1.
x 5 b1 2 m
11
1
x
y
z
(0, 0, 4)
(0, –4, 0)
(2, 0, 0)
272
In 8–11, answers will vary.
8. No fixed point;Starting point 0: 3, 6, 9, 12, 15, 18Each iterate is b more than the previous.
9. Fixed point 100;Starting point 101: 100.8, 100.64, 100.512,100.4096, 100.32768, 100.262144Starting point 99: 99.2, 99.36, 99.488, 99.5904,99.67232, 99.737856Iterates approach the fixed point.
10. Fixed point 3.5;Starting point 4.5: 6.5, 12.5, 30.5, 84.5, 246.5,732.5Starting point 2.5: 0.5, �5.5, �23.5, �77.5,�239.5, �725.5Iterates move away from the fixed point
11. Fixed point 0;Starting point 1: 6, 36, 216, 1,296, 7,776, 46,656Starting point �1: �6; �36; �216; �1,296;�7,776; �46,656Iterates move away from the fixed point.
Enrichment Activity 9-10: Graphing StepFunctions
a.
Hours Cost Hours Cost Hours Cost
$4.00 2 $6.00 $38.00
$4.00 $8.00 18 $38.00
1 $4.00 6 $14.00 19 $40.00
$6.00 $28.00 $40.00
b.
Time Time Time Time In Out Cost In Out Cost
9:15 A.M. 9:50 A.M. $4.00 10:00 A.M. 2:30 P.M. $12.00
9:30 A.M. 10:29 A.M. $4.00 10:10 A.M. 10:00 P.M. $26.00
9:30 A.M. 10:35 A.M. $6.00 12:15 A.M. 8:00 A.M. $40.00
10:00 A.M. 12:45 A.M. $8.00 12:40 A.M. 11:00 A.M. $40.00
2013121
2112
214
23
1734
14
c.
d. From 1 to 19 hours, the graph would be astraight line.
Enrichment Activity 9-11: Holes, Holes, andMore Holes: An Exponential InvestigationPart I
Task 1
# of folds 0 1 2 3 4 5
# of holes 1 2 4 8 16 32
# of holes expressed 20 21 22 23 24 25
as a power of 2
a. The total number of holes doubles with eachfold, or the total number of holes is a power of 2,the power being the number of folds.
b. 2n c. H � 2n
Task 2
# of folds 0 1 2 3 4 5
# of holes 2 4 8 16 32 64
# of holes expressed 21 22 23 24 25 26
as a power of 2
48
121620242832
40
2 4 6 8 10 12 14 16 18 20 22 24
36
y
x
Cos
t of p
arki
ng
Number of hours0
48
121620242832
40
2 4 6 8 10 12 14 16 18 20 22 24
36
y
x
Cos
t of p
arki
ng
Number of hours0
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a. The pattern is similar but begins with 21 ratherthan 20.
b. 21 � 2 20, 22 � 2 21, 23 � 2 22, 24 � 2 23,25 � 2 24, 26 � 2 25
c. H � 2 2n
Part IIa.
Part IIIAnswers will vary. Example: Yes. It is appropriatebecause as n, the number of fold, increases, H,the number of holes punched, increases rapidly(or exponentially).
Enrichment Activity 10-5: Solving SystemsUsing Matrices
1. x � 2, y � 12. x � �1, y � 63. x � 4, y � �14. x � �3, y � 55. a. Error message results.
b. There is no solution to the system.
Enrichment Activity 10-6: Systems withThree Variables
1. (3, 5, 2) 2. (1, �2, 2)3. (2, 1, �2) 4. (4, 3, 0.5)5. (10, 3, 7) 6. (�6, 4, 2)
Enrichment Activity 10-8: LinearProgrammingExample
3. B (0, 5), C (6, 2), D (7, 0)4. At B (0, 5), 2x � 3y � 15
At C (6, 2), 2x � 3y � 18At D (7, 0), 2x � 3y � 14
6
0
12182430364248546066727884
1 2 3 4 5
H
n
H = 2n
Number of folds
5. The function is maximized at (6, 2). Themaximum profit of $18 is obtained when 6bracelets and 2 necklaces are produced.
Exercises1. Max 20 at (4, 4), min 0 at (0, 0)2. Max 68 at (12, 4), min 14 at (2, 2)3. a. S � 30c � 40t
b. 2c � 4t � 800, c � t � 300, c 0, t 0c.
d. (0, 0), (0, 200), (200, 100), (300, 0)e. At (0, 0), S � 0; at (0, 200), S � 8,000;
at (200, 100), S � 10,000; at (300, 0), S � 9,000f. The function is maximized at (200, 100). The
maximum sales of $10,000 is obtained when200 chairs and 100 tables are produced.
Enrichment Activity 11-1: FindingPrimes—The Sieve of Eratosthenes
1. 172. Composite numbers can be written as pairs
of factors, such that when the number dividedby a factor is less than the factor, all positiveintegral factors have been found. In this case,the greatest number, 200, divided by 15 yields
. Therefore, all composite numbers between15 and 200 that have a factor greater than 15must also have a factor less than 15 and havealready been crossed out.
3. 19, the largest prime less than 204. a. x � 1, 12 � 1 � 41 � 41
x � 2, 22 � 2 � 41 � 43x � 3, 32 � 3 � 41 � 47x � 4, 42 � 4 � 41 � 53x � 5, 52 � 5 � 41 � 61x � 6, 62 � 6 � 41 � 71x � 7, 72 � 7 � 41 � 83x � 8, 82 � 8 � 41 � 97x � 9, 92 � 9 � 41 � 113x � 10, 102 � 10 � 41 � 131
b. x � 41, 412 � 41 � 41 � 1,681 � 41 � 41
1313
400
300
(0, 200)
100
100 200 (300, 0) 400 500(0, 0)
y
x
x + y = 300
2x + 4y = 800(200, 100)
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Enrichment Activity 11-5: Differences ofSquares
1. a. 92 � 82 b. 72 � 52 c. 302 � 292
d. 162 � 142 e. 362 � 352 f. 262 � 242
g. 512 � 502 h. 732 � 722 i. 492 � 472
j. 1522 � 1512 k. 972 � 952 l. 5032 � 5022
2. , which is between n � 1 and n,so using the rule, 2n � 1 can be written as n2 � (n �1)2. To check, n2 � (n � 1)2 �n2 � (n2 � 2n � 1) � 2n � 1.
3. 4n � 4 � n, which is between n � 1 and n � 1,so using the rule, 4n can be written as (n � 1)2 � (n � 1)2. To check, (n � 1)2 � (n � 1)2
� n2 � 2n � 1 � (n2 � 2n � 1) � 4n.
Enrichment Activity 11-7: FactoringTrinomials
1. (1) 4x2(�10) � �40x2
(2) 8 and �5(3) 4x2 � 8x � 5x � 10(4) 4x(x � 2) � 5(x � 2)(5) (x � 2)(4x � 5)
2. a. (x � 2)(2x � 5) b. (x � 3)(4x � 3)c. (x � 5)(3x � 1) d. (x � 4)(3x � 2)e. (2x � 3)(3x � 2) f. (3x � 4)(4x � 1)g. (x � 5)(8x � 3) h. (x � 5)(5x � 2)i. (x � 3)(7x � 1)
Enrichment Activity 12-2: Square Root:Divide and Average
1. 7.07 2. 6.32 3. 2.454. 9.38 5. 10.39 6. 24.787. 3.32 8. 23 9. 24.78
10. 0.77 11. 0.55 12. 50.25
13. is rational because 232 � 529.14. 3.162 15. 10.100
Enrichment Activity 12-4: EquivalentRadical ExpressionsIn 1 and 3, answers will vary depending on the numberof decimal places displayed on the calculator.
1. Side (square Side (decimal Square Area root of area) approximation)
a 2 1.414213562
b 8 2.828427125
c 18 4.242640687
d 32 5.6565854249
e 50 7.071067812"50
"32
"18
"8
"2
"529
2n 2 12 5 n 2 12
2. Square Equation Relating Sides
a —
b
c
d
e
3.
Side (square Equationroot of Side (decimal Relating
Square Area area) approximation) Sides
f 5 2.236067977 —
g 20 4.472135955
h 45 6.708203932
Enrichment Activity 12-7: Operations withRadicals
1. a. b.
c. d.
e. f.
g. 0 h.
i. 0 j.
k. l.
m. n.
o. p. 3
q. r.
s. t.
u. v.
w. 3 x.
y.
2. Common Common Matches Answer Matches Answer
a, c k, q
b, u l, t
d, n m, o
f, j p, w 3
g, i 0 v, x
h, r
3. a. 100
b. Yes
2"5
23"5
5 2 2"3
11"76"3 1 3"5
4 1 2"38"11
10"63"2
5"2
23"5
23"58"11
4 1 2"3"5
2"510"6
11"7
6"3 1 3"511"7
4 1 2"310"6
5 2 2"3
2"5
5 2 2"32"10
6"3 1 3"53"2
8"113"2
"45 5 3"5"45
"20 5 2"5"20
"5
"50 5 5"2
"32 5 4"2
"18 5 3"2
"8 5 2"2
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Enrichment Activity 13-2: Carpet Squares1. 4 with fringe on two sides, 8 with fringe on one
side, 4 with no fringe.
2.
Dimensions Fringe on Fringe on No Total of Carpet Two Sides One Side Fringe Squares
2 ft by 2 ft 4 0 0 4
3 ft by 3 ft 4 4 1 9
4 ft by 4 ft 4 8 4 16
5 ft by 5 ft 4 12 9 25
6 ft by 6 ft 4 16 16 36
7 ft by 7 ft 4 20 25 49
8 ft by 8 ft 4 24 36 64
9 ft by 9 ft 4 28 49 81
10 ft by 10 ft 4 32 64 100
3. The number of squares with fringe on two sides is 4for any size because all squares have 4 corners.Thenumber of squares with fringe on one side startsat 0 and increases by 4 each time the length of theside increases by 1 foot.The number of squareswith no fringe starts at 0 and is the sequence ofsquare numbers.The total number of squares isalso the square numbers starting with 4.
4.646056524844403632282420161284
1 2 3 4 5 6 7 8 9 10x
y
* * * * * * * * *
Num
ber
of e
ach
type
of s
quar
e
Length of side of square0
5. The graphs for squares with fringe on two sidesand one side are linear. The graph for squareswith no fringe is steepest.
6. a � 4, b � 4(x � 2), c � (x � 2)2
7. 4 with fringe on two sides, 192 with fringe on oneside, 2,304 with no fringe.
8. The formulas are graphed in Exercise 4 for thedomain of 2 � x � 10, where x is an integer. Ingeneral, the formulas hold true for the domainx 2. The graph of c is quadratic.
9. No. For the number of squares to be equal,4(x � 2) � (x � 2)2. Simplifying gives x2 � 8x �12 � 0, which factors to (x � 6)(x �2) � 0. Theonly solutions for x are 2 and 6.
10. (n � 1)2 � n2 � 2n � 3
Enrichment Activity 13-4: QuadraticInequalities
1.
2. (0, 0) is not in the region, (0, 3) is in the region3. See graph4. All points inside of the graph of y � x2 � 6x � 85. 2 � x � 46.
1
–1–1
O1
x
y
y
xO1
1–1
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7. All points outside the graph of y � x2 � x � 128. x � �3 or x 4
Enrichment Activity 14-2: Is It Magic or IsIt Math?
1. a. ; yesb. ; yes
2. Student-generated results. Each result shouldequal 0.6 when rounded to the nearest tenth.
3.
4.
or
Therefore, to the nearest tenth.5. a. 0.62
b. The seventh term in the sequence beginning
with . Apply this
algebraic fraction to the rule illustrated
in Exercise 4, placing terms in a correct
order, to discover: ,
which is equivalent to or
0.615384615 � � 0.625. Therefore,
� 0.62 to the nearest hundredth.
Enrichment Activity 14-5: Operations withFractions
A � B A � B A � B A � B
1.
2.
3. (2) 4. (4) 5. (4)6. (2) 7. 18. If the sum and the difference in Exercise 1 are
equal, then w � 0. This would contradict thegiven statement, w 0.
9. No. If x � 1 and y � 1, then becomes
1 � w � w (an impossible statement),or 1 � 0 (also impossible).
10. w � 2 11. 12.
Enrichment Activity 14-7: FractionsBetween Fractions
1. a.
b.
c.0.875 � 0.9285714… � 0.95
78 , 7 1 19
8 1 20 , 1920, 78 , 26
28 , 1920,
914 , 9 1 14
4 1 5 , 145 , 94 , 23
9 , 1455, 2.25 , 2.5 , 2.8
23 , 2 1 4
3 1 5 , 45, 23 , 6
8 , 45, 0.6 , 0.75 , 0.8
y 5 23x 5 5
4
x 1 wy 5 xw
y2
xwy
xwy
x 2 wyy
x 1 wyy
xw
xwy2
x 2 wy
x 1 wy
8y 1 5x13y 1 8x
8y 1 5x13y 1 8x
813 ,
8y 1 5x13y 1 8x ,
58
8y13y ,
8y 1 5x13y 1 8x ,
5x8x
xy (x . 0, y . 0) is
5x 1 8y8x 1 13y
3x 1 5y5x 1 8y 5 0.6
0.6 ,3x 1 5y5x 1 8y , 0.625
3x5x ,
3x 1 5y5x 1 8y ,
5y8y is equivalent to
35 ,
3x 1 5y5x 1 8y ,
58
xy S y
x 1 y S x 1 yx 1 2y S x 1 2y
2x 1 3y S 2x 1 3y3x 1 5y S 3x 1 5y
5x 1 8y
213 S 13
15 S 1528 S 28
43 S 4371 S 71
114
35 S 5
8 S 813 S 13
21 S 2134 S 34
55
2. Student-generated examples.3. No. Cite any one example to show that is
not the average of .4. a.
b.c.
5. a. If , then (a 0, b 0, c 0)b. Yes. The average of is equal to
.
Enrichment Activity 14-8: EstimatingSolutions to Quadratic Equations
1. a. x � 1.32 or 1.33 b. x � 1.316 or 1.3172. y � �2; x � �5.32 or �5.33 (to the nearest tenth),
x � �5.316 or �5.317 (to the nearest hundredth)3. Calculator check. 4. x � 1.45, x � �3.455. x � 6.54, x � 0.46 6. x � 1.09, x � �10.097. x � 2.27, x � 5.738. The product of x and x � 4 equals 7, not each
factor. If each factor were to equal 7, then theirproduct would be 49.
Enrichment Activity 15-3: Probability anda Digital Clock
1. Answers will vary; student guesses2. a. P(0) � � .3125
b. P(1) � � .5
c. P(2) � � .375
d. P(3) � � .3125
e. P(4) � � .3125
f. P(5) � � .3125
g. P(6) � � .175
h. P(7) � � .175
i. P(8) � � .175
j. P(9) � � .1753. The digit appearing most often on a digital clock
is 1, and the digit appearing the second mostoften is 2. The digits 0, 3, 4, and 5 appear the samenumber of times over a 24-hour period. The digits6, 7, 8, and 9 also appear the same number oftimes over a 24-hour period, and these digitsappear less than all the others.
4. The answers will be exactly the same for any12-hour period as for the 24-hour period discussedearlier because, in a 24-hour period, a 12-hourcycle is repeated twice. Doubling (or halving) thenumerator and the denominator of the probabilityfraction does not change the value of the fraction.
2521,440 5 7
40
2521,440 5 7
40
2521,440 5 7
40
2521,440 5 7
40
4501,440 5 5
16
4501,440 5 5
16
4501,440 5 5
16
5401,440 5 3
8
7201,440 5 1
2
4501,440 5 5
16
a 1 cb 4 2 5 a 1 c
2b
ab and cb
ab , a 1 c
2b , cb
ab , c
b
92 , 9 1 10
2 1 2 , 102 , 92 , 19
4 , 102 , 4.5 , 4.75 , 5
16 , 1 1 5
6 1 6 , 56, 16 , 6
12 , 56, 0.16 , 0.5 , 0.83
34 , 3 1 9
4 1 4 , 94, 34 , 12
8 , 94, 0.75 , 1.5 , 2.25
ab and cd
a 1 cb 1 d
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Enrichment Activity 15-6: Probability andArea: And, Or, Not
1. a. P(D) � � .28b. P(not D) � � .72
2. a. P(D) � � .28b. P(not D) � � .72
3. a. P(D) � � .28b. P(not D) � � .72
4. Answers will vary. Example: The areas of thethree shaded regions in Exercises 1–3 are equal;each is 7 square units. The probability that apoint on the 5-by-5 grid lies in the shaded regionis for each region.
5. a. A and B intersect in two squares. Example:
b. P(A or B) � � .46. a. A and B intersect in three squares. Example:
b. P(A or B) � � .367. a. B is a subset of A. A and B intersect in four
squares. Example:
b. P(A or B) � � .328. P(A or B) is found by either:
(1) counting the number of square units in the5-by-5 grid that are shaded by A, by B, orby both, and then dividing this area by 25,which is the area of the grid; or
(2) using the formula P(A or B) � P(A) � P(B) �P(A and B)
9. No. P(A and B) cannot exceed P(A), and sinceP(A) � � .16, the statement P(A) .16 is false.4
25
825
A
B
925
A
B
1025 5 2
5
BA
725
1825
725
1825
725
1825
725
Enrichment Activity 15-7: Probability ona Dartboard
1. a. �b. 3� (from 4� � �)c. 5� (from 9� � 4�)d. 7� (from 16� � 9�)e. 9� (from 25� � 16�)
2. a. � .04 b. � .12
c. � .2 d. � .28
e. � .36
3. a. � .4 b. � .56
c. � .6 d. � .96
e. � .36
4. a. � .16 b. � .36
c. � .0016 d. � .9216
e. � .1296 f. � .4096
g. � .0784 h. � .1296
Enrichment Activity 15-11: Expectation1. a. Responses will vary.
b. Most should say “left” or “negative.”2.
3. The expectation of indicates that, on average,the player expects to move 1 place in a negativedirection for every 3 turns taken.
4. a.b. The expectation of 0 indicates that, on
average, a player does not expect to advanceor fall behind over a long period of play.After many turns, the player should still be inthe START box, although the marker mayhave moved to the left and to the right duringdifferent turns.
5. Answers will vary. To win on this board in anaverage of 24 turns, a player must advance8 spaces to the right in that time. Therefore,E(game) should equal . Here is onepossible set of rules to obtain E(game) �(1) If a player guesses correctly, move 7
spaces in the positive direction (to theright).
(2) If a player guesses incorrectly, move 1space in the negative direction (to the left).
Thus:E(game) � A 1
6 B 3 (17) 1 A 56 B 3 (21) 5 2
6 5 13
13:
18 24, or 11
3
E(game) 5 A 16 B 3 (15) 1 A 5
6 B 3 (21) 5 0
21 3
E(game) 5 A 16 B 3 (13) 1 A 5
6 B 3 (21) 5 3
6 1 25 6
5 22 6 5 21
3
81625
49625
256625
81625
576625
1625
925
425
925
2425
1525 5 3
5
1425
1025 5 2
5
925
725
525 5 1
5
325
125
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Enrichment Activity 15-12: A “Pick Six”Lottery
1. a.b. .0000000387
2. a.b. .0000112
3. a.b. .000655
4. a.b. .000837
5. a. Add the solutions to Exercises 1–4 part a:
b. Add solutions to Exercises 1–4 part b: .00156. Yes, the claim is essentially correct:
By choosing two sets of 6 numbers, the probabil-ity of winning a prize is doubled, so we multiplythe probability found in Exercise 5 by two:2 .0015 � .003, which is close to the givenprobability.
Enrichment Activity 16-1:Taking a Survey:Designing a Statistical StudyIn 1–4, student responses will vary.
Enrichment Activity 16-4: Locating theMedian Value
a. 4thb. 5thc. 8thd. 20the. 41stf. 119thg.h. 4th and 5thi. 5th and 6thj. 9th and 10thk. 21st and 22ndl. 44th and 45th
m. 115th and 116thn.
General rule: Arrange the data values in order. Let thenumber of data values be N.If N is odd, the median is the value that is fromeither end.If N is even, the median is the average of the valuesthat are from either end.N
2 and N2 1 1
N 1 12
N2 and N2 1 1
N 1 12
1333 5 .003
38,82925,827,165
6C3 ?
1C1 ?
47C2
54C65
20 ? 1 ? 1,08125,827,165 5
21,62025,827,165
6C4 ?
48C2
54C65
15 ? 1,12825,827,165 5
16,92025,827,165
6C5 ?
48C1
54C65 6 ? 48
25,827,165 5 28825,827,165
6C6 ?
48C0
54C65 1 ? 1
25,827,165 5 125,827,165
Enrichment Activity 16-7:TheMedian-Median Line
1. Yes
2. See graph 3. (58, 71); see graph4. (69, 84); see graph 5. (85, 91); see graph6.
In 7 and 8, answers will vary but should be close tothose given.
7. Using the points (55, 70) and (75, 85),y � 0.75x � 28.75
8. a. 0.75 b. 28.759. a. 82 b. 74 c. 92 or 93
10. The regression equation (values rounded to thenearest hundredth) is y � 0.75x � 30.24, which isclose to the median-median line.
11. Student results will vary. The median-median lineshould be close to the regression line.
50 55 60 65 70 75 80 85 90 95 100
55
50
60
6570
75
8085
9095
100
0
Post
test
Pretest
+
+
+
50 55 60 65 70 75 80 85 90 95 100
55
50
60
6570
75
8085
9095
100
0
Post
test
Pretest
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280
Chapter 11. a. 12 b. 108
c. 19 d. 1122. a. Infinite b. Finite
c. Infinite d. Empty3. True4. False5. True6. True7. False8. False9. a. b.
c. such that b � 0
d. e.10. 18.0511. a. 0.05 b. 0.666 . . . �
c. 0.444 . . . �
d. �3.1666 . . . �12. a. 4 b. 17
c. 6 d. 313. a. Rational b. Irrational
c. Irrational d. Rational14. a. 3.742 b. 1.183
c. 3.142 d. 0.14215.
16. a. A and B b. J and K c. H and Id. F and G e. D and E
17. 15.2 mg18. P � 30.88 cm
A � 58 cm2
Chapter 21. (4) 2. 63. a. True; distributive property
b. True; commutative property of additionc. True; addition property of zerod. True; associative property of multiplicatione. False
4. a. � b. � c. �5. a. �4 b. �5.7
c. d. �606. �127. a. {�3, �1, 0, 2, 4, 5}
b. {0, 1, 5} c. {�1, 1, 4, 8} d. {�3}8. A(�3, 1), B(3, �3), C(5, 4), D(�3, �5), E(�4, 0),
F(0, �2)
135
A B C D E F G H I J K
–1 –1 0 1 1 2 2 3 312
12
12
12
12
12–
23.160.4
0.6
2 21100
731
01, or 0b
163
310
9.
10. a.
b. 10 sq units
Bonus I: The commission
Bonus II: (3 � 3 � 3 � 3) 3 � 3
Chapter 31. a. n � 12 b. n � 7 c. 2n � 4
d. e.2. a. x � 5 b. 10c c. 60h � m
d. 25q � 10d3. a. The product of 7 and a number n, decreased by 2
b. 9 times the sum of c and 3c. 36 minus the product of 2 and a number x,
plus 54. 1, 17, a, b, 17a, 17b, ab, 17ab5. Base � x, exponent � 176. a. 3a2 b. 5x3y5 c. (5a)3
7. a. Coefficient � 6, base � h, exponent � 2b. Coefficient � 1, base � w, exponent � 5c. Coefficient � , base � r, exponent � 3d. Coefficient � �1, base � m, exponent � 1
8. a. 1 b. 1 c. 64d. 4 e. 3 f. 37g. 16 h. 0
9. 100.3275 10. c � 4n � p
4 1 n3
12n 2 6
y
x
A B
C
y
xA
B
C
D
E
F
Answers for Suggested Test ItemsAnswers for Suggested Test Items
14271AKTI.pgs 9/25/06 10:43 AM Page 280
11. c � 35 � 0.15(m � 100) 12. 77 cm2
13. {�3} 14. {4, 5}
Bonus: $7; Jennifer gives 5 dollars to Lisa and 2 dollarsto Ramon. Each person would have Lisa’s originalamount plus 5 dollars.
Chapter 41. x � x � 1 2. a � 5 3. b � 224. y � 42.5 5. x � 5 6. n � �107. a � 7.2 8. x � �5.5 9. c � 9
10. x � 87 11. 412. Length: 10 cm; width: 7.5 cm 13.14. a. b. C � �515. 716. x � 3
17. x � 3
18. x � �2
19. �
20. 1 x 6
21. (2) 22. (1) 23. 3 hr24. 20 quarters, 30 dimes, 40 nickels25. 6 stamps at 63¢, 18 stamps at 39¢, 30 stamps at 4¢26. 5 hr 27. 9:40 A.M.28. 6 mph going, 4 mph returning
Bonus: 72
Chapter 51. �3a3 2. 3x � 2 3. �27x � 634. 4x4 � 5 5. �4a3b3 6. 2x2 � 2x �27. �3x � 22 8. �27x6y3 9. 14y � x
10. 2x2 � 7x � 4 11. 4x2 � 4x � 1 12. 4x2 � 113. �48r3s4 14. a3 � 2a2 � a 15. 13x2 � 316. 7a 17. 5 18. 5y2 � 419. a2b2 20. 3 21.
22. n � 7 23. n � 2 24. n � 325. 19 26. 3.2 103 27. 9.3 107
28. 5.4 10�2 29. 2 10�6 30. 80,00031. 1,700,000,000 32. 0.073 33. 0.000000534. a. 8x � 6 b. 3x2 � 5x � 28 c. x �
35. 4a2 � 12a � 9 36. 4x � 17 37. 2x � 138. y � 10 39. x � 9 40. 10x � 6
73
9x2
–2 –1 0 1 2 3 4 5 6 7 8 9
–2 –1 0 1 2 3 4 5 6 7 8 9
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–2 –1 0 1 2 3 4 5 6 7 8 9
–2 –1 0 1 2 3 4 5 6 7 8 9
C 5 59(F 2 32)
h 5 3VA
Bonus: a. 17 and 18b. No. The numbers whose squares differ by
24 are not counting numbers.(x � 1)2 � x2 � 24x2 � 2x � 1 � x2 � 242x � 1 � 242x � 23x � 11.5
Chapter 61. 1 : 3 2. 6 : 1 3. 4 : 54. 1 to 400 5. 1.2 boxes per sec6. 384 mi 7. x � 8 8. a � 69. y � 18 10. x � 9 11. 3 cans
12. 10°, 40°, 100° 13. 25% 14. x � 2115. 25 persons 16. 6.25%17. 32 cm, 40 cm, 56 cm 18. 200 wads19. $3,680 20. 64 hydrants 21. 45 hr22. 2.4 lb 23. min 24. $121 per hr
Bonus I: 20 typists; 5 pages
Bonus II: 1: 2; Amanda, 12; Daniel, 21; Latanya, 24
Chapter 7
1. a.b. �AEC, �BEC, �AED, �BEDc. �ACE, �BCEd. �ACF, �BCFe. AE f. �CAEg. �BCF, �ACF
2. a. 95 b. 85c. 85 d. 95
3. m�A � 76, m�B � 56, m�C � 484. a. 118 b. 118
c. 62 d. 625. a. x � 5
b. m�A � m�C � 20, m�B � m�D � 1606. 80°, 100° 7. 238 cm2 8. 114 sq in.9. 12 cm 10. 52%
11. a. 5.9% b. 11.4% c. 16.6%
Bonus: m�ECD � 70
Chapter 81. 25 in. 2. 40 cm
3. a. in. b. 6.4 in.4. 43° 5. 35° 6. 41°7. 14° 8. 15.7 in. 9. 11.3 cm
10. 19 ft 11. 21°, 69° 12. 37°, 53°, 90°13. 85° 14. 87 ft 15. 17 in.
Bonus: 25°, 75°, 79°
"41
EBh
813
281
14271AKTI.pgs 9/25/06 10:43 AM Page 281
Chapter 91. a. No
b. Yesc. Yes
2. No3. k � �54. a.
b.
c.
d.
5. a. x-int � 2, b � 10b. x-int � �3, b � �12c. x-int � �3, b � 4.5
6. �47. y � 3x � 18. �3; 1
3
y
xO1
1–1–1
y
xO1
1–1–1
–1
1
1
O
y
x
y
xO1
1–1–1
9. a. 0, horizontalb. , rises to the rightc. �1, falls to the right
10. a. 5 b. y � 5xc. d. 5
11. a. m � 2, b � 1 b. m � �1, b � 7c. m � , b � �3 d. m � 0, b � 3
12. a. (0, 0), (4, �5) b. (4, �5)c. (0, 0), (�1, 3)
13. a. y � 2x � 2b.
14. a.
b. y
xO1
–1 1
y
xO1
–1 1
y
x
O
1
–1–1
12
–1
1
1
O
y
x
35
282
14271AKTI.pgs 9/25/06 10:43 AM Page 282
c.
15. y � �x� � 416. a.
b.
17. a. s � 0.10p � 200b.
c. $1,700
y
xO500
5,000
–1 1
y
xO3
–1
1
1
y
xO
y
xO1
1–1–1
18. a.b.
c. 2 min
Bonus: (9, 2), (9, 6), (1, 6)
Chapter 101. y � x � 3 2. y � 2x � 23. x � 2y � 6 4.5.
6. a.
b. (1, 1)7. a. (3, �2) b. (�6, 8)8. a. or (2.5, 3)
b. y
xO1
–1
( 5 2– , 3)
–1
A 52, 3 B
y
xO
1
–1–1
y
xO
1
–1
(2 , 1)
y 5 2 23x 1 6
y
xO1
1–1
y 5 8 A 12 B
x
283
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9. Any value except 10.11. Last year’s garden: 3 ft by 14 ft
This year’s garden: 6 ft by 7 ft12. First meeting: 7 girls and 5 boys
Second meeting: 14 girls and 15 boys13.
Bonus: a. 94 spectatorsb. 58 students, 24 parents, 12 faculty
Chapter 111. 32 � 5 � 112. 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 903. a. 6ab b. 3x2 c. 4x2y4. 25a2b4
5. 4x2 � 4x � 16. 2x2 � 3x � 97. y2 � 498. 6a(2a � b)9. (a � 3)(a � 2)
10. (s � 5)(s � 5)11. 2r2(4r � 1)12. Prime13. (x � 10)(x � 2)14. (y � 3)(y � 3)15. (2x � 3)(x � 2)16. (4b � 3)(b � 3)17. 4(c2 � 4)18. (3b � 4)(3b � 4)19. 5(y � 2)(y � 2)20. 5(x � 1)(x � 1)21. Prime22. ab(a � 7)(a � 5)23. 6x2 � x � 124. 2s � 325. 4; (5x � 2)(5x � 2)
Bonus I: a. Let the odd integers be 2n � 1 and 2n � 1. Their product is (2n � 2)(2n � 1) � 4n2 � 1. Their average is
, whichsquared is 4n2.
b. Yes. Let the even integers be 2n and 2n � 2. Their product is 2n(2n � 2) �
4n2 � 4n. Their average is
, which squared is 4n2 �4n � 1.
Bonus II: n2 � 1 � (n � 1)(n � 1); n � 1 and n � 1are consecutive even integers if n is odd and consecu-tive odd integers if n is even.
4n 1 22 5 2n 1 1
2n 1 (2n 1 2)2 5
(2n 2 1) 1 (2n 1 1)2 5 4n
2 5 2n
A 1a, 1b B
212
72
Chapter 121. a. Rational
b. Irrationalc. Irrationald. Rationale. Rationalf. Rationalg. Irrational
2. a.
b.
c.d. 75e. 4f. 5
g.3. a. 11a2 b. 0.6x c. 3y3
d. 3a4. 4.55. a. x � ±1.2 b. x � ±76. a. 37 b. �6.08 c. 2.47. 15 and 16
In 8 and 9, part a, answers will vary according to thenumber of digits in the calculator display.
8. a. 9.591663047b. 9.592
9. a. �46.9041576b. �46.90
10. (3)11. (3)12. a. Length
b. (1) in.(2) 62.61 in.
c. (1) sq in.(2) 244.95 sq in.
Chapter 131. x � 4 or x � 32. x � 6 or x � �13. a � 5 or a � �54. y � 0 or y � 55. b � or b � 46. x � 7 or x � �57. �9, �8 or 8, 98. Base � 16 cm, height � 10 cm9. a. 400 ft at t � 1 sec
b. t � 6 sec10. a. x � 2 b. (2, �2) c. Down
11. a. b. c. UpA232, 214 Bx 5 23
2
292
100"6
10"10 1 8"15
2"2
12"6
3"2
4"3
284
14271AKTI.pgs 9/25/06 10:43 AM Page 284
12.
13. x � �1, x � 3
14. No real roots15. {(�3, 6), (4, 13)}
y
x
(–3 , 6)
(4 , 13)
O1
–11
y
xO1
1–1
y
xO1
1
16. {(�1, 5), (3, �3)}
17. {(�4, 24), (1, 4)}
18. {(�2, 4), (�1, 3)} 19. {(�4, 11), (2.5, 1.25)}20. 7 ft, 12 ft
Bonus: 9 chords
Chapter 141. x � �4 2. (x � 0, y � 0)
3. 4. (c � 0)
5. (x � 0, y � 0) 6. (x � 0)
7. (y � 0) 8. 9x (x � 0)
9. (x � 0) 10. (x � 0)
11. 12. (x � 0)
13. (a � 0) 14. (a � 0)
15. 16. 17. x � 4
18. a � 5 19. y � 54 20. x � 1921. b � 4 22. d � 25 23. No solution24. x � �3, x � �4 25. b � �1, b � 426. 18 27. 10 quarters28. 10 free throws
23b10
8b3
2723a 1 21
10a
49
11x 2 1615
15x2 1 15x
35
y 2 2y
2x5
2x 1 110y
3c 2 13c2
a 1 b2
x3y
(–4 , –24)
(1 , 4)2
–2–1 1
y
x
y
xO–1
–11
(–1 , 5)
(3 , –3)
285
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Bonus: a. b. 60
Chapter 151. a. b. c.2. a. b. c.3. 57%4. a. b. c.5. .96.7. 25 red, 75 green8.9. a. 24 b. 5,040 c. 120
d. 60 e. 9 f. 21010.11. a. b.12. 1,32013. 6014.15. 916. 2417. a.
{(H, H, H, H), (H, H, H,T), (H, H,T, H),(H, H,T,T), (H,T, H, H), (H,T, H,T),(H,T,T, H), (H,T,T,T), (T, H, H, H), (T, H, H,T),(T, H,T, H), (T, H,T,T), (T,T, H, H), (T,T, H,T),(T,T,T, H), (T,T,T,T)}
b. 16 c. d.18. 24
36 5 23
1516
616 5 3
8
H
HH H
H
H
H
H
HH
H
H H
H
T
T
T
T
T
T
T
H
T
T
T
T
T
T
T
T
122,652 5 1
221
3066 5 5
111566 5 5
22
17
1552
436 5 1
9
192380 5 48
9556
380 5 1495
132380 5 33
95
1252 5 3
131352 5 1
4452 5 1
13
36 5 1
256
16
3160
19. a.
Let A � Adam, B � Bert, C � Clara,D � Doris, E � Elaine, and F � Flora.{(A, B), (A, C), (A, D), (A, E), (A, F), (B, A),(B, C), (B, D), (B, E), (B, F), (C, A), (C, B),(C, D), (C, E), (C, F), (D, A), (D, B), (D, C),(D, E), (D, F), (E, A), (E, B), (E, C), (E, D),(E, F), (F, A), (F, B), (F, C), (F, D), (F, E)}
b. c. d.20. a. b.21. a. b. c.22. 1,12023. a. 1,225 b.
Bonus: 10216 5 5
108
991,225
54132 5 9
2272
132 5 611
6132 5 1
22
49
69 5 2
3
2030 5 2
31630 5 8
152
30 5 115
A
B
C
D
E
F
BCDEF
ACDEF
ABDEF
ABCEF
ABCDF
ABCDE
286
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Chapter 161. a. 129 g b. 128 g c. 127 g2. 51, 53, 55, 57, 593. 834. 14.4 lb5. a. 5 b.6. 187. 3a � 28. a. Stem Leaf
5 3
6 5, 7
7 0, 1, 5, 7, 8, 9
8 0, 1, 3, 3, 6, 8, 9
9 1, 1, 4, 5, 8
b. Min � 53, Q1 � 73, med � 81, Q3 � 90,max � 98
c. 81st percentile
9. a. Days Cumulative Absent Frequency Frequency
6 1 25
5 3 24
4 3 21
3 1 18
2 2 17
1 8 15
0 7 7
b.
Days absent
Freq
uenc
y
9
8
7
6
5
4
3
2
1
00 1 2 3 4 5 6
50 55 60 65 70 75 80 85 90 95 100
Key: 5 � 3 � 53
1425
c.
d. 1 e. 1.92 f. 0g. 4 h. i.j. 1
10. a.
b. 1,850–1,899 c. 1,750–1,799d. 1,900–1,949
11. a.
b. (39.8, 66) c. y � 055x � 44.1d. y � 0.45x � 48 e. $61,500
Calls
Sale
s ($
1,00
0)
80
70
60
50
0 10 20 30 40 50 60 70
Calories
Cum
ulat
ive
Freq
uenc
y
40
35
30
25
20
15
10
5
0
1,550
–1,59
9
1,600
–1,64
9
1,650
–1,69
9
1,700
–1,74
9
1,750
–1,79
9
1,800
–1,84
9
1,850
–1,89
9
1,900
–1,94
9
1,950
–1,99
9
125
325
Days absent
Cum
ulat
ive
Freq
uenc
y
0–0 0–1 0–2 0–3 0–4 0–5 0–6
262422201816141210 8 6 4 2 0
287
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288
Chapter 11. C 2. D 3. C4. B 5. C 6. B7. B 8. E 9. B
10. A 11. B 12. D13. D 14. D 15. B16. D 17. B 18. E19. 8 20. 198 21. 022. 76 23. or 0.5 24.
Chapter 21. A 2. D 3. B4. C 5. E 6. A7. B 8. D 9. B
10. D 11. B 12. D13. B 14. B 15. B16. E 17. 15 18. 2019. 6 20. 72 21. 222. 3
Chapter 31. D 2. A 3. B4. B 5. B 6. C7. E 8. B 9. E
10. E 11. C 12. D13. D 14. C 15. B16. C
Chapter 41. B 2. D 3. D4. A 5. E 6. B7. A 8. C 9. C
10. A 11. E 12. A13. E 14. D 15. E16. C 17. D 18. B19. A 20. B 21. $822. 3 23. 131 24. 2725. 4 26. $2.40 27. 428. 45
Chapter 51. A 2. D 3. B4. C 5. C 6. B7. D 8. C 9. A
10. D 11. E 12. E13. C 14. D 15. C16. C 17. A 18. B19. 49 20. 0.52 21. 24
38
12
22. 8 23. 256 24. 825. 0
Chapter 61. A 2. E 3. B4. B 5. A 6. B7. A 8. C 9. B
10. A 11. B 12. C13. A 14. C 15. A16. B 17. B 18. A19. D 20. E 21.22. 23. 48 24. 8025. 26. 50 27. $64028. 29. 75 30. $2,20031. 72
Chapter 71. E 2. D 3. E4. B 5. A 6. B7. D 8. A 9. C
10. A 11. C 12. C13. D 14. C 15. C16. 24 17. 65° 18. 8:30 P.M.19. 30 20. 24 21. 12822. 36 23. 55 24. 1425. 508 26. 3,664
Chapter 81. C 2. E 3. C4. B 5. A 6. B7. C 8. D 9. A
10. E 11. C 12. A13. D 14. C 15. E16. D 17. 60 18.19. 4.5 20. 37.3 21. 0.58122. 36 23. 5 24. 7.1
Chapter 91. E 2. B 3. B4. D 5. B 6. D7. E 8. E 9. B
10. E 11. A 12. E13. C 14. D 15. B16. B 17. 24 18.19. $358 20. 10 21. 1722. 624 23. 0 � x � 2 24. 20 m25. 20 min 26. 4 cups
32
718
116
13
215
353
Answers for SAT Preparation ExercisesAnswers for SAT Preparation Exercises
Chapter 101. A 2. E 3. D4. A 5. C 6. B7. E 8. E 9. C
10. B 11. E 12. B13. C 14. A 15. C16. B 17. 5 18. 6519. 13 ft 20. 14 21. 123°22. $2,000 23. 1 24. 60 cc25. 10
Chapter 111. A 2. C 3. A4. B 5. A 6. A7. C 8. A 9. D
10. E 11. E 12. C13. C 14. B 15. D16. C 17. D 18. A19. 16 20. 8 21. 14422. 5 23. 20 24. 025. 64 26. 13 27. 1,15628. 15
Chapter 121. B 2. C 3. C4. B 5. D 6. C7. C 8. D 9. D
10. A 11. A 12. E13. A 14. D 15. D16. B 17. C 18. B19. E 20. A 21. 222. 7 23. 14 yd 24. 825. 9 in. 26. 15 ft 27. 1628. 0.85
Chapter 131. A 2. C 3. B4. A 5. B 6. D7. D 8. E 9. C
10. B 11. B 12. D13. C 14. E 15. C
16. A 17. A 18. C19. D 20. C 21. A22. 72 23. 24. or 13.525. 12 26. 16 in. 27. 9828. 100 29. 9 30. or 7.531. 27 32. 33. 2.5 sec
Chapter 141. E 2. C 3. A4. B 5. D 6. C7. B 8. B 9. E
10. E 11. E 12. D13. B 14. E 15. C16. 17. 120 min 18.19. 4 20. 20% 21. or 0.522. 9 23. or 1.5 24. 1025. 0.04
Chapter 151. D 2. C 3. D4. C 5. C 6. D7. D 8. B 9. C
10. B 11. C 12. C13. D 14. D 15. A16. E 17. D 18. B19. D 20. D 21. 022. or .25 23. 24.25. 26. or .125 27. or .528. .42
Chapter 161. C 2. C 3. B4. D 5. C 6. B7. A 8. B 9. C
10. B 11. A 12. D13. C 14. B 15. C16. D 17. 43% 18. 1019. 14.4° 20. 80.4 21. $9022. 1,470
12
18
12
56
23
14
32
12
37
415
23
152
272
103
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290
Introductory Page (page 1)3 buses; 2.6 hr; 2 casesIn each problem, 125 is divided by 48. However, thequotients are rounded differently; the first problemrequires that the quotient be rounded up to the nearestwhole number, the second that the quotient be roundedto a decimal place, and the third that the quotient berounded down to the nearest whole number.
1-1 The Integers (pages 9–10)Writing About Mathematics
1. Yes. If the real number is positive or zero, thenit is equal to its absolute value. If the number isnegative, then its absolute value is positive so itis greater than the original number.
2. If a is less than b, then a must be the negativenumber and b must be the positive number sinceall positive numbers are greater than all negativenumbers.
Developing Skills3. a. 10.4 4. a. 7
b. �10.4 b. 75. a. 21 6. a. 13
b. �21 b. 137. a. 20 8. a.
b. 20 b.9. a. 10. a. 1.45
b. b. 1.4511. a. 2.7 12. a. 0.02
b. �2.7 b. 0.0213. True 14. True 15. False16. True 17. False 18. True19. True 20. True 21. 1222. 6 23. 10 24. 525. 0 26. 10 27. 228. 5 29. 0 30. 1431. True; on the number line, �5 is to the right of �2.32. True; on the number line, �3 is to the left of 0.33. False; on the number line, �7 is to the left of �1
and therefore is less than �7.34. True; on the number line, �2 is to the right of �10.35. �8 � �6 36. �8 � 037. �5 � �2 38. �5 � �2539. 16 � 3 � 9 � 240. 6 � 7 � 100 � 241. �7 is greater than �7.
2334
334
2112
112
42. �20 is less than �3.43. �4 is less than 0.44. �9 is greater than or equal to �90.45. �5 � �4 � 846. �6 � �3 � �3 � �647. �4 � �2 � 0 � �348. �8 � �2 � 0 � �849. 8 � 14, 8 � 14, 8 1450. 9 � 3, 9 � 3, 9 351. 15 15, 15 �15, 15 1552. 6 � �2, 6 � �2, 6 �253. 1-f, 2-b, 3-c, 4-h, 5-j, 6-l, 7-e, 8-a, 9-g, 10-i, 11-d, 12-kApplying Skills54. a. Three 34-passenger buses and two 25-passenger
buses should be scheduled.b. 1 empty seat
55. Fill the 4-pound bag with rice. Use this rice tofill the 3-pound bag, leaving 1 pound remainingin the 4-pound bag. Empty the 3-pound bag. Pourthe pound of rice from the 4-pound bag into the3-pound bag. Refill the 4-pound bag.1 pound � 4 pounds � 5 pounds
56. Answers will vary. Examples: Loss of yardagein a football game; temperatures below zerodegrees; decline in a stock price; loss of weight;time before liftoff; etc.
1-2 The Rational Numbers (pages 15–17)Writing About Mathematics
1. is a rational number, and every rational numbercan be expressed as a repeating decimal; therefore,
can be expressed as a repeating decimal.2. Imagine a smallest positive rational number, n.
We can take the mean of n and 0 and get a valuesmaller than n. This process can be repeatedforever, so there is no smallest positive rationalnumber.
Developing Skills
3. 4. 5.
6. 7.
8. or where x is any counting number
9. 10. 11.
12. 13. 14.15. 16. 17. 5
215
56
113
72223
10
71,000210
3112
0x
01
231
91
2 21100
950
710
117
117
Chapter 1. Number Systems
Answers for Textbook ExercisesAnswers for Textbook Exercises
14271AKTE1.pgs 9/25/06 10:37 AM Page 290
18. 19. 20.21. 1.275 22. 0.6
In 23–32, the mean is given. Answers will vary.23. 24. 25.
26. 27. 28.
29. �2.15 30. 31.32. 3.075 33. 34.35. 36. 37.38. 39. 40.41. 42. 43.
44. 45. 46.
47. 48. 49.
50. 51. 52.53. True; integers are a subset of the rational numbers.54. False; whole numbers are the counting numbers
and 0. Negative integers, such as �1, �2, and �3,are the opposites of the positive whole numbers.These numbers are not whole numbers.
55. True; on a number line, the numbers increase invalue from left to right.
56. True; every rational number can be written aseither a terminating decimal or a repeatingdecimal. If the decimal terminates, then afterthe last nonzero digit, zero repeats.
57. True; between 0 and 1, another rational numbercan be found by taking their mean (or average).Since this process has no end, there is an infinitenumber of fractions (which are rational numbers)between 0 and 1.
58. True; �2 and �1 are both rational numbers.There is an infinite number of rational numbersbetween any two different rational numbers.
59. True; if x is any rational number, then x � 1 is arational number that is greater than x.
Applying Skills
60. 61.62. a. 63. a. 64.8 in. 64. a.
b. b. b.
1-3 The Irrational Numbers (pages 23–24)Writing About Mathematics
1. Answers will vary. Examples:
� � (��) � 0,0.121221222 . . . � 0.212112111 . . . �
0.333333333 . . . �
2. No, is closer to � than 3.14:� � ≈ 3.1428571 � 3.1425927 or 0.0012644
� � 3.14 ≈ 3.1415927 � 3.14 or 0.0015927
227
227
0.3 5 13
"2 1 (2"2) 5 0
25
3340
13
12
14
110
35
2 310
799800
7100
101400
18
1111,000
32521
5111200
1220.830.050.180.71.60.5831.625025.502.2500.625021 7
242 916
21724
1116
38
212231
2512
2 512
136213
6Developing Skills
3. Rational 4. Rational 5. Rational6. Irrational 7. Irrational 8. Irrational9. Irrational 10. Rational 11. Irrational
12. Rational 13. Rational 14. Irrational15. Rational 16. Rational 17. Irrational18. Irrational 19. Irrational 20. Rational21. Rational 22. Irrational 23. (1), (2), (4)
In 24–43, part a, answers will vary depending on thenumber of digits in the calculator display.24. a. 2.236067977 25. a. 2.645751311
b. 2.236 b. 2.646c. 2.24 c. 2.65
26. a. 4.358898944 27. a. 8.660254038b. 4.359 b. 8.660c. 4.36 c. 8.66
28. a. 7.937253933 29. a. 9.486832981b. 7.937 b. 9.487c. 7.94 c. 9.49
30. a. �3.741657387 31. a. �4.69041576b. �3.742 b. �4.690c. �3.74 c. �4.69
32. a. .4472135955 33. a. .5477225575b. 0.447 b. 0.548c. 0.45 c. 0.55
34. a. 3.464101615 35. a. 4b. 3.464 b. 4.000c. 3.46 c. 4.00
36. a. 1.374368542 37. a. 1.047197551b. 1.374 b. 1.047c. 1.37 c. 1.05
38. a. .4123105626 39. a. �9.055385138b. 0.412 b. �9.055c. 0.41 c. �9.06
40. a. 2.549509757 41. a. �7.416198487b. 2.550 b. �7.416c. 2.55 c. �7.42
42. a. 41.61730409 43. a. 15.5241747b. 41.617 b. 15.524c. 41.62 c. 15.52
44. a. 2.999824b.
45. a. 9.998244 b. 10.004569c. The better approximation for is 3.162.
; the value obtained in part a is0.001756 from 10, which is closer than thevalue in part b, which is 0.004569 from 10.
46. a. 7� 47. a. 15� 48. a. 72�b. 21.99 b. 47.12 b. 226.19
49. a. � 50. a. 3 � or �
b. 1.57 b. 10.47
103
13
12
A"10 B 2 5 10"10
"3
291
14271AKTE1.pgs 9/25/06 10:37 AM Page 291
51. False; since � 2, we know that � �
2 � 2 � 4, and 4 � .52. False; is approximately 4.24, so �
is approximately 8.48. � 6, which is less than8.48.
Hands-On Activitya. 1 sq ft b. 2 sq ft c. ft; irrationald. About 1 ft 5 inches (1.4 ft or 17 inches); rationale. The answers should be very close but not exactly
the same. It is impossible to measure an irrationallength, so the measurement in part d is actuallyan estimate.
1-4 The Real Numbers (pages 27–28)Writing About Mathematics
1. The first statement includes only countingnumbers since you would not have a fractional(or negative) number of persons in your family.The second statement includes all positiverational numbers less than 6, since measurementscan be fractional. Both statements includenumbers less than 6 but greater than 0.
2. No. The calculator approximates as3.16227766 and � as 3.141592654. Looking atthe hundredths digit, we can see that isgreater than �.
3. Irrational. The decimal does not terminate andthe pattern of decimal digits does not repeatinfinitely; therefore, the number is irrational.
Developing Skills4. a. 1, 2 b. 0, 1, 2 c. �2, �1, 0, 1, 2
d. , �2, �1, �0.63, 0, , 1, 2
e. , , ,f. All numbers shown
5. a.b.c. All
6. a. None b. All c. All7. 2.5 8. 8 9.
10. 0.23 11. 12. �5.613. 0.43 14. 0. 15. �
16. 17. 18.19. 0.202 � 0.2022 � 20. � 0.4499 � 0.4521. � � 22. � �1.5 �23. 0.5 � � 24. � � �25. False 26. True 27. True28. False 29. True 30. False31. True 32. False 33. False34. True
"103.150.5"0.32"22"30.670.6670.6
0.40.2
227"2"0.5
20.7
0.2
"2, "3, "5, "6, "7, "8"0, "1, "4, "9
"6p2"0.52"3
1322.7
"10
"10
"2
"36"18"18"18
"8"4"4"4 Hands-On Activity
a. Results will vary.b. Results will vary.c. Results should be close to �.
1-5 Numbers as Measurements (page 33)Writing About Mathematics
1. The measure 12.50 in. is more precise because12.50 is correct to the nearest hundredth with anerror of 0.005 in. whereas 12.5 ft is only correctto the nearest tenth with an error of 0.05 in.12.50 in. is more accurate because it has threesignificant digits whereas 12.5 in. has only twosignificant digits.
2. Mario is correct.To calculate the distance he rodehis bike, Mario multiplied 10(2� � 63 m).The onlyinexact measurement involved was 63 meters,which has two significant digits, so his distancetraveled should also have two significant digits.
Developing Skills3. a. 2 significant digits 4. a. 3 significant digits
b. ones b. hundredthsc. 0.5 in. c. 0.005 cm
5. a. 2 significant digits 6. a. 3 significant digitsb. hundreds b. onesc. 50 ft c. 0.5 lb
7. a. 2 significant digits 8. a. 3 significant digitsb. ten-thousandths b. hundredthsc. 0.00005 kg c. 0.005 yd
9. a. 4 significant digits 10. a. 1 significant digitb. thousandths b. hundredsc. 0.0005 m c. 50 mi
11. a. 57 in. 12. a. 2.50 ftb. 4,250 in. b. 2.50 ft
13. a. 0.0003 g 14. a. 0.055 mb. 32 g b. 0.055 m
Applying Skills15. a. 18.1 ft 16. a. 59.0 ft
b. 328 sq ft b. 6,200 sq ft17. 33.0 in. 18. 250 cm2
Review Exercises (pages 35–36)1. 434.06 2. 4.22 3. 149.574. 14.70 5. 37.70 6. �5 � �1 � 37. True 8. False 9. False
10. False 11. 12.13. 14. 15.16. 17. Answers will vary.
Example: 19.9518. Rational 19. Irrational 20. Rational21. Irrational 22. Irrational
2631
13
141
172
920
910
292
14271AKTE1.pgs 9/25/06 10:37 AM Page 292
In 23–27, part a, answers will vary depending on thenumber of digits in the calculator display.23. a. 3.31662479 24. a. .8366600265
b. 3.32 b. 0.8425. a. 30.08321791 26. a. 39.98749805
b. 30.08 b. 39.9927. a. 3.141592654 28. 5
b. 3.1429. �12� � ��8� 30. 3.2 31. 0.432. 0.12 33. True 34. False35. False 36. True 37. True38.
39. A � �1, B � �0.5, C � 0, D � 0.5, E � 1,F � 1.5, G � 2, H � 2.5, I � 3, J � 3.5
–3 –1.5 0 1 4�
40. a. F and G b. A and B c. E and Fd. I and J e. G and H
41. a. 1400 cm b. No
Exploration (page 36)Answers will vary.a. 5.566566656666 . . .b. 5.556555655556 . . . , 5.565565556 . . . ,
5.6565565556 . . .c. 5.555655556555556 . . . , 5.556555655556 . . . ,
5.556655666556666 . . .d. 5.5565565556 . . . , 5.55665666566665 . . . ,
5.5566655666555666 . . .
293
2-1 Order of Operations (pages 43–44)Writing About Mathematics
1. Any even number greater than 2 will have atleast three factors, itself, 1, and 2, and is thereforecomposite.
2. No. Numbers that end in a multiple of 3 are notnecessarily divisible by 3, so it is possible forthem to have only two factors. Examples include:13, 19, 23, and 29.
Developing Skills3. a. The sum of 6 and 1 is to be added to 20; 27
b. Add 20 and 6, then add 1 to that result; 274. a. The sum of 4 and 3 is to be subtracted from
18; 11b. Subtract 4 from 18, then add 3 to that result; 17
5. a. The difference of 3 and 0.5 is to be subtractedfrom 12; 9.5
b. Subtract 3 from 12, then subtract 0.5 from thatresult; 8.5
6. a. Multiply 15 by the sum of 2 and 1; 45b. Multiply 15 by 2, then add 1; 31
7. a. The sum of 12 and 8 is to be divided by 4; 5b. The quotient of 8 and 4 is to be added to 12; 14
8. a. Divide 48 by the difference of 8 and 4; 12b. Divide 48 by 8, then subtract 4 from that
result; 29. a. Add the square of 5 to 7; 32
b. Square the sum of 7 and 5; 14410. a. Multiply 4 by the square of 3; 36
b. Square the product of 4 and 3; 14411. No.The expression in the numerator is divided by
the expression in the denominator: (10 � 15) �(5 � 3) � 10. Noella’s expression, 10 � 15 � 5 � 3,equals 90.
12. a. 25, 125, 625 13. a. 0.25, 0.125, 0.0625b. 54 b. 0.52
14. a. 0.25, 0.36, 0.49 15. a. 1.21, 1.44, 1.69b. 0.72 b. 1.32
16. a. 1, 2, 41, 82 17. a. 1, 101b. Composite b. Prime
18. a. 1, 71 19. a. 1, 3, 5, 15b. Prime b. Composite
20. a. 1 21. a. 1, 2, 4, 8, 101, 202,b. Neither 404, 808
b. Composite22. a. 1, 67 23. a. 1, 397
b. Prime b. PrimeApplying Skills24. 2 � 0.28 � 3 � 0.28 or (2 � 3) � 0.28; $1.4025. 2 � 0.30 � 3 � 0.25; $1.3526. 30 � � 55 � 1 ; 105 mi27. 2 � 0.38 � 3 � 0.69; $2.8328. 5 � 0.29 � 3 � 0.75 � 1.75; $5.4529. $3.21 30. 9 years31. Answers will vary.
a. 3 � 2 � 1 � 4 b. 1 � 3 � 1 � 4c. (1 � 2) � 3 � 4 � 5 d. (4 � 3 � 2) � 1 � 5e. (6 � 6 � 6) � 6 � 5 f. (6 � 6) � 6 � 6 � 6
2-2 Property of Operations (pages 53–54)Writing About Mathematics
1. a. y � 1; the multiplication property of one saysthat the product of any number and one is thenumber itself.
b. x � 0; the multiplication property of zero saysthat the product of zero and any number is zero.
2. The distributive property of multiplication overaddition allows us to say that 0.75(2 � 3) �0.75(2) � 0.75(3).
12
34
Chapter 2. Operations and Properties
14271AKTE1.pgs 9/25/06 10:38 AM Page 293
Developing Skills3. a. 9 b. 0 c. 9
d. 0 e. f.
g. 1 h. 4.5 i.
j. 0 k. l. 04. a. 8 b. Commutative property of addition5. a. 5 b. Commutative property of
multiplication6. a. 15 b. Associative property of
multiplication7. a. 6 b. Distributive property of
multiplication over addition8. a. 0.2 b. Associative property of addition9. a. 4 b. Addition property of zero
10. a. 7 b. Commutative property ofmultiplication
11. a. 9 b. Commutative property ofmultiplication
12. a. 0 b. Addition property of zero13. a. 1 b. Multiplication property of one14. a. �17 15. a. �1 16. a. 10
b. b. 1 b.17. a. �2.5 18. a. 1.8 19. a.
b. or 0.4 b. b. 920. a. 21. a. � 22. a.
b. b. b.23. a. �1.78 24. a. 25. a.
b. b. 11 b.26. Correct27. Incorrect;28. Incorrect; addition is not distributive over
multiplication.29. Correct 30. Correct 31. Correct32. a. True 33. a. False 34. a. False
b. Yes b. No b. No35. a. True 36. a. True 37. a. False
b. Yes b. Yes b. No38. a. False 39. a. True
b. No b. Yes40. Answers will vary.
a. 3 � (2 � 1) � 3 � 3b. 4 � 3 � (2 � 2) � 3c. (8 � 8 � 8 � 8) � 8 � 8d. 3 � 3 � 3 � (3 � 3) � 1e. 3 � (3 � 3) � (3 � 3) � 0f. 0 � (12 � 3 � 16 � 8) � 0
Applying Skills41. Steve can calculate a 15% tip by adding 10% of
the fare and 5% of the fare using the distributiveproperty of multiplication over addition.
10 A 12 1 15 B 5 10 3 1
2 1 10 3 15
71218
5089
23 571
111
73 or 21
321p
37
237221
3
259 or 20.52
5
219
2 110
117
"5
"7
23
23
42. a. Commutative property of multiplicationb. $17.50
2-3 Addition of Signed Numbers (pages 58–59)Writing About Mathematics
1. Negative.The sum is the difference of the absolutevalues of the numbers.
2. Positive. The sum of two negative numbers isnegative, so one number must be positive.
3. Positive.A positive number must be added in orderfor this sum to be larger than the negative number.
4. Negative. The sum of two positive numbers ispositive, so one number must be negative.
5. Yes.Answers will vary. Example: (�5) � (�8) ��13
Developing Skills6. �10 7. �5 8. 09. �10 10. �2 11. �45
12. �12 13. 0 14. �0.6915. 16. 17.18. �9 19. �35 20.21. 22. 23. 024. �1 25. �7 26. 0.627. 25.4Applying Skills28. �2° C 29. 19th floor 30. �3 yd31. �$230 32. �$0.98
2-4 Subtraction of Signed Numbers(pages 62–63)Writing About Mathematics
1. The expression can have either meaning since �8 � 12 � �8 � (�12) � �4.
2. The difference plus the subtrahend should equalthe minuend.
Developing Skills3. �12 4. �174 5. �146. �5.1 7. �3.43 8. �539. 10. 11. �32
12. �1.56 13. �81 14. �5915. 16. �3 17. �11118. �64 19. �25 20. �2721. �3 22. �12 23. �2624. �15 25. a. False
b. False26. a. No
b. Yes. The values are equal, both 0, when x � y.c. They are opposites.d. No
215912
157123
4
232121121
2
23858
157123
421323
294
14271AKTE1.pgs 9/25/06 10:38 AM Page 294
27. a. False 28. No. Problem 27 gives two b. False counter-examples.
Applying Skills29. a. �3° 30. �110 m 31. 185 points
b. �28°c. �12°d. �16°
32. 114° 33. 0.25 km
2-5 Multiplication of Signed Numbers (page 67)Writing About Mathematics
1. Javier must first multiply two of the negativenumbers, which will result in a positive number.Then he must multiply this positive number bythe third negative number, and the product ofa positive number and a negative number isnegative; (�5)(�4)(�2) � (�20)(�2) � �40
2. Since �3(�4) is the opposite of 3(�4), its productis the opposite of 12, or �12.
Developing Skills3. �102 4. �162 5. �2436. �345 7. �16 8. �3.249. �0.13 10. 11.
12. 13. �360 14. �0.37215. �112 16. �120 17. �518. �2 19. �9 20. �921. �125 22. �16 23. �1624. �16 25. �0.25 26. �0.2527. �0.25 28. 2729. Distributive property of multiplication over addition30. Commutative property of addition31. Associative property of addition32. Zero property of addition33. Commutative property of multiplication34. Associative property of multiplication
2-6 Division of Signed Numbers (pages 70–71)Writing About Mathematics
1. The quotients are reciprocals.2. Yes, if x and y are opposites.
Developing Skills
3. 4. 5. 16. �1 7. �2 8. �109. 10. 11. �7
12. �3 13. �1 14. 015. 16. �30 17. �0.118. �10 19. �40 20. Undefined21. �2 22. 50 23. �3624. 25. 26. 1
224321
8
113
11x24
3
21511
6
24623
1278255
9
27. a. False 28. a. Trueb. False b. Truec. False c. False
d. False
2-7 Operations with Sets (page 74)Writing About Mathematics
1. Yes. If two lines are parallel, their intersection isthe empty set.
2. No. The number 1 is a member of the countingnumbers. However, it is neither a prime numbernor a composite number, so it does not belong totheir union.
Developing Skills3. {3} 4. {1, 2, 3, 4, 5, 6} 5. {1, 3}6. {1, 2, 3, 4, 6} 7. {3, 4, 6} 8. {1, 3, 4, 5, 6}9. {3, 4, 5, 6} 10. or { } 11. {1, 2, 3, 4, 5, 6}
12. {2, 3, 4} 13. {1, 3, 4} 14. {1, 3, 4, 5}15. {1, 2, 5} 16. {5} 17. {1, 3, 4, 5}18. {3, 4} 19. {1, 2, 3, 4} 20. {2, 4, 8}21. {2, 6, 8} 22. or { } 23. a. 5 b. 3
c. 2 d. 024. a. � {4, 6, 10} b. � {2, 8, 12}
c. � {2, 6, 8, 12} d. � {4, 10}25. a. (1) A y B � ; y � ; x � U
(2) A y B � ; y � {1, 3, 5, 7}; x � U(3) A y B � ; y � ; x � U(4) A y B � ; y � {1, 3, 5, 6, 7, 8};
x � Ub. The set y is the intersection of the
complement of A and the complement of B.c. If A and B are disjoint, x is equal to the
universe, U.
2-8 Graphing Number Pairs (page 80–81)Writing About Mathematics
1.
The length of AC � 5 � (�2) � 5 � 2 � 7,so the length of BC is 7 units long. Since �C
y
x
1O
1–1–1
B(5, 3)
C(5, –4)A(–2, –4)
BA
BABA
BABABA
BABABABA
BBAA
295
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is a right angle, BC must be perpendicular toAC, or parallel to the y-axis. 7 units verticallyfrom point C is (5, 3) or (5, �11). Point B isin quadrant I, so it has coordinates (5, 3).
2.
Phyllis can find the areas of �DEF and �DGF.In �DEF, base DF � 3 � (�2) � 3 � 2 � 5and height OE � 5 � 0 � 5. Then the area of�DEF � (5)(5) � 12.5. In �DFG, base DF � 5 and height OG � 4 � 0 � 4. Then thearea of �DFG � (5)(4) � 10. The area ofDEFG � area of �DEF � area of �DFG �
12.5 � 10 � 22.5 square units.Developing Skills
3. A(1, 2)B(�2, 1)C(�2, �1)D(2, �2)E(2, 0)F(0, 1)G(�1, 0)H(0, �2)O(0, 0)
4–15.
16. I 17. III 18. II19. IV 20. I
(5, 7)(1, 6)
(0, 4)
(0, 0) (5, 0)
(4, –4)
(2, –6)(0, –6)(–4, –5)
(–3, 0)
(–3, 2)
(–8, 5)
x
y
O–1–1
1
1
12
12
y
x
E(0, 5)
D(3, 0)
G(0, –4)
F(–2, 0) O1
–1–1
1
21. Graphs will vary; 0
22. Graphs will vary; 0
23. (0, 0)Applying Skills24. a.
b. Right triangle c. 14 sq units25. a.
b. Rectangle c. 20 sq units
y
x
S R
P Q
1
1O
y
x
C
BAO
1
–1–1 1
y
x
(0, 2)
(0, 0)
(0, –5)
O–1–1
1
1
y
x
(4, 0)(1, 0)(–3, 0) 1
–1–1 1O
296
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26. a.
b. Parallelogram c. 20 sq units27. a.
b. Square c. 16 sq units28. a.
b. Isosceles triangle c. 21 sq units29. a.
b. Trapezoid c. 24 sq units
y
x
R
F
A
M
–1 1O–1
y
x
N
H
E
–1 1O
y
x
E M
H O
1
–1
y
x
L A
F C
1
–1–1 1O
30. a.
b. Rectangle c. 20 sq units31. a.
b. Parallelogram c. 6 sq units32. a.
b. Right triangle c. 10 sq units33. a.
b. Square c. 16 sq units
y
x
L
M
K
I
1
1O
y
x
A
M
RO
–1–1
1
1
y
x
N
O
D
P –1–1
y
x
N
A
R
B
–1
1
1O
297
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34. (1, 4)
35. R(�1, �1), S(2, �1) or R(�1, �7), S(2, �7)
36. a.
b. 24 sq units37. a.
b. 6 sq units
y
x
LA
N P
E T
y
xSA
T
R
O1
–1–1 1
y
x
QP
O1
–1–1 1
y
x
C
A BO
1
–1–1 1
Review Exercises (pages 83–84)1. 8 2. �68 3. �64. �4 5. 0.0256 6. 1007. 196 8. 569. 2; commutative property of addition
10. 2; associative property of addition11. 1; multiplication property of one12. 0; multiplication property of zero13. 5; distributive property of multiplication over
addition14. 5; commutative property of multiplication15. {2, 4} 16. {1, 2, 4, 5, 6} 17. {3, 6}18. {6} 19. or { } 20. {1, 2, 3, 4, 5, 6}21. {1, 5} 22. {1, 2, 3, 4, 5} 23. {3}24. {1, 3, 5, 6} 25. 0 26. 1227. �1.3 28. �1 29. 2330. 2 31. 125 32. 733. g 34. h 35. a36. e 37. b 38. d39. f 40. c 41. a. 23,232
b. 12,22142. a. B(3, �4), D(�2, 5)
b. 45 sq units43. a. �15
b. 8c. No. Since R is an integer and �4 is an integer,
or must be an integer.There is nocombination of R and W that will make S � �4.
d. Yes. There are many possible combinations:R � 8, W � 48; R � 7, W � 44; R � 6, W � 40;R � 5, W � 36; etc.
44. 30 � 5 � 8 � 17 � 3 � 47-yard line
Exploration (page 85)STEP 1. 23,568 � 20,000 � 3,000 � 500 � 60 � 8
� 2 � 10,000 � 3 � 1,000 � 5 � 100� 6 � 10 � 8
STEP 2. 2 � 10,000 � 3 � 1,000 � 5 � 100 � 6 � 10 � 8� 2 � (9,999 � 1) � 3 � (999 � 1) �
5 � (99 � 1) � 6 � (9 � 1) � 8
60 2 R4
W4
y
x
CD
A B
O1
–1–1 1
298
14271AKTE1.pgs 9/25/06 10:38 AM Page 298
STEP 3. 2 � (9,999 � 1) � 3 � (999 � 1) � 5 � (99 � 1)� 6 � (9 � 1) � 8� (2 � 9,999 � 2) � (3 � 999 � 3) �
(5 � 99 � 5) � (6 � 9 � 6) � 8STEP 4. (2 � 9,999 � 2) � (3 � 999 � 3) � (5 � 99 � 5)
� (6 � 9 � 6) � 8� (2 � 9,999 � 3 � 999 � 5 � 99 � 6 � 9)
� (2 � 3 � 5 � 6 � 8)STEP 5. The expression involving the digit terms and
the original number have the same digits.STEP 6. The expression involving the 9s is divisible
by 3 because it is the sum of products thatare divisible by 3. Each product is divisibleby 3 because it is a multiple of a numbercomposed entirely of 9s.
STEP 7. Yes. The expression involving 9s will alwaysbe divisible by 3. The sum of two numbersdivisible by 3 is also divisible by 3.Therefore, if the expression involving thedigit terms is divisible by 3, the entireexpression will be divisible by 3.
STEP 8. A number is divisible by 3 if the sum of itsdigits is divisible by 3.
Cumulative Review (pages 85–87)Part I
1. 3 2. 3 3. 34. 1 5. 4 6. 37. 2 8. 1 9. 3
10. 4Part II11. {$4.91, $4.92, $4.93, . . . , $4.99}; this is a finite set
because there are exactly 9 members. Mrs. Lingcannot spend a partial cent.
12. 16 players; seven teams have the minimumnumber of players and the eighth team has therest: 7 � 12 � 84, 100 � 84 � 16
Part III13. a. Even counting numbers or add 2 to the
previous number in the sequence
b. Sum of the previous two numbers in thesequence
c. Sum of the previous three numbers in thesequence or sum of all previous numbers inthe sequence
14. 4(�7 � 3) � 8 � (�2 � 6)2
� 4(�4) � 8 � (4)2
� 4(�4) � 8 � 16� �16 ��
Part IV15. In �ABC, base � 5 � (�2) � 5 � 2 � 7 and
height � 4 � (�3) � 4 � 3 � 7. Therefore, the area of �ABC � square units.
16. Diagram is optional.
a. 50 personsb. 90 personsc. 70 persons
6:00 11:00
9050
70
40
y
x
C
A B
O1
–1 1
12(7)(7) 5 24.5
21612
12
299
3-1 Using Letters to Represent Numbers(page 91)Writing About Mathematics
1. Addition is commutative.2. Subtraction is not commutative.
Developing Skills3. y � 8 4. 4 � r 5. 7x6. x � 7 or 7x 7. or x � 10 8. or 10 � x10
xx10
9. c � 6 10. 11. xy
12. d � 5 13. or 8 � y 14. y � 10 or 10y
15. w � t 16. 17. 2(p � q)18. m � 4 19. 5x � 2 20. 10 � 2a21. n � 2 22. n � 20 23. 8 � n24. n � 6 25. n � 2 26. 3n27. 28. 4n � 3 29. 2n � 330. 10n � 2 31. b � 100 32. h � 233. 0.39n 34. d
12
34n or 3n
4
13z or z3
8y
110w or w
10
Chapter 3. Algebraic Expressions and Open Sentences
14271AKTE1.pgs 9/25/06 10:38 AM Page 299
3-2 Translating Verbal Phrases Into Symbols(pages 93–94)Writing About Mathematics
1. a. d � bb. No; Examples will vary: If d � 5 and b � 2,
d � b � 2.5.2. a. c � x
b. n(c � x)c. No. In part a, if x does not divide into c evenly,
you will get a decimal answer. In part b, if xdoes not divide into c or n evenly, you willget a decimal answer. However, these answersare only applicable if we allow for fractions ofa penny.
Developing Skills3. x � 200 4. 1,000 � d 5. 5x6. 7. c � 12 8. 100 � x9. 0.39x 10. tg 11. 45 � x
12. c � d 13. 250 � y 14. c � 2515. w � 8 16. 2x � 3 17. 550h18. 5r 19. a. 7w � 5
b. 7w � d
Applying Skills20. or c � m 21. 6x � 14y22. 0.45 � 0.09(m � 3)23. 0.75 � 0.06(c � 8)24. 5.00 � 0.75(6) �0.55(g � 9)
3-3 Algebraic Terms and Vocabulary (pages 96–97)Writing About Mathematics
1. Yes; (ab)2 � (ab)(ab) � (ab)(ba) � a(bb)a �(aa)(bb) � a2b2
2. No. Examples will vary: (1 � 2)2 � 32 � 9;12 � 22 � 5
Developing Skills3. x, y, xy 4. 3, a, 3a5. 7, m, n, 7m, 7n, mn, 7mn6. s, t, st 7. 8 8. 79. 10. 1 11. �1.4
12. 7 13. 3.4 14. �115. Base � m, exponent � 216. Base � s, exponent � 317. Base � t, exponent � 118. Base � �a, exponent � 419. Base � 10, exponent � 620. Base � 5y, exponent � 421. Base � x � y, exponent � 522. Base � c, exponent � 323. b5 24. �r2 25. a4b2
26. 7r3s2 27. (6ab)3 28. (a � b)3
12
cm
l2
29. (m � 2n)4 30. r � r � r � r � r � r31. 5 � x � x � x � x32. 4 � a � a � a � a � b � b33. 3y � 3y � 3y � 3y � 3y34. Coefficient � �3, base � k, exponent � 135. Coefficient � �1, base � k, exponent � 336. Coefficient � �, base � r, exponent � 237. Coefficient � 1, base � ax, exponent � 5
38. Coefficient � , base � y, exponent � 139. Coefficient � 0.0004, base � t, exponent � 1240. Coefficient � , base � a, exponent � 441. Coefficient � 1, base � �b, exponent � 3Applying Skills42. Cost of 5 cans of soda43. 3 times the speed of a car or distance traveled
at r mph in 3 hours44. Number of Alice’s CDs after she gave away 545. Number of weeks until the end of the year46. Perimeter of the square47. Diameter of the circle48. Number of months in the school year49. Cost of 1 bottle of water50. Score for 7 field goals
3-4 Writing Algebraic Expressions inWords (pages 99–100)Writing About Mathematics
1. a. (4 � n) � (4 � n) � 8 booksb. The domain is the set of whole numbers
greater than or equal to 0 but less than orequal to 4 so that both 4 � n and 4 � n willbe positive whole numbers.
2. Yes, 0.01 represents one cent and x would allow usto have whole number multiples of one cent. Every-thing we buy costs some multiple of one penny.
Developing SkillsIn 3–14, answers will vary.
3. a. The second route is 0.2 miles shorter than thefirst.
b. The set of real numbers greater than 0.24. a. The cans of soda in the machine are $0.15
more expensive than the cans at the store.b. The set of whole numbers greater than 0
5. a. Alexander did homework for 10 minutes morethan 3 times the number of minutes he spentreading.
b. The set of real numbers greater than or equalto 0
6. a. The length of a rectangle is 8 meters morethan twice its width.
b. The set of rational numbers greater than 0
32
"2
300
14271AKTE1.pgs 9/25/06 10:38 AM Page 300
7. a. Abby’s lunch is the length, in hours, of herclasses and her sports are the length, inhours, of her classes.
b. The set of rational numbers greater than orequal to 0
8. a. Jen’s drive to and from work takes thenumber of hours she spends at work.
b. The set of rational numbers greater than orequal to 0
9. a. Her son’s golf score was 10 strokes higherAlicia’s.
b. The set of whole numbers greater than orequal to 18
10. a. Tom paid 30 cents more than 5 times the costof a pen for a notebook.
b. The set of whole numbers greater than 011. a. Dominic’s essay had 80 more than the
number of words in Seema’s.b. The set of whole number multiples of 4
12. a. Anna read 5 fewer than 3 times the number ofbooks Virginia read.
b. The set of whole numbers greater than orequal to 2
13. a. Mario’s score is 220 less than Pete’s.b. The set of integers
14. a. Agatha now walks 10 fewer than 3 times thenumber of minutes she used to walk.
b. The set of rational numbers greater than 3
3-5 Evaluating Algebraic Expressions (pages 103–104)Writing About Mathematics
1. a and b represent different values in differentproblems. 12 always has the same value.
2. Parentheses were needed to clearly indicatemultiplication. Otherwise, one might interpretthe problem as 50 � 37 as opposed to 50 � 3 � 7.
Developing Skills3. 40 4. �2 5. 0.156. 11 7. �8 8. 1289. �6 10. �21 11. �3.5
12. �36 13. 2 14. �1.215. �48 16. 2.25 17. 218. 91 19. 64 20. 621. �26 22. �0.72 23. 8.524. �12 25. �16,807 26. 127. 31Applying Skills28. a. $35.50 29. a. $66.25 30. a. 46 ft
b. $75.50 b. $403.75 b. 124 ftc. $42.20 c. 234 ft
13
34
112
13
16 31. a. $9.60 32. a. 60 shrubs
b. $14.60 b. 112 shrubsc. $26.70
3-6 Open Sentences and Solution Sets(pages 106–107)Writing About MathematicsIn 1–3, answers will vary
1. {0, 1, 2, 3} or {real numbers less than or equal to 5}2. {1, 2, 3, 4, 5, 6}3. {real numbers greater than 5}
Developing Skills4. True sentence 5. Open sentence6. Algebraic expression 7. True sentence8. Open sentence 9. True sentence
10. Algebraic expression 11. Open sentence12. x 13. y 14. r 15. a16. {4} 17. {�5, �4, �3, �2, �1,
0, 1, 2, 3, 4, 5}18. {3} 19. or { }20. {�5, �4, �3, �2, �1, 21. or { }
0, 1, 2, 3}22. {�5, �4, �3, �2, �1} 23. {�5, �4, �3}24. a. {1, 2, 3, . . . , 10} 25. a. {0, 1, 2, 3, 4, 5}
b. {1, 2, 3, 4, 5, 6, 7} b. {0, 1, 2, 3, 4, 5}c. 1, 2, 3, 4, 5, 6, or c. 0, 1, 2, 3, 4, or
7 pencils 5 cans26. a. {whole numbers}
b. {0, 1, 2, 3, 4, 5}c. 0, 1, 2, 3, 4, or 5 times
3-7 Writing Formulas (pages 108–110)Writing About Mathematics
1. Agree. In a recipe, adding specific ingredientstogether equals a new product.
2. a. No. An algebraic expression does not state anequality; there is no equal sign.
b. Yes. Every sentence that contains a variable isan open sentence.
Developing Skills3. l � 10m 4. S � c � m 5. P � 2l � 2w6. m � 7. A � bh 8. A � s2
9. V � e3 10. S � 6e2 11. S � 4�r2
12. r � or 13. F � 32 � 14. C � (F � 32)r � d � t
15. D � dq � r 16. T � 0.08v 17. F � s � 0.02v
Applying Skills18. a. C � 20 � dn b. $80 c. $20
59
95Cd
t
12
a 1 b 1 c3
301
14271AKTE1.pgs 9/25/06 10:38 AM Page 301
19. a. C � x if m � 3;C � x � y(m � 3) if m � 3
b. $0.25 c. $0.6020. a. D � a if p � 1;
D � a � b(p � 1) if p � 1b. $1.00 c. $3.40
21. a. P � 0.12n if n � 25,000;P � 3,000 � 0.15(n � 25,000) if n � 25,000
b. $2,520 c. $3,75022. a. B � 25s if s � 20
b. B � 400 � 65(s � 20) if s � 20c. $450 d. $725
23. a. E � 0 b. E � 0.25(c � 36)c. 44 cookies
Review Exercises (pages 111–113)1. An algebraic expression has no sign of equality
or inequality, whereas an open sentence does.2. In the term 2a2, 2 is not squared because it is the
coefficient of the power a2. In the term (2a)2, 2 issquared because it is part of the base.
3. or x � b 4. r � 4 5. q � d6. 2g � 3 7. 475 8. 259. 35 10. 0 11.
12. 13. 14. 2515. C � 5n � 25q 16. 2 17. 318. y 19. d � rt 20. 48.521. 49 22. 48.423. a. 5 fewer than twice the winning team’s runs
b. 4 to 3 or 3 to 1; yes c. 3 and 424. a. {counting numbers} b. {1}25. Let C represent the cost of delivery and
m represent the number of miles.C � 25 � 3m if m � 10;C � 55 � 4.5(m � 10) if m � 10
26. a. 14b. Each term in the list is found by adding 3 to
the term before it.c. 74
27. 3 is prime; 6 is even; 9 is a perfect square; 35 isdivisible by 5.
28. 2
Exploration (page 113)STEP 1. Results will vary.STEP 2. Results will vary.STEP 3. If the sum of the hundreds digit and the
ones digit is equal to the tens digit, or if thesum minus 11 is equal to the tens digit, thenthe number is a multiple of 11.
STEP 4. Results will vary.STEP 5. Results will vary.
21423
8
365 or 71
5
xb
STEP 6. Add the hundreds digit and the ones digit.Add the thousands digit and the tens digit.If either sum is greater than or equal to 11,subtract 11. If the two revised sums areequal, then the number is a multiple of 11.
STEP 7. Add the even digits and the odd digits. Ifeither sum is greater than or equal to 11,subtract 11. Repeat this process as neededuntil the sum is greater than or equal to 0and less than 11. If the two revised sums areequal, then the number is divisible by 11.
Cumulative Review (pages 114–115)Part I
1. 4 2. 2 3. 3 4. 3 5. 46. 4 7. 4 8. 3 9. 1 10. 1
Part II11. 32 students; diagram is optional.
12. 716,539; 83 � 89 � 97 � 716,539Part III
13. 68.7 cm2;14. a. Add 1 to the first number, 2 to the second
number, 3 to the third number, . . .b. Double the previous number, or add 1 to the
sum of all numbers before it.c. Add 1 to the first number, add (or multiply)
2 to the second number, add 1 to the thirdnumber, . . .
Part IV15. 15; � 3(b � 2) � � 3(�5 �2)
� � 3 � � 7 � 36 � �21� 15
16. Let M represent all the material. Then represents the skirt. The vest is represented by .
a. yd;
b.
c. 334 yd; 12 A 15
2 B 5 154 or 33
4
212 yd; 13 A 15
2 B 5 52 or 21
2
M 2 12M 2 13M 5 54
16M 5 54
M 5 152 or 71
2
712
23 A 1
2M B 5 13M
12M
1213
7 2 (25)13
a 2 bc
"34 (12.6)2 5
"34 ? 158.76 < 68.7
Fiction Nonfiction
23
22
32 3
302
303
4-1 Solving Equations Using More ThanOne Operation (pages 121–122)Writing About Mathematics
1. No. The sum of 2 times a positive number and 5cannot be 0; 2x � 5 � 5 � 0 � 5, so 2x � �5 andx must be negative.
2. a. Yes, x � � ; the next step would be toadd to both sides (or subtract fromboth sides).
b. Answers will vary.Developing SkillsIn 3–4, answers will vary.
3. 3x � 5 � 35 Given(3x � 5) � (�5) � 35 � (�5) Addition property
of equality3x � [5 � (�5)] � 35 � (�5) Associative prop-
erty of addition3x � 0 � 30 Additive inverse
property3x � 30 Addition property
of zero Multiplicationproperty of equalityAssociative property ofmultiplication
1x � 10 Multiplicativeinverse property
x � 10 Multiplicativeidentity property
4. GivenDefinition ofsubtractionAddition propertyof equalityAssociative prop-erty of additionAdditive inversepropertyAddition propertyof zeroMultiplicationproperty of equalityAssociative propertyof multiplication
1x � 32 Multiplicativeinverse property
x � 32 Multiplicativeidentity property
C2 A 12 B Dx 5 2(16)
2 A 12x B 5 2(16)
12x 5 16
12x 1 0 5 16
12x 1 f(21) 1 1g 5 15 1 1
C12x 1 (21) D 1 1 5 15 1 1
12x 1 (21) 5 15
12x 2 1 5 15
C13(3) Dx 5 13(30)
13(3x) 5 1
3(30)
157215
7
717
157
5. a � 8 6. c � 3 7. x � 28. t � 9. a � 12 10. d � �2
11. y � 7 12. a � 32 13. x � �1214. y � 16 15. t � 18 16. m � �5017. m � 9 18. a � �16 19. y � �2520. d � 2 21. a � 1.2 22. t � 1.423. x � �24 24. y � �12 25. t � 5026. c � �40 27. x � 28. c �
29. x � 13 30. r � 1.25 31. w � �2432. m � 9Applying Skills33. 15° C 34. $4.95 35. 187 mi36. 24 stamps
4-2 Simplifying Each Side of an Equation(pages 127–128)Writing About Mathematics
1. Irene uses x to represent the smaller number whileHenry uses x to represent the larger number.
2. It is impossible to determine the answer.If x � 13.5, then x is the smaller number and27 � x is the larger number. If x � 13.5, thenx and 27 � x have the same value. If x � 13.5,then x is the larger number and 27 � x is thesmaller number.
Developing Skills3. x � 13 4. x � 25 5. x �
6. c � 20 7. x � �4 8. x � �139. x � 2 10. y � 19 11. c � 4
12. c � �2 13. y � �5 14. c � 715. t � 8 16. x � 3 17. m � 1518. x � 11 19. b � 9 20. c � 321. r � 2 22. b � 4 23. m �24. y � 1 25. a � �17 26. r � 527. a � 2 28. x � 4Applying SkillsIn 29–33, answers to parts a–e and g will vary.29. a. 6 yd of material; difference between lengths of
pieces is 1.5 ydb. x � shorter piece of materialc. x � 1.5 � longer piece of materiald. x � (x � 1.5) � 6e. x � 2.25f. 2.25 yd, 3.75 ydg. 2.25 yd � 3.75 yd � 6 yd;
3.75 yd � 2.25 yd � 1.5 yd30. a. Won 8 games more than lost; played 78 games
b. x � games lostc. x � 8 � games won
73
35
67
12
13
213
Chapter 4. First-Degree Equations and Inequalities in One Variable
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d. x � (x � 8) � 78e. x � 35f. 35 gamesg. 78 games � 35 games lost � 43 games won;
43 games won � 35 games lost � 831. a. Saved $20 more this month than last month;
saved total of $70b. x � saved last monthc. x � 20 � saved this monthd. x � (x � 20) � 70e. x � 25f. $25 last month, $45 this monthg. $45 this month � $25 last month � $20;
$25 � $45 � $7032. a. 100 passengers on 3 buses; 2 buses carry
4 fewer passengers than third busb. x � passengers on the third busc. x � 4 � passengers on other busesd. x � 2(x � 4) � 100e. x � 36f. 36 passengers, 32 passengers, 32 passengersg. 36 � 32 � 32 � 100 passengers;
32 passengers � 36 � 4 passengers33. a. of the seats were filled; 31,000 empty seats
b. x � seating capacityc. x � filled seatsd. x � 31,000 � xe. x � 155,000f. 155,000 seatsg. (31,000 seats) � 124,000 filled seats;
155,000 seats � 124,000 filled seats � 31,000empty seats
4-3 Solving Equations That Have theVariable in Both Sides (pages 132–133)Writing About Mathematics
1. Yes. Milus is using the multiplication property ofequality.
2. Yes;Developing Skills
3. x � 2 4. x � 4 5. c � 76. y � �10 7. d � �12 8. y � �89. m � 40 10. y � 9 11. x � �18
12. x � 96 13. a � 20 14. c � �2715. x � 9 16. m � �50 17. c � �818. r � 10 19. y � 11 20. x � �721. x � 0 22. x � 7 23. y � 424. c � 7 25. d � �18 26. y � �1727. m � �0.5 28. x � 10 29. b � 230. t � 15 31. n � �1 32. c � �533. a � 20 34. x � 10 35. m � 936. a � �1
x 5 x4 1 15
45
45
45
45
In 37–42, answers to part a will vary.37. a. 8n � n � 35 38. a. 6n � 3n � 24
b. 5 b. 839. a. 3n � 22 � 7n � 14 40. a. 3(2s �1) � 5s �10
b. 9 b. 7, 1541. a. f � (f � 6) � 2f � 26 42. a. t � (2t � 1) � 5
b. 5, 11, 10 b. 12, 11, 6Applying Skills43. 500 mi, 425 mi 44. $660, $79045. 14 five-dollar bills, 46. $40, $60
7 ten-dollar bills47. 348 mi 48. 2 hr49. 1.5 mi 50. $15
4-4 Using Formulas to Solve Problems(pages 137–141)Writing About Mathematics
1. 2.5r represents the total time, 2.5 hours, Sabrinadrove on her trip times her speed, r, on localroads. 30 represents the additional mileageSabrina accrued by driving for two hours at thehigher interstate speed.
2. Antonio should represent the local road speedas r � 15 since it is 15 miles per hour slower thanSabrina’s interstate speed, r.
Developing Skills3. P � perimeter of a triangle, a, b, c � sides;
c � 48 in.4. P � perimeter of a square, s � side; s � 8.0 m5. P � perimeter of a square, s � side; s � 1.7 ft6. P � perimeter of a rectangle (parallelogram),
l � length, w � width; w � 5 yd7. P � perimeter of an isosceles triangle, a � leg,
b � base; b� 20 cm8. P � perimeter of an isosceles triangle, a � leg,
b � base; a � 6.4 m9. A � area of a rectangle (parallelogram), b � base,
h � height; b � 16 cm10. A � area of a rectangle (parallelogram), b � base,
h � height; h � 4.0 m11. A � area of a triangle, b � base, h � height;
h � 6.0 ft12. V � volume of a rectangular prism, l � length,
w � width, h � height; w � 8.0 yd13. d � distance, r � rate, t � time; r � 40 mph14. I � interest, p � principal, r � rate, t � time;
p � $1,80015. I � interest, p � principal, r � rate, t � time;
r � 2.3%16. T � total cost, n � number of items, c � cost of
one item; n � 4 items
304
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17. T � total cost, n � number of items, c � cost ofone item; c � $0.49
18. S � salary earned, n � number of hours worked,w � hourly wage; w � $6.40
19. S � salary earned, n � number of hours worked,w � wage; n � 37.5 hr
20. P � 2l � 2w; l � 11.6 cm21. P � a � b � c; a � 21 in., b � 19 in., c � 33 in.22. P � 2a � b; a � 30 cm, b � 33 cm23. P � 2l � 2w; l � 19 m, w � 14 m24. P � 2l � 2w; l � 34 yd, w � 34 yd25. P � 3a, P � 4s; a � 20 cm, s � 15 cm26. P � 4s, P � 2l � 2w; l � 5 cm, w � 3 cm27. A � bh; h � 9 cm 28. A � bh; b � 3.5 m29. P � 2l � 2w; l � 64 in., w � 32 in.30. P � 2l � 2w; l � 10 ft, w � 6 ft31. P � 4s; P � 3a; a � 8 m32. P � 6a, P � 4s; a � 8 in., s � 12 in.Applying Skills33. l � 57 m, w � 16 m 34. l � 78 ft, w � 36 ft35.
Number Value of Total of Coins One Coin Value
Dimes x $0.10 $0.10x
Quarters 25 � x $0.25 $0.25x
a. 25 � x b. 0.10x c. 0.25(25 � x)d. 4.90 � 0.10x � 0.25(25 � x); 9 dimes, 16 quarterse. 9 � 16 � 25; 0.10(9) � 0.25(25 � 9) � 4.90
36. There is either a mistake in the value of the coinsor in the total number of coins.
37. 59 dimes
Number Value of Total of Coins One Coin Value
Nickels x $0.05 $0.05x
Dimes 84 � x $0.10 $0.10(84 � x)
38. 17 stamps39.
Hours Wages Worked per Hour Earnings
Monday–Friday x $8.50 $8.50x
Saturday 38 � x $12.75 $12.75(38 � x)
a. 38 � x b. 8.50xc. 12.75(38 � x) d. 4 hr
40. 7 hr
Hours Wages Worked per Hour Earnings
Monday–Friday x $6.00 $6.00x
Saturday 42 � x $9.00 $9.00(42 � x)
12
12
41. Week 1, 34 hr; week 2, 38 hr42.
Rate Time Distance
First part of the trip r 1.5 hr 1.5r
Last part of the trip r � 26 3 hr 3(r � 26)
a. 1.5r b. r � 26c. 3(r � 26) d. First, 66 mph; last, 40 mph
43. Walked, 20 min ; drove, 30 min
Rate Time Distance
Walked 3 mph t 3t
Rode 30 mph
44. 65 mph, 2.4 hr (2 hr, 24 min);55 mph, 0.6 hr (36 min)
45. Shelly, 45 mph; Jack, 60 mph46. Carla, 56 mph; Candice, 42 mph47. mi 48. hr (12 min); 9 mi
4-5 Solving for a Variable in Terms ofAnother Variable (page 143)Writing About MathematicsIn 1 and 2, answers will vary.
1. 2(x � 1) � 8 2. 10y � 12 � 4y
Developing Skills3. x � 4. x � 5. y �
6. y � 7. x � 2r 8. x � 3a
9. y � 8c 10. x � k � 4 11. y � 9 � d
12. x � 2q 13. x � 3r 14. y �
15. x � 16. x � 17. y �
18. x � 19. x � 9b 20. x �
21. x � s 22. x � 15 23. x � 6a24. x � 2a
4-6 Transforming Formulas (pages 145–146)Developing SkillsIn 1–14, answers will vary depending on simplification.
1. s � 2. h � 3. r �
4. l � 5. r � 6. t �
7. h � 8. B � 9. g �
10. b � P � 2a 11. a � 12. w �
13. C � (F � 32) 14. a �
15. a. h �
b. (1) 27.9 cm(2) 17.8 cm(3) 7.0 cm
c. 27.9; it is the only container height greaterthan or equal to 20 cm.
Vpr2
2Sn 2 l
59
P2 2 lP2b
2
2st
3Vh
2Ab
Ipr
Pb
vwh
dt
Ab
P4
c2
m2 2 n
t 2 rs
8cd
2ba
5dc
mh
sr
8s
b5
1521
2
30 A 56 2 t B5
6 2 t
A 12 hr BA 1
3 hr B
305
14271AKTE1.pgs 9/25/06 10:38 AM Page 305
16. a. t � ; 1.5 hr, 1.7 hr, 2.9 hr
b. Arrival Departure
Buffalo — 9:00 A.M.
Rochester 10:30 A.M. 11:00 A.M.
Syracuse 12:42 P.M. 1:12 P.M.
Albany 4:06 P.M. —
4-7 Properties of Inequalities (page 150)Writing About Mathematics
1. No. If x is a positive number, then 5x � 4x.However, if x is a negative number, then 5x � 4x,and if x is zero, then 5x � 4x.
2. Yes. By the addition property of inequality,x � a � x � b and x � b � y � b. By thetransitive property of inequality, x � a � y � b.
3. No. If the difference of x and y equals thedifference of a and b, then the x � a � y � b.If the difference of x and y is less than thedifference of a and b, then the x � a � y � b.
Developing Skills4. � 5. � 6. �7. � 8. � 9. �
10. � 11. � 12. �13. � 14. � 15. �16. � 17. � 18. �19. � 20. � 21. �22. � 23. � 24. �25. � 26. � 27. �28. � 29. � 30. �31. �
4-8 Finding and Graphing the Solution Setof an Inequality (pages 155–156)Writing About Mathematics
1. If the inequality represents the minimum numberof cans you need to buy to get the sale price,then the solution set has a smallest value becausethe domain is the set of whole numbers. If theinequality represents the minimum number ofmiles you need to run to earn money at a fund-raiser, then the solution set does not have asmallest value because the domain is the set ofreal numbers.
2. No. The solution set does not include 4.Developing Skills
3. x � 6
3 4 5 6 7 8 9 10 11 12
dr 4. z � 10
5. y �
6. x � 5
7. x � 3
8. y � 2
9. d � 3
10. c � �4
11. y 8
12. d 3
13. t � 2
14. x 6
15. y 5
16. h �2.5
17. y � �4
18. y � �3
19. x � 2
–4 –3 –2 –1 0 1 2 3 4
–6 –5 –4 –3 –2 –1 0 1 2
–6 –5 –4 –3 –2 –1 0 1 2
–4 –3 –2 –1 0 1 2 3 4
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
–1 0 1 2 3 4 5 6 7
–1 0 1 2 3 4 5 6 7
4 5 6 7 8 9 10 11 12
–6 –5 –4 –3 –2 –1 0 1 2
–1 0 1 2 3 4 5 6 7
–1 0 1 2 3 4 5 6 7
–1 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7 8
–1 0 1 2 3 4 5 6 7
212
4 5 6 7 8 9 10 11 12
306
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20. r �10
21. x � 6
22. z �9
23. x � 2
24. z �4
25. x � 3
26. y 6
27. y 2
28. x � 1
29. y 6
30. x � �6
31. m 1.5
32. �3 � x � 3
33. �7 x � 5
34. 2 � x 5
35. x � 4
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
–4 –3 –2 –1 0 1 2 3 4
–2 –1 0 1 2 3 4 5 6
–8 –7 –6 –5 –4 –3 –2 –1 0
1 2 3 4 5 6 7 8 9
–3 –2 –1 0 1 2 3 4 5
–1 0 1 2 3 4 5 6 7
2 3 4 5 6 7 8 9 10
–1 0 1 2 3 4 5 6 7
–8 –7 –6 –5 –4 –3 –2 –1 0
–2 –1 0 1 2 3 4 5 6
–15–14–13–12–11–10 –9 –8 –7
2 3 4 5 6 7 8 9 10
–15–14–13–12–11–10 –9 –8 –7
36. (x � 1) or (x 5)
37. all real numbers
38. (2) 39. (1) 40. (3)41. (2) 42. (1, �) 43. (��, �1)44. (��, 2] 45. [�3, �) 46. (�3, 2]47. (��, �2] or (3, �)48. a.
b. x 2
4-9 Using Inequalities to Solve Problems(pages 159–160)Writing About Mathematics
1. Yes. The inequality �x � �3 is equivalent to x � 3. Because x � 3, x is also greater than everyvalue less than 3, including �3.
2. No.The solution set of x � �3 includes all valuesgreater than �3, while the solution set of �x � �3includes only the values greater than 3.
Developing Skills3. x 15 4. y 4 5. x 506. x � 50 7. 3y 30 8. 5x � 2x 709. 4x � 6 54 10. 2x � 1 13
11. 3x(x � 1) � 35 12. � 713. n � 6 � 4; n � 10 14. n � 6 � 4; n � 1015. 6n � 72; n � 12 16. n � 10 � 50; n � 4017. n �15 � 35; n � 50 18. 2n � 6 � 48; n � 2119. 5n � 24 � 3n; n � 12Applying Skills20. m � 100 550; $45021. b � (b � 80) 250; 165 balcony tickets22. s � (3s � 20) 120; $2523. 3n � 8 n � 40; 1624. 2w � 2(5w � 8) 104; 10 m25. 2w � 2(3w � 12) 176; 63 cm26. 8s � 300 1,500; $15027. n � (n � 2) � 98 � 2(n � 2); 24, 2628. 6h � 5 29; 4 hr29. 2 x � x � 2x 3
a. 31 min b. 44 min c. 89 min
Review Exercises (pages 162–164)1. Answers will vary. 2. w � 53. w � 15 4. h � 4 5. y � �16. a � �2 7. b � �2 8. x � �39. 10. x � bc � a 11.
12. 13. 14. x 5 cbax 5 2b
a 1 cx 5 6a 1 bc
x 5 b 2 acx 5 1
3
x3
0 1 2 3 4 5 6 7 8
–4 –3 –2 –1 0 1 2 3 4
–1 0 1 2 3 4 5 6 7
307
14271AKTE1.pgs 9/25/06 10:38 AM Page 307
15. 16. a. 17. w � 3.5b. h � 12
18. C � 2019. x � �3
20. x �1
21. x � 3
22. x �4
23. �2 � x 3
24. 3 x � 7
25. (x �2) or (x � 0)
26. 5 x � 9
27. Always; addition property of inequality28. Sometimes; if a � 0, then ax � ay; if a � 0, then
ax � ay29. Always; transitive property of inequality30. Never; multiplication property of inequality31. (1) 32. (1) 33. (3)34. 3s2 35. 7w � 436. Length � 24.5 ft, width � 6.5 ft37. At most, 22,310 lb38. a. More than 150 lb b. Answers will vary.39. a. At most 42 prints b. At least 43 prints40. a. Positive rational numbers with two decimal
placesb. 76 � 44h � 450; h � 8.5 c. {8.50}
2 3 4 5 6 7 8 9 10
–4 –3 –2 –1 0 1 2 3 4
0 1 2 3 4 5 6 7 8
–4 –3 –2 –1 0 1 2 3 4
–7 –6 –5 –4 –3 –2 –1 0 1
0 1 2 3 4 5 6 7 8
–5 –4 –3 –2 –1 0 1 2 3
–5 –4 –3 –2 –1 0 1 2 3
h 5 2Abx 5 c 2 2b
aExploration (page 164)If the circle has a radius of r, then the side of thesquare has a length of 2r. It follows that the area ofthe circle is �r2 and the area of the square is r2. Thefigure shows that the area of the circle is less thanthe area of the square, so �r2 � r2.
Cumulative Review (pages 164–166)Part I
1. 4 2. 4 3. 44. 1 5. 2 6. 47. 4 8. 4 9. 4
10. 3Part II11. 5.0 m, 5.0 m, 5.0 m, 13.0 m; let each of the three
equal sides � x and the fourth side � x � 8;3x � (x � 8) � 28, 4x � 8 � 28, x � 5, x � 8 � 13
12. a. C � (F � 32)b. 37° C; C � (98.6 � 32) � 37
Part III
13. No. Suppose that a prime number, p, dividedby 4 has a remainder of 2. This can be writtenas , where k is the whole numberpart of the quotient. The equation simplifies top � 4k � 2, which equals p � 2(k � 2). Thismeans p is even, but no prime number greaterthan 2 is even. Therefore, this is false.
14. 5 plums; let plum � x, pineapple � y, and peach �z; x � y � 3z, 2x � z, x � y � 3(2x), y � 5x
Part IV15. 58.4 cm2; A � (12.75 � 9.50) � 2.625(22.25) �
58.416. a. $12.67; 50 � 3s 12, s ; the smallest
rational number with two decimal placesgreater than or equal to is 12.67
b. $16.66; 50 � 3s 0, s ; the largestrational number with two decimal placesless than or equal to is 16.66162
3
1623
1223
1223
5.252
p4 5 k 1 12
59
59
308
5-1 Adding and Subtracting AlgebraicExpressions (pages 172–173)Writing About Mathematics
1. a. 3x � x � (3 � 1)x � 2xb. Answers will vary. Example: Let x � 2,
3(2) � 2 � 6 � 2 � 4 � 3
2. To add like terms, add the numerical coefficientsand keep the common term; to add like fractions,add the numerators and keep the commondenominator.
Developing Skills3. 15c 4. �10a 5. �15r
Chapter 5. Operations with Algebraic Expressions
14271AKTE1.pgs 9/25/06 10:38 AM Page 308
6. 0 7. �4ab 8. 7x9. �4y 10. 0 11. �3x
12. 13a � 3 13. 11b � 6 14. �c � 715. �x � 4 16. 3r � s 17. 14d2 � 4d18. �x � 8 19. �3y 20. �15y � 721. 7a � b 22. 2x2 � 13 23. 9y2 � 3y � 424. x3 � 5x2 � 9 25. �5d2 � 9d � 226. x � 15 27. x3 � 4x2 � 21x28. Binomial 29. Monomial30. Trinomial 31. None of these32. Answers will vary.
a. (x � 4) � (2x � 6) � 3x � 10b. (3x � 2) � (5x � 2) � 8xc. (x2 � 3x) � (4x � 5) � x2 � 7x � 5d. (x3 � 3x2) � (x � 3) � x3 � 3x2 � x � 3e. No
Applying Skills33. 6s 34. or 3.5x 35. 6p36. 14x � 30 37. 12x � 3 38. 5x � 339. 0.02h � 0.40 40. 3b � 11 41. 32z2 � 10
5-2 Multiplying Powers That Have theSame Base (pages 176–177)Writing About Mathematics
1. Yes; �53 � 53 � (�1)(53)(53) � (�1)(5 � 5 � 5)(5 � 5 �5) � (�1)(5 � 5)(5 � 5)(5 � 5) � (�1)(5 � 5)3 � �253
2. Yes; �24 � 4 � (�1)(24)(4) � (�1)(24)(22) �(�1)(2 � 2 � 2 � 2)(2 � 2) � (�1)(2 � 2 � 2 � 2 � 2 � 2)� �26
Developing Skills3. a5 4. b7 5. r11
6. r6 7. z11 8. t14
9. x2 10. a3 11. e10
12. 25 � 32 13. 37 � 2,187 14. 56 � 15,62515. 44 � 256 16. 210 � 1,024 17. x6
18. a8 19. z14 20. x4y6
21. a4b8 22. r3s3
23. 26 � 36 � 46,656 24. 54 � 212 � 2,560,00025. 1035 26. a11 27. x3a
28. yc � 2 29. cr � 2 30. xm � 1
31. (3y)a � b 32. True 33. False34. False 35. True 36. False37. True 38. False 39. TrueApplying Skills40. a. 32 persons b. The sixth meeting41. 105 or 100,000 cm
5-3 Multiplying by a Monomial (pages 180–182)Writing About Mathematics
1. Write the constant and the variable (or the vari-ables) together without an operation sign.
312x
2. The multiplication of 3 and 7y is performedfirst in accordance with the order of operations;2 � 3(7y) � 2 � 21y
3. The addition of 2 � 3 is performed first becausethe order of operations states that terms inparentheses must be simplified first; (2 � 3)(7y) �(5)(7y)
4. 5y is multiplied by y and 3 according to thedistributive property of multiplication overaddition; 5y(y � 3) � 5y2 � 15y
5. No. x2 and x3 are not like terms. Although theexpression can be factored, the two terms cannotbe simplified.
6. Yes. Terms with the same base may be multiplied;x2(x3) � x6
Developing Skills7. 24b2 8. 30y2 9. 20ab
10. 16r2 11. �42xyz 12. �3xy13. �6ab 14. xyz 15. �15abc16. �35rst 17. 12cde 18. �18x2y19. �60s2t 20. �20a4 21. 18x7
22. �140y5 23. �90r7 24. 12z3
25. �40y6 26. �72z8 27. �24x6y5
28. �35a5b3 29. �8a3b5 30. 18c � 9d31. �20m � 30n 32. �16a � 12b 33. 20x � 2y34. 8m � 48n 35. �32r � 2s 36. �12c � 10d37. 20x2 � 24x 38. 5d3 � 15d2 39. �55c3
40. m2n � mn2 41. �a2b � ab2
42. 15a3b � 21ab3 43. 2r7s4 � 3r3s7
44. 20ad � 30cd � 40bd 45. �16x2 � 24x � 4046. 3x3y � 3x2y2 � 3xy3 47. �10r4s2 � 15r3s3 � 20r2s4
48. 15y2 49. 15xy50. 24c2 � 6c 51. a. 12x2 � 24x
b. 2x2 � 4xc. 10x2 � 20x
52. 5d � 5 53. 6 � 4c 54. 14x � 355. 2x � 4 56. 12 � 24a � 32s57. 25 � 12e 58. 4e � 6 59. b60. 4b � 4 61. 3 � 5t 62. 2 � 8s63. �6x � 21 64. 10x2 � 6x 65. 24y � 6y2
66. 13x � 11 67. �c � 2d 68. 3a � 7a2
69. 6b 70. 29x � 14 71. 5x � 3y72. 11x � 18x2 73. �2y
Applying Skills74. 175x cents 75. 40z mi76. 10n cents 77. 10x � 1578. 15y2 � 21y 79. 6b2 � 4b80. (9x � 21) mi 81. (3xy � 6y) dollars82. (3w2 � 2w) 83. (12x � 18) dollars84. (5x � 35) dollars 85. (5x � 35) dollars86. (8x � 19) dollars
309
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5-4 Multiplying Polynomials (pages 185–186)Writing About Mathematics
1. Answers will vary.a. The product has four terms when all terms in
the binomials are unlike.b. The product is a trinomial when at least one
term in the first binomial is a like term with aterm in the second binomial.
c. The product is a binomial when the first termsof the binomial factors are the same and thesecond terms are opposites.
2. (a � 3)2 � (a � 3)(a � 3) � a2 � 3a � 3a � 9 �a2 � 6a � 9
Developing Skills3. a2 � 5a � 6 4. x2 � 8x � 15 5. d2 � 6d �276. x2 � 9x � 14 7. m2 � 4m � 21 8. t2 � 9t � 909. b2 � 18b � 80 10. 30 � 11y � y2
11. 48 � 14e � e2 12. 72 � 6r � r2
13. x2 � 25 14. 4y2 � 49 15. 25a2 � 8116. 2x2 � 11x � 6 17. 15y2 � 11y � 218. 4x2 � 9 19. 9d2 � 64 20. x2 � 2xy � y2
21. a2 � b2 22. a2 � 2ab � b2
23. a2 � 2ab � b2 24. x2 � 16y2
25. x2 � 8xy � 16y2 26. 18x2 � 17xy � 15y2
27. r4 � 3r2 � 10 28. x4 � y4
29. x3 � 5x2 � 11x � 10 30. 4c3 � 4c2 � c � 131. 15 � 16a � a2 � 2a 32. 6x3 � 13x2 � 19x � 1233. x3 � 12x2 � 48x � 64 34. a3 � 15a2 � 75a � 12535. x3 � 3x2y � 3xy2 � y3 36. 5x � 1437. 12x2 � 6 38. r2 � 3r � 5 39. �2x � 340. 14x � 2 41. y2 � 33y � 8 42. 14y � 743. a3 � 8a
Applying Skills44. (2x � 5)(x � 7) � 2x2 � 9x � 3545. (11x � 8)(3x � 5) � 33x2 � 31x � 4046. a. (15x � 100)(x � 3) � 15x2 � 145x � 300
b. 130 kph c. 650 km
5-5 Dividing Powers That Have the SameBase (pages 187–188)Writing About Mathematics
1. No; 54 � 5 � 54 � 1 � 53 � 125 � 12. a. First, simplify the denominator by using the rule
for multiplying powers with like bases.Then,use the rule for dividing powers with like bases.
b. No. Using the rules for multiplying and
dividing powers with like bases, �
38 � (5 � 2) � 3. The second expression,38 � 35 � 32, simplifies to 38 � 5 � 2 � 35. Thetwo expressions are not equal becausesubtraction is not associative.
38
35? 32
Developing Skills3. x6 4. a5 5. c1 or c6. x0 7. e6 8. m8
9. n1 or n 10. r0 11. x7
12. z9 13. t4 14. 23 � 815. 102 � 100 16. 32 � 9 17. 52 � 2518. 103 � 1,000 19. x3a 20. y8b
21. rc � d 22. sx � 2 23. a0
24. 2a � b 25. a. 25 26. a. 53
b. 32 b. 12527. a. 101 28. a. 33 29. a. 100
b. 10 b. 27 b. 130. a. 100 31. a. 68 32. a. 42
b. 1 b. 1,679,616 b. 1633. False 34. True 35. False
5-6 Powers with Zero and NegativeExponents (page 191)Writing About Mathematics
1. No. If x is a fraction between �1 and 1, then x�2 is greater than 1. For example, if x � ,then x�2 � � 4.
2. Brandon’s statement is true only for values ofn greater than or equal to 0. If n 0, 10n iswritten as 1 followed by n zeros. For example,102 � 100, which has three digits. If n � 0, 10n
has �n digits. For example, 10�2 � � .01,which has two digits.
Developing Skills3. 4. 5.
6. 7. 8. 1
9. 1 10. �1 11.
12. � 0.1 13. � 0.01 14. � 0.004
15. � 0.04 16. � 0.0015 17.
18. 19. � �0.5 20. 103 � 1,000
21. 3�6 � 22. 102 � 100 23. 40 � 1
24. 34 � 81 25. 4�2 � 26. 36 � 729
27. 6 28.
5-7 Scientific Notation (pages 195–197)Writing About Mathematics
1. Yes. All significant digits are represented in a.No new significant digits result from the multi-plication of a by 10n. If n � 0, the only possiblenew digits in a � 10n are non-significant zerosbetween the decimal point and the ones digit of a. If n 0, the only possible new digits in a � 10n are non-significant zeros trailing afterthe last digit in a.
512
116
1729
2121 1
27 5 1.037
1 136 5 1.0273
2,000125
11,000
1100
110
19 5 0.1
1r3
1m6
A 32 B
2121
1104
1100
A 12 B
22
12
310
14271AKTE1.pgs 9/25/06 10:38 AM Page 310
2. No. The key is used for subtraction. The
key needs to be used to represent the
negative sign of the exponent �5.Developing Skills
3. 102 4. 104 5. 10�2
6. 10�4 7. 109 8. 10�7
9. 10,000,000 10. 10,000,000,00011. 0.001 12. 0.00001 13. 300,00014. 400,000,000 15. 0.6 16. 0.000000917. 13,000 18. 0.0000000008319. 1,270 20. 0.0614 21. 222. 3 23. 3 24. �325. �5 26. 4 27. �928. 8 29. �1 30. 031. �3 32. 6 33. 8.4 � 103
34. 2.7 � 104 35. 5.4 � 107 36. 3.2 � 108
37. 6.1 � 10�4 38. 3.9 � 10�6 39. 1.40 � 10�8
40. 1.56 � 10�1 41. 4.53 � 105 42. 3.81 � 10�3
43. 3.75 � 108 44. 7.63 � 10�5 45. a. 8.7 � 100
b. 8.746. a. 7.65 � 10�5 47. a. 3.0 � 101
b. 0.0000765 b. 3048. a. 2.00 � 103
b.Applying Skills49. 9.5 � 1012 km 50. 1.2 � 1022 mi51. 5 � 10�13 cm 52. 8 � 10�4 in.53. 2,000,000,000 light-years 54. 240,000 mi55. 0.06 sec 56. 0.00000002 sec57. 5,900,000,000,000,000,000,000,000 kg
5-8 Dividing by a Monomial (page 199)Writing About Mathematics
1. Mikhail did not divide the second term of thepolynomial by the monomial. The quotientshould be 2b � 1.
2. Yes;Developing Skills
3. �2x2y2 4. �6y8 5. 9x4
6. �x2 7. �7c2b 8. 8x9. �7 10. �3y 11. 2x � 1
12. m � n 13. t � 1 14. �2c2 � 3d2
15. 1 � rt 16. �y � 5 17. 3d2 � 2d18. 3r3 � 2r 19. �3y6 � 2y3 20. �2a � 121. 3b � 4a 22. c � 3d 23. 2a2 � 3a � 124. �4y2 � 2y � 1 25. 2a � 4 26. 2 � 5c
Applying Skills27. 3y cents 28. 8b 29. 6r mi30. 8b chairs
15cd 1 11c5c 5 15cd
5c 1 11c5c 5 3d 1 2.2
2,00#0
(-)
� 5-9 Dividing a Polynomial by a Binomial(page 201)Writing About Mathematics
1. No. Nate split up the terms of the denominator,which is incorrect:
2. a. Yes. The expressions are equivalent by theaddition property of zero.
b. x2 � x � 1Developing Skills
3. b � 2 4. y � 1 5. m � 76. w � 3 7. y � 4 8. x � 29. a � 2 10. 3t � 8 11. 5y � 8
12. 3c � 4 13. x � 11 14. x � 815. x � 3Applying Skills16. x � 9 17. y � 2
Review Exercises (pages 203–204)1. Answers will vary. Example: Scientific notation
makes writing and computing very larger smallnumbers easier.
2. No, an � bn are not like terms.3. 4bc 4. 4y2 � 10y � 25. 13t � 4 6. �24mg2
7. 12x4 � 6x3 � 3x2 8. 8x2 � 2x � 39. 36a2b6 10. 36a2 � 12ab � b2
11. 4a2 � 25 12. 4a2 � 20a � 2513. 7x � 2x2 14. �5bc5 15. 2y16. 3w2 � 4w � 1 17. x � 6 18. 39 � 19,68319. 76 � 117,649 20. 23 � 106 � 8,000,00021. 1 � 12�1 � 22. 5.8 � 103
23. 1.42 � 107 24. 6 � 10�5 25. 2.77 � 10�6
26. 40,000 27. 0.00306 28. 970,000,00029. 0.000103 30. w2 � 5w; w2 � 8w; 2w2 � 13w31. a. 8h � 12 32. 16px
b. 4h2 � 12h � 933. a. x � 2 34. 24x � 40
b. 4x � 14
Exploration (page 204)To find the square of a two-digit number ending in 5,multiply the first digit by the next larger integer andby 100, and then add 25. This method can be appliedto find the square of a three-digit number ending in 5.
Given the square of an integer, the square of the nextlarger integer equals the square of the integer plusthe integer plus the next larger integer. Alternately,the square of the next larger integer equals the squareof the integer plus twice the integer plus 1; (n � 1)2 �n2 � n � (n � 1) � n2 � 2n � 1
1 112
x3 21x 1 1 5 x3
x 1 1 1 21x 1 1
311
14271AKTE1.pgs 9/25/06 10:38 AM Page 311
Cumulative Review (pages 204–206)Part I
1. 4 2. 3 3. 1 4. 1 5. 46. 2 7. 2 8. 4 9. 2 10. 4
Part II11. h �12. Answers will vary. 2 is the only number that is not
a multiple of 7; 7 is the only odd prime; 77 is theonly multiple of 11; 84 is the only multiple of 12.
Part III13. 3(x � 4) � 5x � 8 Given
3x � 12 � 5x � 8 Distributive property� 3x � 3x Subtraction property
of equality�12 � 2x � 8 Additive inverse
property� 8 � � 8 Subtraction property
of equality�20 � 2x Additive inverse
propertyDivision property ofequality
�10 � x, x � �10
2202 5 2x
2
3VB
14. 4a � 7 � (7 � 3a) Given4a � (�7) � (7 � 3a) Definition of
subtraction4a � [(�7) � 7] � 3a Associative property4a � 0 � 3a Additive identity
property4a � 3a Addition property of
zero(4 � 3)a Distributive property1a � a Multiplicative
identity propertyPart IV15. a. 1,986 ft; 2(525) � 2(468) � 1,986
b. 246,000 sq ft; 525 � 468 � 245,700c. 10 sacks; 245,700 � 25,000 � 9.84
16. 390 single-dip cones, 110 double-dip cones2.25d � 1.75(500 � d) � 930
0.5d � 875 � 9300.5d � 55
d � 110500 � 110 � 390
312
6-1 Ratio (pages 210–211)Writing About Mathematics
1. This week’s test. Last week, Melanie’s ratioof correct test questions to total test questions was . This week, her ratio was , which is equivalent to
2. The numerator and denominator are not wholenumbers, and they have a common factor other than 1;
Developing Skills3. a. 4. a. 5. a.
b. 3 : 1 b. 2 : 1 b. 8 : 56. a. 7. a. 8. a.
b. 4 : 1 b. 5 : 4 b. 1 : 49. a. 10. a. 11. a.
b. 8 : 1 b. 1 : 40 b. 60 : 112. a. 13. 10 times 14.
b. 3 : 515. 3 : 1 16. 3 : 1 17. 1 : 218. 3 : 1 19. 24 : 1 20. 5 : 121. 3 : 4 22. 1 : 2 23. 3 : 524. 3 : 14 25. 1 : 2 26. 3 : 1
18
35
601
140
81
14
54
41
85
21
31
1.54.5 5 1.5 4 1.5
4.5 4 1.5 5 13
2530; 25
30 . 2430.
2024
2430
27. 6 : 1 28. 12 : 1 29. 3 : 130. 4 : 1 31. 12 : 1Applying Skills32. a. 5 : 9 33. a. 3 : 2 34. 5 : 7
b. 5 b. 335. 2 : 1 36. b : (b � g) 37. 1 : 538. 12 : 7 : 6 39. 4 : 2 : 1 40. 45 or 80
6-2 Using a Ratio to Express a Rate (page 213)Writing About Mathematics
1. A rate, like a ratio, is a comparison of two quanti-ties, but these quantities may have different unitsof measure.
2. A ratio is written without units of measure, but arate contains units. When 1 is the second term ofa rate, it is usually omitted.
Developing Skills3. 2 apples per person (2 apples/person)4. 8 patients per nurse (8 patients/nurse)5. $0.50 per liter ($0.50/L)6. 6 cents per gram (6¢/g)7. $0.33 per ounce Q dollar/ozR8. 0.62 mile per kilometer (0.62 mi/km)
13
Chapter 6. Ratio and Proportion
14271AKTE1.pgs 9/25/06 10:38 AM Page 312
Applying Skills9. 57.5 miles per hour (57.5 mi/hr)
10. 4.5 miles per hour (4.5 mi/hr)11. 32 miles per hour (32 mi/hr)12. 124 miles per hour (124 mi/hr)13. 3 balls per can (3 balls/can)14. 8 cents per ounce (8¢/oz)15. a. 3.5 cents per ounce (3.5¢/oz)
b. 3.3 cents per ounce (3.3¢/oz)c. Giant size
16. Johanna17. Ronald
6-3 Verbal Problems Involving Ratio(pages 215–216)Writing About Mathematics
1. If x � 0, 2x : 3x cannot be reduced to 2 : 3 becausedivision by 0 is undefined. Also, 2x : 3x wouldequal 0 : 0, which is not equivalent to 2 : 3.
2. No.The perimeter of a rectangle is twice its lengthplus twice its width: 2(7x) � 2(4x) � 22x.Thereforethe ratio of the length to the perimeter is 7 : 22.
Developing Skills3. 40, 30 4. 100, 605. 42, 30 6. 48, 127. 12 cm, 12 cm, 10 cm 8. 12 cm, 16 cm, 20 cm9. 132 cm, 48 cm 10. 36°, 54°
11. 80°, 100° 12. 40°, 40°, 100°13. 25 in., 25 in., 15 in. 14. 9, 2115. 12, 20Applying Skills16. 12 cm, 20 cm 17. 275 boys18. Carl, $70; Donald, $3019. Sam, 21 points; Wilbur, 27 points20. 7.5 liters of water, 5 liters of acid
6-4 Proportion (pages 220–221)Writing About Mathematics
1. Yes. Switching the means and the extremes of a proportion will not affect the product of themeans or the product of the extremes. Therefore,the resulting ratios are still proportions. In ,ad � bc, and in , bc � ad (a, c � 0).
2. No.Adding a constant to all terms changes thevalues of the ratios in different ways. For example,
, but if you add 1 to each term, .Developing Skills
3. Yes 4. No 5. No6. No 7. Yes 8. Yes
23 2
35
12 5 2
4
ba 5 d
c
ab 5 c
d
9. 4 10. 30 11. 2412. 28 13. 20 14. 515. 36 16. 12 17. x � 918. x � 15 19. x � 18 20. x � 421. x � 3 22. x � 7 23. x � 324. x � 36 25. x � 8 26. x �
27. x � 6r 28. x �
Applying Skills29. 30.
31. 32.
33. 29 34.
35. 36. 24 shots
6-5 Direct Variation (pages 224–226)Writing About Mathematics
1. Yes. Natasha’s speed stays constant; therefore,.
2. No. The ratio of cost to time changes dependingon the number of hours a car is parked. Forexample, 1 hour costs $5.50, 2 hours cost $8.25,3 hours costs $11.00, and .
Developing Skills3. 4. 5.6. 7. 8.9. 10. 11.
12. Yes; s � P 13. Yes; c � 2n 14. No15. Yes; d � 20t 16. Yes; y � �3x 17. No18. A � 5h; h � 5, A � 1019. S � ; h � 10, S � 1220. l � 2w; l � 14, w � 421. No. If R increases, T decreases.22. Yes. The ratio is the constant .23. Yes. The ratio is the constant .24. No. If h increases, b decreases.25. a. Directly
b. The cost of 9 articles will be 3 times as muchas the cost of 3 articles.
c. C is doubled.26. a. Directly
b. The rectangle with a length of 8 inches willhave 2 times the area of the rectangle with alength of 4 inches.
c. A is tripled.27. d � 360 28. Y � 140 29. h � 330. N � 15Applying Skills31. $4.45 32. $30.80 33. lb34. 140 shots 35. 420 calories 36. 1,620 calories
114
201
ei
115
TD
32h
13
rs 5 3
2AP 5 53
50st 5 4
5
It 5 17
2tn 5 3
2Ps 5 4
yz 5 1
9dt 5 40x
y 5 4
5.51 2
8.252 2
113
dt 5 65
1
1219
129
3612
2050
2560
1220
2mrs
bca
313
14271AKTE1.pgs 9/25/06 10:38 AM Page 313
37. 170 calories 38. hr 39. 7.65 kg40. cups 41. 25 bags 42. $1,32043. 5.6 km 44. $21.00 45. 4 in.
46. a. $12.82 47. 48.b. $15.58c. $0.89 per lbd. 22.4 lb
6-6 Percent and Percentage Problems(pages 231–233)Writing About Mathematics
1. No. To write a percent as a decimal, move thedecimal point two places to the left and removethe decimal sign (that is, divide by 100); 3.6% �0.036
2. 104%. Ms. Edward’s new salary, in terms ofher old salary, s, is equal to s � 4%s � 100%s �4%s � (100% � 4%)s � 104%s.
Developing Skills3. 0.72 4. 9 5. 7.26. 10 7. 33.6 8. 7.59. 16 10. 24 11. 27
12. 200 13. 80 14. 20015. 72 16. 36 17. 16018. 81 19. 62 20. 50%21. 30% 22. 60% 23. 80%24. 25. 100% 26. 150%27.Applying Skills28. 25% 29. 30 students 30. a. $1.92
b. $25.9231. 90 planes 32. $8,000,000 33. $814.5034. 6 kg 35. 52,464 parts 36. 15 games37. 15% 38. $3,500 39. $12040. 25% 41. 5% 42. $78.0043. 12% 44. 8% 45. $6046. a. Coat, $111.99; blouse, $35.19; shoes, $71.99;
jeans, $26.39b. original price � 0.2(original price);
0.8(original price)47. Both plans are the same. Let C � original cost.
Plan 1: 0.7C � 0.08(0.7C) � 0.756C; Plan 2:0.7(1.08C) � 0.756C.
48. ABC, 99%; XYZ, 99%49. 1.6% 50. 0.8% 51. 130.9 lb52. No. Responses will vary. If Isaiah had 50 questions
on his midterm and 100 questions on his final,he answered 40 correctly on the midterm and 90correctly on the final; 0.8 � 50 � 40, 0.9 � 100 � 90.He answered or % correctly.862
3130150
12%331
3%
hqd
ndp
14
212
512
53. Not necessarily;Amy’s total gain is guaranteed tobe 20% only if she bought an equal number ofshares from each company at the same price.
6-7 Changing Units of Measure (pages 236–237)Writing About Mathematics
1. Using dimensional analysis, Sid can multiply the ratio of feet to miles by the ratio of yards to feet to
find the number of yards in a mile; �1760 yd/mi
2. Abigail can multiply the cups of butter by the
ratio of tablespoons to cups; � 6 tbsDeveloping Skills
3. a. 4. a. 5. a.b. 2.25 ft b. 1.75 m b. 2.5 lb
6. a. 7. a. 8. a.b. 1.5 mi b. 85 cm b. 1.5 gal
9. a. 10. a. 11. a.b. 378 in. b. 42 in. b. 4 ft
12. a. 13. a. 14. a.b. 150 cm b. 19.2 oz b. 13,200 ft
15. a. 16. a.b. 440 mm b. 10 qt
Applying Skills17. a. 2.62 ft
b. Miranda should buy 3-foot boards from whichto cut her 0.8-meter lengths. The 2-foot boardsare too short, and the 4-foot boards wouldleave unnecessary waste.
18. a. Yes b. $5.99c. The pre-cut piece
19. a. No b. 0.5 mi/hr20. a. Smaller b. Taylor will lose $5,936.51.
c. 74.8%
Review Exercises (pages 239–241)In 1 and 2, answers will vary.
1. No. The ratios and are not equal;8 � 5 � 3 � 12.
2. No. The sale price of the dress is 85% of theoriginal price or 0.85p. With the coupon, theprice is 80% of the sale price or 0.8(0.85p) �0.68p � 68%p. This means the original priceis reduced by 32%.
3. 6 : 7 4. 2 to 3 5. 3 to 56. 1 : 2 7. x � 3 8. x � 59. x � 6 10. 8, 32 11. (3)
12. (3) 13. (1) 14. 30015. 125% 16. 17. 165 g7
10
125
83
4 qt1 gal
10 mm1 cm
5,280 ft1 mi
16 oz1 lb
100 cm1 m
3 ft1 yd
12 in.1 ft
36 in.1 yd
1 gal8 pt
1 cm10 mm
1 mi5,280 ft
1 lb16 oz
1 m100 cm
1 ft12 in.
3 cup8 3 16 tbs
1 cup
5280 ft1 mi 3
1 yd3 ft
314
14271AKTE1.pgs 9/25/06 10:38 AM Page 314
18. 500 words 19. 3 : 420. 1,700 students 21. 21 in.22. 25 ft 23. 4 days 24. 50 girls25. a. Yes 26. 6.71 � 108
b. 2 km/hr over27. The 25% discount is the better buy. The price
of a package does not matter; 0.75(5x) � 3.75x,3.75x � 4x
28. 5.9% 29. 5.3%
Exploration (page 241)a. No.The $585 investment earns $11.70 in interest
and the $360 investment earns $5.40 in interest, soMark earns a total of $17.10 in interest. However,3.5% of $945 is $33.08.
b. No.Taking 10% off each $30 clothing item savesyou $3 a piece or $6 in total. However, taking 20%off the total of $60 would save you $12.
c. No. Some students may participate in both a cluband a sport, so the total percentage of studentswho are in an after-school activity is 59% �41% � (% of students in a club and sport).
d. Yese. It makes sense to add percents when the bases are
the same, the bases describe the same set, and thepercents describe different members of the set.
Cumulative Review (pages 242–244)Part I
1. 3 2. 1 3. 34. 2 5. 2 6. 37. 2 8. 4 9. 2
10. 3
Part II11. 12, 13; (x � 1)2 � x2 � 25, 2x � 1 � 25, x � 1212. x � 3; 4(2x � 1) � 5x � 5, 8x � 4 � 5x � 5, 3x � 9,
x � 3Part III13. a. 30 cm; 20 cm � 10 cm � 30 cm
b. 2 : 3; area of AEFD � (20 cm)2 � 400 cm2,area of ABCD � 30 cm � 20 cm � 600 cm2;400 : 600 � 2 : 3
14. hr at 40 mph, hr at 60 mph; 40x � 60(7 � x) �410, 420 � 20x � 410, 10 � 20x, x � , 7 � x �
Part IV15. a. 3 : 5 : 2
b. Rita, $162; Fred, $270; Glen, $108; (540) � 162,(540) � 270, � 108
c. Rita, $381; Fred, $635; Glen, $254; (1,270) �381, (1,270) � 635, (1,270) � 254
16. a. 490 ft; T � (64 � 5)� � 2(150) ≈ 485.35b. 55 trips; 5,280 ft � 5 � 26,400 ft; 26,400 ft �
485.35 ft ≈ 54.87
210
510
310
210(540)5
10
310
612
12
612
12
D F C
H
BEA
20 cm G
20 cm 10 cm20 + 10 = 30 cm
10 cm
10 cm
315
7-1 Points, Lines and Planes (pages 249–250)Writing About Mathematics
1. AB refers to the length of a line segment,refers to the line segment with endpoints A andB, and refers to the line containing thepoints A and B.
2. A half-line consists of the set of points that lie toone side of a point on a line.A ray consists of a pointon a line and all points to one side of this endpoint.Therefore, a ray is a half-line plus an endpoint.
Developing Skills3. Acute 4. Obtuse 5. Straight6. Obtuse 7. Right 8. Acute9. a. �AOB or �BOA b. �y
c. �x or �y (�AOB, �BOA or �BOC, �COB)d. �AOC or �COA
ABvh
AB
Applying Skills10. 30° 11. 120°12. 180° 13. 15°
In 14 and 15, answers will vary.14. 12:00, 1:05 15. 3:00, 9:00
7-2 Pairs of Angles (pages 255–258)Writing About Mathematics
1. If �x and �y are supplementary, then m�x �m�y � 180. If m�x equals 90, m�y � 180 �90 � 90, so both angles are right angles. If m�xis less than 90, m�y � 180 � m�x must begreater than 90, so �x is acute and �y is obtuse.If m�x is greater than 90, m�y � 180 � m�xmust be less than 90, so �x is obtuse and �y isacute.
Chapter 7. Geometric Figures,Areas, and Volumes
14271AKTE1.pgs 9/25/06 10:38 AM Page 315
2. No. Complementary refers to exactly two angleswhose measures sum to 90°, and there are threegiven angles.
Developing Skills3. a. 75° 4. a. 53°
b. 165° b. 143°c. 90° c. 90°
5. a. 23° 6. a. (90 � x)°b. 113° b. (180 � x)°c. 90° c. 90°
7. m�A � 15, m�B � 758. m�A � 35, m�B � 559. m�A � 65, m�B � 25
10. m�A � 20, m�B � 7011. m�ABD � 45, m�DBC � 13512. m�ABD � 50, m�DBC � 13013. m�DBC � 105, m�ABD � 7514. m�DBC � 144, m�ABD � 3615. 60°, 120° 16. 84°17. 110° 18. 30°, 60°19. 81° 20. 140°21. 130° 22. 35°23. 95° 24. 41°25. (3) 26. m�RTM � 2527. m�MTS � 60 28. m�NTS � 7229. a. m�FGH � 90 30. a. m�JLN � 42
b. m�HGI � 28 b. m�MLK � 90c. m�KLO � 42d. m�JLO � 138
31. a. m�HKI � 56 32. a. m�PMO � 90b. m�HKG � 124 b. m�OML � 50c. m�GKJ � 146
33. a. m�ABD � 70 34. a. m�FKJ � 40b. m�DBC � 110 b. m�FKG � 40
c. m�GKH � 100d. m�JKI � 140
35. a. m�CEB � 20b. m�BED � 160c. m�CEA � 160
36. a. m�RQS � 60b. m�SQT � 120c. m�PQT � 150
37. a. m�LRP � 50b. m�LRQ � 130c. m�PRM � 130
38. a. m�FEB � 10b. m�CEF � 80c. m�AEF � 170
39.
7-3 Angles and Parallel Lines (pages 261–262)Writing About Mathematics
1. Two lines in different planes that do not intersect2. Yes. Responses will vary. Example:
(1) Let l �� m and p⊥ l(2) m�1 � m�2 � 90
(Definition of per-pendicular lines)
(3) m�1 � m�3 (�1 and �3 are corre-sponding angles)m�2 � m�4 (�2 and �4 are corresponding angles)
(4) m�3 � m�4 � 90 (Substitution of �3 for �1and �4 for �2 in (2))
(5) p ⊥ m (Definition of perpendicular lines)Developing Skills
3. m�1 � m�5 � m�7 � 80, m�2 � m�4 � m�6 �m�8 � 100
4. m�1 � m�3 � m�5 � m�7 � 30, m�2 � m�4 �m�8 � 150
5. m�1 � m�3 � m�7 � 60, m�2 � m�4 � m�6 �m�8 � 120
6. m�3 � m�5 � m�7 � 75, m�2 � m�4 � m�6 �m�8 � 105
7. m�1 � m�3 � m�5 � m�7 � 125, m�4 �m�6 � m�8 � 55
8. m�1 � m�3 � m�5 � m�7 � 170, m�2 �m�6 � m�8 � 10
9. m�1 � m�3 � m�5 � 2, m�2 � m�4 � m�6 �m�8 � 178
10. m�1 � m�3 � m�5 � m�7 � 1, m�2 � m�4 �m�6 � 179
11. m�1 � m�3 � m�5 � m�7 � 66, m�2 � m�4 �m�6 � m�8 � 114
12. m�1 � m�3 � m�5 � m�7 � 70, m�2 � m�4 �m�6 � m�8 � 110
13. m�1 � m�3 � m�5 � m�7 � 56, m�2 � m�4 �m�6 � m�8 � 124
14. m�1 � m�3 � m�5 � m�7 � 50, m�2 � m�4 �m�6 � m�8 � 130
15. m�1 � m�2 � m�3 � m�4 � m�5 � m�6 �m�7 � m�8 � 90
ABCvh
' BDvh
316
1 2
3 4
p
l
m
A
D
E
C
B20°160°
160°
20°
RS
Q
P T150°
30°60° 120°
130°
50°130°
50° R
M
Q
P
L
80°
EA
D
B
F
C
10°
14271AKTE1.pgs 9/25/06 10:38 AM Page 316
16. m�1 � m�3 � 150, m�2 � m�11 � 30, m�6 �m�8 � 140, m�9 � 40, m�10 � 110
17. Never 18. Sometimes 19. Always20. Always 21. Sometimes 22. Sometimes23. Answers will vary. Example:
(1) m�1 � m�x (�1 and �x are corre-sponding angles)
(2) m�2 � m�x (�2 and �x are vertical angles)
(3) m�1 � m�2 (Substitution of �2 for �x in (1))
7-4 Triangles (pages 267–269)Writing About Mathematics
1. Let x and y represent the meas-ures of two of the angles in atriangle and their sum, x � y,represent the measure of thethird angle.Then the sum of themeasures of the angles in the triangle is x � y �(x � y) � 180° or 2x � 2y � 180°. Dividing bothsides by 2, you get x � y � 90°. Since the third anglehas a measure of 90°, the triangle is a right triangle.
2. If two angles of a triangle each measure 60°, themeasure of the third angle is 180° � 60° � 60° �60°. Therefore, the triangle is equilateral.
Developing Skills3. Yes 4. No 5. Yes 6. 80°7. 60° 8. 43.2° 9. ° 10. 60°
11. a. No.The sum of the measures of two right anglesis 180°, which equals the sum of the angles in atriangle, so there could be no third angle.
b. No. The sum of the measures of two obtuseangles is greater than 180°, which exceeds thesum of the angles in a triangle.
c. No. The sum of the measures of one obtuseand one right angle is greater than 180°, whichexceeds the sum of the angles in a triangle.
12. 90°13. a. Isosceles b. �A and �C
c. If two sides of a triangle are equal in measure,then the angles opposite these sides are equalin measure.
d. Legs: , ; base: ; base angles: �A, �C;vertex: �B
14. a. 70° b.c. If two angles of a triangle are congruent, the
sides opposite the angles are congruent.d. Isoscelese. Legs: , ; base: ; base angles: �R, �S;
vertex: �TRSSTRT
RT > ST
ACBCAB
7414
15. a. 20° 16. a. 80° 17. 45°b. 70° b. 65°c. 96° c. 52°d. 135° d. 40°e. 77° e. 57.5°
18. 60°Applying Skills19. 84°, 84°, 12° 20. 45°, 45°, 90° 21. 36°, 36°, 108°22. 55°, 55°, 70° 23. 58°, 58°, 64° 24. 79°, 79°, 22°25. a. m�A � 30, m�B � m�C � 75
b. Acute, isosceles26. a. 60°, 60°, 60°
b. Equilateral, equiangular, acute27. 20°, 60°, 100° 28. 18°, 72°, 90°; right, scalene29. 35°, 65°, 80° 30. 35°, 75°, 70°31. a. m�x � 50, m�y � 40, m�z � 90
b. Right, scalene32. a. m�R � 60, m�S � 90, m�T � 30
b. Right, scalene33. a. m�K � m�L � m�M � 60
b. Equilateral, equiangular, acuteHands-On Activity 1: Constructing a LineSegment Congruent to a Given Segment
a. Construct a segment congruent to the given seg-ment on a ray. Starting from the newly constructedpoint, construct a second (non-overlapping) seg-ment congruent to the given segment on the sameray. Repeat to construct a third (non-overlapping)segment on the same ray.The length of the seg-ment whose endpoints are the endpoint of the rayand the newest constructed point is three timesthe length of the given segment.
b. Construct a segment congruent to the longersegment on a ray. On this segment, constructa segment congruent to the shorter segment,starting from the endpoint of the ray. The lengthof the segment whose endpoints are the twoconstructed points on the ray has length equal tothe difference of the two given segments.
c. Construct a segment congruent to the firstsegment on a ray. On the same ray, startingfrom the newly constructed point, construct asegment congruent to the second segmentwithout overlapping the first. The length of thesegment whose endpoints are the endpoint ofthe ray and the second constructed point is thesum of the lengths of the given segments.
A
C
B
D
317
1
x
2
x
x + y y
14271AKTE1.pgs 9/25/06 10:38 AM Page 317
Hands-On Activity 2: Constructing an AngleCongruent to a Given Angle
a. Construct an angle congruent to the given angleon a ray. Using the newly constructed ray, con-struct a second angle congruent to the given angleand adjacent to the first constructed angle. Repeatto construct a third angle congruent to the givenangle and adjacent to the second constructedangle.The measure of the angle whose sides arethe original ray and the newest construct ray isthree times the measure of the given angle.
b. Construct an angle congruent to the larger angleon a ray. On the same ray, construct an anglecongruent to the smaller angle such that the newray lies inside the larger angle.The measure of theangle whose sides are the two constructed rays isequal to the difference of the two given angles.
c. Construct an angle congruent to the included angleon a ray. Construct a segment congruent to a givenside on each of the rays of this angle. Connect thenew points lying on the sides of this angle to form atriangle congruent to the given triangle.
Hands-On Activity 3: Constructing a PerpendicularBisector
a. �EAB � �FAB; �EAB � �ECBb. Draw a segment on a line. Construct the perpen-
dicular bisector of the segment. Starting fromone of the endpoints of the segment, constructan angle congruent to the given angle along thesegment such that the new ray intersects the per-pendicular bisector. From the same point on thesegment, construct a second angle congruent tothe given angle on the other side of the segmentsuch that the new ray intersects the perpendicularbisector.The triangle formed by the vertex of thetwo congruent angles and the intersection of theconstructed rays and the perpendicular bisectoris isosceles with a vertex angle that is twice themeasure of the given angle.
A C
F
E
B
S
T
RM
N
L
c. Draw a line segment and construct its perpen-dicular bisector. Construct a segment congruentto half the line segment on the perpendicularbisector starting from the point of intersectionwith the original segment. Connect an endpointof the original segment with the newly con-structed point on the bisector to form anisosceles right triangle.
Hands-On Activity 4: Constructing an AngleBisector
a. Construct an angle congruent to the given angle.Construct its angle bisector. Consider the smallerangle whose sides are the bisector and one of thesides of the bisected angle. Construct an anglecongruent to this angle on one of the rays of thebisected angle such that it is adjacent to thebisected angle. The measure of the angle whosesides enclose the other rays is one and a halftimes the measure of the given angle.
b. Bisect a straight angle. Each resulting anglemeasures 90 degrees.
c. Construct an angle congruent to the given angle.With the point of the compass on the vertex, drawan arc that intersects both sides of the angle.Connect the intersections.The triangle formed isisosceles with the vertex angle congruent to thegiven angle.
7-5 Quadrilaterals (pages 277–279)Writing About MathematicsIn 1 and 2, responses will vary.
1. Yes. Since all the angles are equal, then each anglemust be 90°; x � x � x � x � 360°, 4x � 360°,x � 90°.A parallelogram with four right angles isa rectangle.
2. Yes. Consecutive angles of a parallelogram are sup-plementary. In parallelogram ABCD, if m�A � 90,then m�B � 180 � 90 � 90, and so on.
Developing Skills3. True 4. True 5. True6. False 7. True 8. True9. a. x � 50° b. m�B � 50
10. a. x � 90° b. m�E � 90, m�F � 90,m�G � 70, m�H � 110
11. a. x� 60° b. m�K � 60, m�L � 50,m�M � 130, m�N � 120
PS
T
R
318
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12. a. x � 43.75° b. m�Q � 53.75, m�R � 111.25,m�S � 107.5, m�T � 87.5
13. AB � DC � 1414. m�A � m�C � 110, m�B � �D � 7015. BC � AD � 11 16. x � 20°17. AB � BC � CD � DA � 4218. KL � LM � MN � NK � 18Applying Skills19. a. 900°
b. There are 5 triangles;5(180°) � 900°
c. m�A � m�G � 90,m�B � m�F � 135,m�C � m�E � 165,m�D � 120
d. 900°20. a. Since each triangle is isosceles, its base angles
are equal. We know that one of the angles is60° since it is cut from an equilateral triangle.If the 60° angle is the vertex angle, then forbase angle x, x � x � 60° � 180°, so x � 60°.If the 60° angle is a base angle, then for vertexangle x, x � 60° � 60° � 180°, so x � 60°.Since the triangles have three equal angles,they are equilateral.
b. 120° c. Regular hexagon21. a. 4.2 sq mi b. 2,700 acres22. a. Not possible; if the four angles of a quadri-
lateral, w, x, y, and z, are all acute, then w � 90, x � 90, y � 90, and z � 90. Therefore,w � x � y � z � 360.
b. Not possible; if the four angles of a quadri-lateral, w, x, y, and z, are all obtuse, then w � 90, x � 90, y � 90, and z � 90. Therefore,w � x � y � z � 360.
c. Possible. Example:
d. Possible. Example:
60° 60°
60°
110°
30°
110°
110°
110°
319
B
A G
F
ED
C
e. Not possible; if three angles of a quadrilateralare right angles, then the fourth angle x mustalso be a right angle: 90 � 90 � 90 � x � 360,so x � 90°.
7-6 Areas of Irregular Polygons (pages 280–282)Writing About Mathematics
1. 35 sq units; both triangles have the same base( ) and the same height (the distance between
), so their areas are equal.2. 76 sq in.; the area of �ABD plus the area
of �BDC equals the total area of ABCD;57 � 62 � 119 square inches. The total areaof ABCD minus �ABC equals the area of�ADC; 119 � 43 � 76 sq in.
Developing Skills3. 139 cm2 4. 108 sq ft 5. 585 m2
6. 303 sq in. 7. 44 sq ft 8. 102 cm2
9. 400 sq yd 10. 50 mm2
Applying Skills11. Triangle, 162 sq in.; pentagon, 1,134 sq in.12. 76 m2
7-7 Surface Areas of Solids (pages 285–286)Writing About Mathematics
1. a. 8b. Rectanglesc. a � 2a; 2a2 sq units
2. a. sq units
b. � 12a2 sq unitsDeveloping Skills
3. 7,464.4 sq in. 4. 2,754 sq in. 5. 312 sq ft6. 1,800 cm2 7. 144.5 cm2
8. 7,257.1 mm2 or 72.6 cm2 9. 1,570.8 sq ftApplying Skills10. 118 sq in. 11. 94.11 sq in. 12. sq in.13. 522 sq ft
7-8 Volumes of Solids (pages 290–292)Writing About Mathematics
1. a. The amount left over is the same, 64 � 16� ≈13.7 sq in.
b. The volume of the pie with the 8-inch radiusis approximately four times the volume ofthe pie with the 4-inch radius; 75.40 cu in. :18.85 cu in. � 4 : 1
2. Each radius is the height of the can.16
660#
3"3a2
3"32 a2
AB and CDAB
14271AKTE1.pgs 9/25/06 10:38 AM Page 319
Developing Skills3. 140 cu ft 4. 210 cm3
5. 690,000 cm3 or 6. cu in. or 0.069 m3 0.0806 cu ft
7. 636.056 cm3 8. 1,520.875 cm3
9. a. 24 cm2 10. a. 32 sq ftb. 84 cm3 or b. 192 cu ft or
84,000 mm3 7.1 cu yd11. 208 mm3 12. 28 cu in.13. a. 10,048.013� cm3 14. a. 2,343.75� m3
b. 31,600 cm3 b. 7,360 m3
15. a. 16. a.b. 257 cu in.
b. 1 cm3
17. a. 18. a. 4.5� cu ftb. 8,181 cm3 b. 14 cu ft
19. a. 5%b. 9.8%c. 14.3%
Applying Skills20. a. 58 cm3 21. approx. 11,875.22 gal
b. 4 cu in.22. a. 670 sq in. 23. a. approx. 395.69 cm3
b. 1,200 cu in. b. approx. 0.40 Lc. cu ft
24. 576 cu in. 25. 100 cu yd26. a. 2,590,000 m3
b. 91,800,000 cu ft
Review Exercises (pages 294–296)1. m�AEC � 50 2. 90°3. 55° 4. 70°5. 58°, 60°, 62° 6. m�GHD � 737. x � 24 8. m�GHD � 709. a. m�A � 50, m�R � 80, m�T � 50
b. Acutec. Isosceles but not equilateral
10. 65°, 65°, 50° 11. 130°, 50°12. m�A �36, m�B � 54, m�C � 9013. 90°, 30° 14. 80°, 100°15. a.
A B
CD
y
xO
1
–1 1
23
2,60416p cm3
49150p cm3
32623p mm3 or812
3p cu in.
140#
b. (�3, 5)c. AB � 10, CD � 12, AE � 6d. 66 sq units
16. AB � CD � 25, BC � AD � 1917. 59.4 cu in. 18. 127 sq in.19. a. 72.25 cm3 20. a. 987.5 cm2
b. 72 ml b. 489,800 cm3
c. Yes. With c. 489.8 Lsignificant digits,both answers are the same.
Exploration (pages 296–297)a. [area of rectangle � area of 30° sector]b. 2� [area of 180° sector]c. [area of 260° sector]d. [2(area of 60° sector)]e. [2(area of 90° sector � area of rt. triangle
with legs of length 1)]
f. [2(area of 60° sector � area of eq.
triangle with sides of length 2)]
Cumulative Review (pages 297–299)Part I
1. 1 2. 4 3. 24. 3 5. 2 6. 47. 3 8. 3 9. 1
10. 1Part II
11. w � x � 2;
12. x � �5; 7(x � 2) � 3(x � 2), 7x � 14 � 3x � 6,4x � �20, x � �5
Part III13. 130 oz and 142 oz or 8 lb 2 oz and 8 lb 14 oz14. a. x � b. x � 0.17Part IV15. 3 hr at 40 mph, 8 hr at 60 mph; 40x � 60(11 � x) �
600, x � 316. a. 270 sq ft; (9.0 � 12) � (9.0 � 4.5) � (12 � 4.5) �
(15 � 4.5) � 270b. 240 cu ft; (9.0 � 12) � 4.5 � 2431
2
7 2 3ba
x 1 4qx 1 2
x2 1 6x 1 8 x2 1 4x 2x 1 8 2x 1 8 0
43p 2 2"3
12p 2 1
13p
269 p
6 1 34p
320
14271AKTE1.pgs 9/25/06 10:38 AM Page 320
321
8-1 The Pythagorean Theorem(pages 305–306)Writing About Mathematics
1. Yes. The square of the length of the hypotenuse isequal to the sum of the squares of the lengths of
the other two sides: (5k)2 (3k)2 � (4k)2, 25k2
9k2 � 16k2, 25k2 � 25k2
2. Yes. The square of the length of the hypotenuse isequal to the sum of the squares of the lengths of
the other two sides: (2n � 1)2 � (2n2 � 2n)2
(2n2 � 2n � 1)2, (4n2 � 4n � 1) � (4n4 � 8n3 �
4n2) 4n4 � 8n3 � 8n2 � 4n � 1, 4n4 � 8n3 �8n2 � 4n � 1 � 4n4 � 8n3 � 8n2 � 4n � 1
Developing Skills3. c � 5 4. c � 17 5. b � 86. b � 5 7. a � 8 8. a � 159. c � 2 10. c � 2 11. b � 1
12. a. 13. a.b. c � 3.61 b. c � 4.24
14. a. 15. a.b. b � 6.93 b. c � 7.28
16. a. 17. a.b. a � 3.32 b. b � 5.39
18. 19.20. 21.22. 25 in. 23. 41 cm24. 53 ft 25. 145 m26. 25 yd 27. 30 mm28. a. 56 cm 29. 17 in.
b. 1,848 cm2
30. 72.9 mApplying Skills31. 36 ft 32. 15.8 ft33. 26 km 34. 596 ft35. Yes. The distance between their homes is 2,163 ft,
which is less than the device’s maximum range of2,640 ft.
36. 127.3 ft
8-2 The Tangent Ratio (pages 311–312)Writing About Mathematics
1. In a right triangle with a 45° angle, the thirdangle measures 45°, so the triangle is isosceles.Since the legs of an isosceles triangle are congruent, tan 45° � � 1.
oppadj 5 x
x
x 5 "48 5 4"3x 5 "27 5 3"3x 5 "32 5 4"2x 5 "8 5 2"2
b 5 "29a 5 "11
c 5 "53b 5 "48 5 4"3
c 5 "18 5 3"2c 5 "13
5?
5?
5?
5?
2.
Let the lengths of the sides of the equilateraltriangle be equal to 1. Each half of the side cutby the altitude has length . By the Pythagorean
Theorem, the length of the altitude is .There-
fore, tan 30° � .
This matches the calculator result.Developing Skills
3. a. 1 4. a.b. 1 b.
5. a. 6. a.b. b.
7. 8.9. 0.1763 10. 0.4663 11. 2.7475
12. 1.4281 13. 0.0175 14. 57.290015. 0.7265 16. 2.3559 17. 5°18. 20° 19. 29° 20. 45°21. 64° 22. 72° 23. 21°24. 37° 25. 61° 26. 19°27. 8° 28. 71° 29. Increase30. a. tan 20° � 0.3640, tan 40° � 0.8391
b. NoApplying Skills31. a. 1 32. a. 33. a. � 0.5
b. 45° b. 24° b. 27°c. � 2.25 c. � 2d. 66° d. 63°
34. a. 1 35. a.b. 1 b.
c. 1
8-3 Applications of the Tangent Ratio (pages 315–317)Writing About Mathematics
1. The angle of depression is the angle determinedby the horizontal line and the line of sight fromthe top of the building. The angle Zack labeledis the complement of the angle of depression,so it measures 26°.
sr
rs
105
94
12
49 5 0.4
512 5 0.4164
3 5 1.3
kt
65 5 1.2
tk
56 5 0.83
125 5 2.4
512 5 0.416
12
4"3
25 1
2? 2"3
5 1"3
< 0.57735
"32
12
30°
1
12_
60°
_2
"3
Chapter 8.Trigonometry of the Right Triangle
14271AKTE1.pgs 9/25/06 10:39 AM Page 321
2. The angle of elevation from point A to point Brelies on a horizontal line that is parallel to thehorizontal line used in determining the angle ofdepression from point B to point A.The line ofsight from A to B is the same as from B to A,and acts as a transversal of the parallel lines.Thismeans that the angle of elevation from A to B andthe angle of depression from B to A are alternateinterior angles and therefore congruent.
Developing Skills3. 23 ft 4. 28 ft 5. 15 ft6. 7 ft 7. 29 ft 8. 27°9. 34° 10. 50 ft 11. 45°
Applying Skills12. 58 m 13. 22.5 ft 14. 1,470 ft15. 14 m 16. 187 m 17. 78 ft18. 45° 19. 74° 20. 69°21. a. 35°
b. 55°c. 110°d. 70°
8-4 The Sine and Cosine Ratios (pages 320–322)Writing About Mathematics
1. In triangle ABC, sin A � and
cos B � , so sin A � cos B.
2. In right triangle ABC with legs of lengths a and b and hypotenuse of length c, and
. The lengths of the legs of a righttriangle are always less than the length of thehypotenuse, so a � c and b � c. Therefore
, so sin A � 1 and cos A � 1.Developing Skills
3. a. 4. a.
b. b.
c. c.
d. d.
5. a. 6. a.
b. b.
c. c.
d. d.
7. 8.9. 0.3090 10. 0.6691 11. 0.8480
12. 0.9703 13. 0.0175 14. 0.999815. 0.9336 16. 0.8192 17. 0.766018. 0.5150 19. 0.2756 20. 0.034921. 11° 22. 57° 23. 20°
1213 5 0.9230763
5 5 0.6
kp
2129 < 0.7241
rp
2029 < 0.6897
rp
2029 < 0.6897
kp
2129 < 0.7241
513 5 0.3846156
10 5 0.6
1213 5 0.9230768
10 5 0.8
1213 5 0.9230768
10 5 0.8
513 5 0.3846156
10 5 0.6
ac , 1 and bc , 1
cos A 5 bc
sin A 5 ac
adjhyp 5 BC
AB 5 ac
opphyp 5 BC
AB 5 ac
24. 20° 25. 86° 26. 27°27. 63° 28. 87° 29. 3°30. 11° 31. 13° 32. 61°33. 31° 34. 35° 35. 54°36. 73° 37. 7° 38. 30°39. a. sin 25° ≈ 0.4226, sin 50° ≈ 0.7660
b. No40. a. cos 25° ≈ 0.9063, cos 50° ≈ 0.6428
b. No41. a. Increase 42. a. 20° 43. a. 40°
b. Decrease b. 67° b. 73°c. 52° c. 8°d. (90 � x)° d. (90 � x)°
44. a. 0.5 45. a. 0.5b. 30° b. 60°
46. a. 47. a.
b. b.
c. c.
d. d.
e. 62° e. 10°f. 28° f. 80°
48. a. 49. a.
b. b.
c. c.
d. d.
e. 67° e. 19°f. 23° f. 71°
50. a.
b. sin 30° � 0.5, cos 30° ≈ 0.8660c. None
51. 45°
8-5 Applications of the Sine and CosineRatios (pages 325–327)Writing About Mathematics
1. Yes; tan A � , cos A � , so (tan A)(cos A) �
� sin A
2. No. Both sin A and cos A have values less than
1 . If cos A was the reciprocal
of sin A, cos A would equal , but this is greater than 1. Therefore, cos A cannot be the reciprocalof sin A.
Developing Skills3. 14 cm 4. 112 cm 5. 82 cm6. 24 cm 7. 28 cm 8. 18 cm9. 21 cm 10. 26 cm 11. 37 cm
hypopp
A opphyp , 1,
adjhyp , 1 B
A oppadj B A
adjhyp B 5
opphyp
adjhyp
oppadj
cos A 5"3
2sin A 5 12,
1237 5 0.32412
13 5 0.923076
3537 5 0.9455
13 5 0.384615
3537 5 0.9455
13 5 0.384615
1237 5 0.32412
13 5 0.923076
1161 < 0.180315
17 < 0.8824
6061 < 0.98368
17 < 0.4706
6061 < 0.98368
17 < 0.4706
1161 < 0.180315
17 < 0.8824
322
14271AKTE1.pgs 9/25/06 10:39 AM Page 322
12. 40° 13. 30° 14. 44°15. 42°Applying Skills16. 5.7 m 17. 309 ft 18. 10 m19. 7,700 ft 20. 55° 21. 7°22. 8 ft 23. 2.0 m 24. 1,500 ft25. 142 m 26. 33° 27. 26°28 a. 104 ft 29. 1870 cm2 30. 24 ft
b. 48.6 ftc. 282 ftd. 24,300 sq fte. 697 ft
8-6 Using the Three TrigonometricFunctions to Solve Problems (pages 328–331)Writing About Mathematics
1. Use the Pythagorean Theorem to find the lengthof the third side. Calculate sin�1, cos�1, or tan�1
of the ratios of the lengths of appropriate sidesto find the measures of the acute angles.
2. No. Angle measures of a triangle determine theratio of the sides, not their specific measures. Inorder to find the measures of the sides, at leastone side length must be given.
Developing Skills3. 17 ft 4. 29 ft 5. 19 ft6. 16 ft 7. 34° 8. 37°9. 19° 10. 23 ft 11. 9.4
12. 17.9 13. 9° 14. 60°15. 5 16. 50°17. AD � 16, DB � 9, CD � 12; �B � 53°18. a. 52 19. 34° 20. a. 20
b. 23° b. 2821. a. 14 22. l � 18, 23. a. 72
b. 13 w � 13 b. 5424. a. AC � 95, BC � 31
b. 1002 � 10,000; 952 � 312 � 9,986; 10,000 ≈ 9,986Applying Skills25. 131 m 26. 288 ft 27. 520 m28. 40 ft 29. 2,400 ft 30. 26°31. a. 58 m 32. a. 99 ft
b. 104 m b. 203 ft
Review Exercises (pages 332–334)1. Responses will vary.
a. Talia can first use the sine function to findthe measure of hypotenuse AB: AB � 4.5 �sin 43° ≈ 6.6. Then she can find the measureof AC using either the cosine function or thePythagorean Theorem: AC � 6.6 cos 43° ≈ 4.8 or AC � ≈ 4.8"6.62 2 4.52
b. Talia can use sine or cosine ratios to find themeasure of the second side, and then useeither sine or cosine ratios or the PythagoreanTheorem to find the measure of the third side.She can then calculate the ratio to find the tangent of the angle.
2. The sine of an angle is never greater than 1.3. 4. 5.6. 7. 8.9. 27 cm 10. 13 cm 11. 29 cm
12. 101 cm 13. 60° 14. 22515. 34 16. 28 17. 30°18. 29° 19. 38 m 20. 53°21. 320 ft 22. a. 45 yd 23. 6
b. 95 yd24. a. 75°, 75°, 30° 25. a. 42
b. 91.1 cm b. 84c. 47.2 cm c. 36.9°d. 229.4 cm d. 22.6°e. 2076.8 cm2 e. 59.5°
Exploration (page 334)
a. b.
c. d.
e.
Cumulative Review (pages 334–336)Part I
1. 4 2. 3 3. 1 4. 1 5. 26. 4 7. 1 8. 4 9. 1 10. 3
Part II11. a.
b. 54 sq units; responses will vary. Split ABCDEinto two trapezoids using point F(7, 0): area ofABCDE � area of ABFE � area of CDEF.Area of ABFE � � 14 sq units,and area of CDEF � � 40 sq units,so area of ABCDE � 14 � 40 � 54 sq units.
12(5)(7 1 9)
12(2)(5 1 9)
E
A B
D C
y
xO1
1
ns2
4 tan A90 2 180n B
s2
4 tan A90 2 180n Bs
2tan A90 2 180n B
180 2 360n
2 or 90 2 180n
360n
815
817
1517
1517
158
817
oppadj
323
14271AKTE1.pgs 9/25/06 10:39 AM Page 323
324
9-1 Relations and Functions (pages 344–346)Writing About Mathematics
1. For each number of miles driven, there is one andonly one associated cost.
2. The domain would be the set of whole numbers:{x � x � whole numbers}
Developing Skills3. 6 4. 0 5. �46. 5.5 7. �4 8. 99. 10. 4 11.
12. �12 13. Yes 14. No15. Yes 16. No 17. Yes18. Yes 19. No 20. Yes21. No 22. Yes 23. No24. Yes 25. Yes 26. No27. Yes 28. Yes 29. Yes30. No. There are two y-values for x-values �5
and 6.31. No. There are two y-values for x-values 25
and 81.
32. a. {25, 27} 33. a.
b. {3, 15, 27, . . . } b. {0, 1, 2, . . . }34. a. {�13, �14} 35. a. {12, 13}
b. {�2, �8, �14, . . . } b. {1, 7, 13, . . . }Applying Skills36. a. y � 5 � 2.5x
b. Answers will vary but must have domain x � 0. Examples: (1, 7.50), (2, 15)
c. Yes
U116 , 2V
51441
2
37. a. x � 2y � 54b. Answers will vary but must have domain
0 � x � 27. Examples: (2, 26), (6, 12)c. Yes
38. a. y � 265 � 30xb. Answers will vary but must have domain
0 � x � 8.83. Examples: (1, 235), (2, 205)c. No
39. a. y � 2 � 0.4xb. Answers will vary but must have domain
x � 0. Examples: (1, 2.2), (1.5, 2.3)c. No
40. a. Length Width Area
3 176 � 2(3) 3[176 � 2(3)] � 510
4 176 � 2(4) 4[176 � 2(4)] � 672
5 176 � 2(5) 5[176 � 2(5)] � 830
6 176 � 2(6) 6[176 � 2(6)] � 984
7 176 � 2(7) 7[176 � 2(7)] � 1,134
b. Yes; A � x(176 � 2x) for {x � x � 0}41. a. 2x � y � 10
b. (0, 10), (1, 8), (2, 6), (3, 4), (4, 2), (5, 0)c. (0, 10), (5, 0)d. Answers will vary but must have domain
0 � x � 5 with x not an integer. Examples:(2.5, 5), (3.75, 2.5)
e. The domains are different. In problem 1, x andy must be integers with domain 0 � x � 5. Inproblem 2, x and y can be any positive rationalnumbers with domain 0 � x � 5.
42. {(r, h) � r2h � 102, where r and h are both positive}
Chapter 9. Graphing Linear Functions and Relations
12. ; sin 30° �
Part III13. 2.0 mi; tan 57° � , 1.3 tan 57° � x, x � 2.0
57°
State St.1.3 mi
Holt Rd.
Pla
nk R
d.
{
x
x1.3
A
BC
6x2 – 4x 2x
6x2 2 4x2x , 12 5 3x 2 2, 3x 5 5
2, x 5 56
56
14. Benny: lawyer, basketball; Carlos: doctor, soccer;Danny: engineer, baseball
Baseball Basketball Soccer Doctor Engineer Lawyer
Benny X � X X X �
Carlos X X � � X X
Danny � X X X � X
Part IV15. 32 sq ft; radius of the tree: C � 2r, C � 9.5, 2r �
9.5, r � 1.52; area of tree: A � 1.5122 � 7.182;outer radius of garden � 1.512 � 2.0 � 3.512; areaof garden � 3.5122 � 7.182 � 38.749 � 7.182 �31.567
16. Height � 0.5 in., width � 3.75 in.; ;3.75 · 8 � 5x, 30 � 5x, x � 6; 6.5 � 6 � 0.5
3.75x 5 5
8
14271AKTE2.pgs 9/25/06 10:41 AM Page 324
9-2 Graphing Linear Functions Using TheirSolutions (pages 350–352)Writing About Mathematics
1. Any point on the line x � y � 4. The sum of thex-coordinate and y-coordinate of each orderedpair is 4.
2. (3, 3); explanations will vary. Students may plotthe points and note the line that goes throughthree of the points does not go through (3, 3).
Developing Skills3. Yes 4. No 5. No6. Yes 7. Yes 8. No9. Yes 10. 1 11. 4
12. �1 13. Any number 14. 715. 8 16. 3 17. �518. y � �3x � 1 19. y � 4x � 6 20. y � 3x
21. y � 8x 22. y � �2x � 4 23. y �
24. a. x y
0 0
1 4
2 8
25. a. x y
�1 �2
0 1
1 4
26. a. x y
�1 2
2
5 �1
12
2x 2 53
27.
28.
29.
30.
31.
–1–1
1O
y
xx = 2y – 3
1
y
xy = –x
O1
1–1–1
y
x
y = –3x
O
1–1–1
y
xO
1
1–1
y =
5x
y
y = x
xO
1
1–1
325
b. y
xO
1
1–1
y =
4x
b. y
xO
1
1–1
y =
3x
+ 1
b. y
x
x + 2y = 3O1
1–1–1
14271AKTE2.pgs 9/25/06 10:41 AM Page 325
32.
33.
34.
35.
36. y
xO
1
1–1–1
y = –2x + 4
y
xO
1
1–1
y =
3x +
1
y
xO
1–1–1
y =
2x –
1
y
xO
1
1–1–1
y = x
+ 3
y
xO
1
1–1–1x
= –
y +
11 2
37.
38.
39.
40.
41. y
xO1
1–1
x – 2y = 0
y
x
2
2–2–2
3x + y = 12
y
xO
1
1–1
–1
y – x
= 0
y
xO
1
1–1–1
x – y
= 5
y
xO1
1–1
x + y = 8
326
14271AKTE2.pgs 9/25/06 10:41 AM Page 326
42.
43.
44.
45.
46. y
xO1
1–1–1
2x + 3y = 6
y
xO
2
2–2–2
x + 3y = 12
y
xO
1
1–1–1
3x –
y =
–6
y
x
O1
1–1
–1
2x –
y =
6
y
xO
1
1–1–1
y –
3x =
–5
47.
48.
49.
50.
51. a.
b. (2, �1), (�2, �3), or any other point satisfying x � 2y � 4
y
xO
1
1–1–1
(4, 0)
(0, –2)
y
x
4x =
3y
O1
1–1–1
y
xO
1
1–1–1
2x =
y –
4
y
xO
3
3–3–3
x – 3y = 9
y
xO
1
1–1–1
3x –
2y
= –4
327
14271AKTE2.pgs 9/25/06 10:41 AM Page 327
52. a.
b. YesApplying Skills53. a. y � 2x
b.
54. a. y � x � 2b.
55. a. x � y � 6b.
56. a. x � y � 1 b. y
x
O1
1
x –y =
1
y
xO
1
1
x + y = 6
y
x
y = x
+ 2
O1
1
y
xO
1
1
y =
2x
y
xO
1
1–1–1
(–2, 3)
(1, –3)
57. a. 3x � y � 6b.
58. a. 2x � y � 9b.
59. a. y � 2x� 5b.
60. a. x � y � 30b.
9-3 Graphing a Line Parallel to an Axis(pages 353–354)Writing About Mathematics
1. No. The equation of the y-axis is x � 0; all pointswith an x-coordinate of 0, such as (0, 0), (0, 1) and(0, �2), lie on the y-axis. The correct equation ofthe x-axis is y � 0.
5
5O
y
x
x + y = 30
y
x
y =
2x +
5
1
1O
y
xO
2x + y = 9
1
1
y
xO
11
3x + y = 6
328
14271AKTE2.pgs 9/25/06 10:41 AM Page 328
2. No. To intersect the x-axis, the y-coordinate of apoint must be 0. However, every point on the liney � 4 has a y-coordinate of 4.
Developing Skills3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
x
1
1
y
O–1–1
y = 1 12–
x1
y
O–1–1
x =
–1 2
x
y = –7
2
2
y
O–2–2
x
y = –4
1
1
y
O
–1–1
x
y = 0
1
1
y
O
–1–1
x
y = 5
1
1
y
O
–1
x
y = 4
1
1–1
y
O–1
1
1–1
y
xO
x =
–5
–1
1
1–1
y
xOx
= –
3
–11–1
y
xO
x =
0
–1
1
1–1
y
xO x
= 4
–1
1
1–1
y
xO x
= 6
15. 16.
17. 18. a. y � 1b. y � 5c. y � �4d. y � �8e. y � 2.5f. y � �3.5
19. a. x � 3 b. x � 10 c. x � �4.5d. x � �6 e. x � 2.5 f. x � �5.2
20. (2) 21. (3) 22. (4)Applying Skills23. a. $25 b. y � 2524. a. Rectangle on
graphb. y � �1, x � 4,
y � 5, x � �2 c. Dashed lines
on graphd. x � 1, y � 2
25. a. y � 5,432 26. a. y � 74,000b. 5,432; 5,432 b. 74,000; 74,000
9-4 The Slope of a Line (pages 360–363)Writing About Mathematics
1. Yes;2. If any two points have the same x-value, then the
change in x is 0; , which is undefined,
so the line has no slope.Developing Skills
3. a. Positive slope 4. a. No slopeb.
5. a. Zero slope 6. a. Negative slopeb. m � 0 b. m � �1
7. a. Positive slope 8. a. Negative slopeb. m � 3 b. m � �2
m 5 32
y1 2 y
2
x 2 x 5y
12y
20
y1 2 y
2
x1 2 x25 4 2 2
22 2 4 5 226 5 21
3
x
1
1
y
O–1–1
y = 3.5
x
1
1
y
O–1–1x = – –32
x
1
1
y
O–1–1
y = –2.5
329
y
xO
A B
D C
14271AKTE2.pgs 9/25/06 10:41 AM Page 329
9. a. b. m � 1
10. a. b. m � 2
11. a. b. m � �2
12. a. b. m � 2y
xO
1
1–1
(1, 5)
(3, 9)
y
x(0, 0)
O
1
1–1–1
(3, –6)
y
xO
1
–1 (0, 0)
(4, 8)
y
xO
1
1–1
–1(0, 0)
(4, 4)
13. a. b.
14. a. b. m ��1
15. a. b. m ��3
16. a. b. m � 0
17. a. b. m � 0y
xO
1
1–1–1
(–1, 3) (2, 3)
y
xO
1
1–1
(8, 2)(4, 2)
yx
O1–1
(5, –2)
(7, –8)
y
xO
1
1–1–1
(–2, 4)
(0, 2)
m 5 23
–1–1
1O
y
x
(1, –1)
(7, 3)
330
14271AKTE2.pgs 9/25/06 10:41 AM Page 330
18.
19.
20.
21.
22. y
xO
1–1–1
(3, 1)
y
xO
1
1–1
(–4, 5)
y
xO
1
1–1–1
(2, –5)
y
xO
1
1–1
(1, 3)
y
xO
1
–1 (0, 0)
23.
24.
25.
26.
27. y
xO
1
1–1–1
(–1, 0)
y
xO
1
1–1–1
(–2, 3)
y
xO1
1–1–1
(2, 4)
y
xO
1
1–1–1
(1, –5)
y
xO
1–1
(–3, –4)
–1
331
14271AKTE2.pgs 9/25/06 10:41 AM Page 331
28.
29.
Applying Skills
30. Slope of , slope of , slope of
31. a.
b. Parallelogram
c. Slope of , slope of d. The slopes are equal.e. The slopes are equal.f. The slopes are equal.g. Slope of , slope of h. Yes
32. a.b.c. foot vertical rise per horizontal footd. foot vertical descent per horizontal foot
33. $0.02 per minute34. $.075 per hundred cubic feet35. a. m � 0.08 or
b. 0.08 foot vertical rise per horizontalfoot
Aor 225 B
225
23
12
m 5 223
m 5 12
CD 5 0AB 5 0
ADvh
5 23BCvh
5 23
y
xO
A B
CD
1
1–1–1
AC 5 21BC 5 1AB 5 0
y
xO
1–1–1
(–2, 0)
y
xO
1
1–1–1
(0, 2)
9-5 The Slopes of Parallel andPerpendicular Lines (pages 365–366)Writing About Mathematics
1. a. No slope; the negative reciprocal of 0 is ,which is undefined.
b. 0; the negative reciprocal of a fraction withundefined slope (denominator of 0) hasnumerator of 0.
2. y increases unit for each 1-unit increase in x.Developing SkillsIn 3–11, part a, answers will vary. Examples are given.
3. a. (0, 6), (�3, 0) 4. a. (0, �2), (2, 0)b. 2 b. 1c. 2 c. 1d. d. �1
5. a. (0, 7), (1, 4) 6. a. (0, 0), (3, 1)b. �3 b.
c. �3 c.d. d. �3
7. a. (0, �4), (4, 0) 8. a. (0, �2), (3, 0)b. 1 b.c. 1 c.d. �1 d.
9. a. (�1, 0), (1, 1) 10. a. (4, 0), (4, 1)b. b. No slopec. c. No sloped. �2 d. 0
11. a. (0, �5), (1, �5)b. 0c. 0d. No slope
Applying Skills12. a. $6
b. $9c.
d. y � 0.75x � 3 or y �e. $3f. $0.75 per mile
34x 1 3
y
x
10 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 9 10
Trip (miles)
Cos
t ($)
12
12
232
23
23
13
13
13
212
14
210
332
14271AKTE2.pgs 9/25/06 10:41 AM Page 332
13. a. 120°b. 130°c.
d. y � 2x � 100e. 2° per second
9-6 The Intercepts of a Line (page 369)Writing About Mathematics
1. No.The equation is not in slope-intercept form.When transformed to , the y-intercept is seen to be 4.
2. No.The coefficient of x is 1 and the constant termis 0.Written as y � 1x � 0, it can be observed thatthe slope is 1 and the y-intercept is 0.
Developing Skills3. m � 3, b � 1, x-intercept �4. m � �1, b � 3, x-intercept � 35. m � 2, b � 0, x-intercept � 06. m � 1, b � 0, x-intercept � 07. no slope, no y-intercept, x-intercept � �38. m � 0, b � �2, no x-intercept9. m � , b � 4, x-intercept � 6
10. m � 3, b � 7, x-intercept �11. m � �2, b � 5, x-intercept �12. m � 2, b � 3, x-intercept �13. m � , b � �2, x-intercept �14. m � , , x-intercept �15. m � , b � 0, x-intercept � 016. m � , b � 2, x-intercept � �517. m � , b � 3, x-intercept � 218. m � , b � , x-intercept �19. The slope of each equation is 4.20. The y-intercept of each equation is 1.21. The slopes are equal. 22. The lines are parallel.23. Parallel 24. Neither25. Perpendicular 26. Perpendicular27. (3) 28. (1)
5421
225
232
25
43
232b 5 9
432
45
52
232
52
273
223
213
y 5 232x 1 4
Time (seconds)
Tem
pera
ture
(de
gree
s)y
x
150145140135
125120115110105100
605 10 15 20 25303540 50 5545
130
9-7 Graphing Linear Functions Using TheirSlopes (pages 373–374)Writing About Mathematics
1. When x � 0, y � m(0) � b so y � b. Therefore,(0, b) is a point on the graph.
2. Yes. Moving up 2 units and to the left 3 unitsfrom the y-intercept will locate those points onthe line with negative x-values.
3. Yes. In a vertical transla-tion of c units, y � x � c.Therefore, the new y-intercept is c. The newx-intercept is �c, whichmeans that the graphshifted horizontally �cunits. Reflecting acrossthe x-axis has the sameeffect as reflectingacross the y-axisbecause an untranslatedline is symmetricthrough the origin.
Developing Skills4.
5.
6. y
xO
1–1–1
y = –2x
y
xO1
1–1
y =
2x
y
xO
1
1–1–1y
= 2x
+ 3
333
y
x(–c, 0)
(0, c)
y
x
y = –x
y = x
14271AKTE2.pgs 9/25/06 10:41 AM Page 333
7.
8.
9.
10.
11. y
xO
1
1–1
y = – –x + 6
34
y
xO
1
–1–1 y = – –x1
3
y
xO1
–1
y = –x – 112
–1–1
1O
y
x
y = –x + 22
3
1
y
xO
1
1–1
y = –3x – 2
12.
13.
14.
15.
16. y
xO1
1–1–1
3y =
4x
+ 9
y
xO
1
1–1
4x =
2y
y
xO
1
1–1–1
x y
2 3– + – = 1
y
xO
1
1–1–1
3x + y = 4
y
xO
2
2–2–2
y –
2x =
8
334
14271AKTE2.pgs 9/25/06 10:41 AM Page 334
17.
18.
19.
20.
21.
22. Reflected across the x-axis and shifted 2 units up23. Reflected across the x-axis, compressed vertically
by a factor of , and shifted 2 units down24. Stretched vertically by a factor of 2 and shifted
1.5 units up25. Shifted 4 units up
13
y
xO
1
1–1–12x – 3y – 6 = 0
–1–1
1O
y
x1
3y – x = 5
y
xO
1
1–1
2x = 3y – 1
y
xO
1
1–1–1
3x + 4y = 7
y
x
O1
1–1–1
5x –
2y
= 3
Applying Skills26. a.
b. 1c. y � 2x � 1d. Yes
27. a.
b. 3c.d. Yes
28. a. Yesb.c.
d. The y-intercept, , is a fractional value,so points located by counting 3 units up and2 units over from the y-intercept will alsohave fractional values.
9-8 Graphing Direct Variation (pages 376–378)Writing About Mathematics
1. The unit of measure for the flour and for themilk is the same (ounces). Therefore, no unit ofmeasure is associated with ratio.
2. Yes. Since 1 quart � 4 cups, the ingredients arestill being used in a 1 : 4 ratio.
212
y
xO1
1–1
3x –
2y
= 1
(1, 1)
32
y 5 23x 1 3
–1 1
1O
y
x
(3, 5)
y
xO
1
1–1
(–2, –3)
335
14271AKTE2.pgs 9/25/06 10:41 AM Page 335
Developing Skills3. a. 4
b. y � 4xc.
d. 44. a. 45 words/min
b. y � 45xc.
d. 455. a. 16 characters/sec
b. y � 16xc.
d. 16
y
xO
y =
16x
100908070605040302010
1 2 3 4 5 6 7 8
Cha
ract
ers
Seconds
y
xO
135
120
105
90
75
60
45
30
15
1 2 3 4
Wor
ds
Minutes
y =
45x
y
xO
y =
4x
Per
imet
er
Side
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7
6. a.b.c.
d.7. a. 10
b. y � 10xc.
d. 108. a. slices/oz
b.c.
d. 53
y
xO
70
60
50
40
30
20
10
6 12 18 24 30 Ounces
Slic
es
y = – x53
y 5 53x
53
y
xOy
= 10
x
50
45
40
35
30
25
20
15
10
5
1 2 3 4 5 6
Pho
togr
aphs
Negatives
43
y
xO
10987654321
1 2 3 4 5 6 7
y =
– x43F
lour
Sugar
y 5 43x
43
336
14271AKTE2.pgs 9/25/06 10:41 AM Page 336
9. a. lb/personb.c.
d.10. a. slice/oz
b. y � xc.
d.11. a. hit/at bat
b. y � xc.
d.12. a. cal/cracker
b.c.
d. 203
y
xO
Crackers
Cal
orie
s
35
30
25
20
15
10
5
1 2 3 4
y =
– x20 3
y 5 203 x
203
14
y
xO
Hit
s
At bats
9
6
3
3 6 9 12 15 18
y = – x14
14
14
32
y
xO
y = – x32
Slic
es o
f che
ese
Weight (oz)
12
10
8
6
4
2
2 4 6 8 10 12
32
32
15
y
O
People
Poun
ds o
f mea
t963
3 6 9 12 15 18 21
y = – x15
x
y 5 15x
15
Applying Skills
13. a. 30 mphb.
c. 44 feet/secondd.
14. a. 3.5 characters/secb.
c. 42 words/mind.
15. Yes; P � 4s 16. No17. No 18. Yes; I � 0.025P
y
xO
y = 42
x
160
120
80
40
1 2 3 4
y
xO
y =
210x
Cha
ract
ers
Minutes
400
300
200
100
1 2 3
y
x
y =
44x
O
Dis
tanc
e (f
t)
T ime (sec)
88
77
66
55
44
33
22
11
1 2 3 4 5
y
O x
y =
30x
Time (hr)
Dis
tanc
e (m
i)
605040302010
1 2 3 4 5
337
14271AKTE2.pgs 9/25/06 10:41 AM Page 337
19. Yes; c � 0.39i20. Yes; d � s
9-9 Graphing First-Degree Inequalities inTwo Variables (pages 381–382)Writing About Mathematics
1. No. Solving the inequality for y yields y � 2x � 5,which is the region below 2x � y � 5.
2. No. The line x � 2 is excluded from both thegraph of x � 2 and the graph of x � 2. Therefore,it is excluded from their union.
Developing Skills3. y � 2x 4. y � 5. y � x � 36. y � �2x 7. y � 3x � 4 8. y �
9.
10.
11.
12. y
xO
1
1–1
y = –3
y –3
–1
y
xO
1
1–1
y = 5
y 5
y
xO
1
1–1–1
x =
2x –2
y
xO
1
1 2 3–1–1
x =
4
x 4
34x 1 3
52x
13.
14.
15.
16.
17. y
xO
1
1–1–1
x – 2y = 4
x – 2y 4
y
xO
1–1–1
x –y =
–1x – y –1
y
xO
1
1–1x + y = –3x + y –3
y
xO
1
1–1
y = – x + 312
y – x + 312
y
xO
1
–1–1
y x – 2
y = x
– 2
338
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18.
19.
20.
21.
22. y
x
– + – = 1x2
y5
– + – 1x2
y5
1
–1O
y
x
9 – x = 3y
9 – x 3y1
–1–1O
y
x12–y = –x + 1
13
12–y –x + 11
3
1–1
O
y
xO
1
–1 2x – 3y = 6
2x – 3y 6
y
x
2y – 6 x 0
2y –
6 x
= 0
O
1
1–1
23.
24. a. y � x � 3b.
25. a. x � y � 5b.
26. a. y � 3x � 2b.
Applying SkillsIn 27–31, part c, answers will vary. Examples are given.27. a. y � x
b.
c. (4, 6)
1
1
O
y
x
y = x
y x
y
x
y –
3x =
2y – 3 x 2
O1–1
y
x
x + y = 5
x + y 5
O
1–1–1
y
xO
1–1–1
y x + 3
y = x
+ 3
y
x
2x + 2y – 6 = 0
2x + 2y – 6 0
1–1O
1
339
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28. a. y �
b.
c. (50, 20)29. a. y � x � 2
b.
c. (6, 9)30. a. y � x � 4
b.
c. (12, 15)31. a. x � y � 5
b.
c. (2.75, 3.00)
9-10 Graphs Involving Absolute Value (page 387)Writing About Mathematics
1. Yes. For x � 0, �2x� � 2x, so y � �2x� � 1 is the graphof y � 2x � 1. For x � 0, �2x� � �2x, so y � �2x� � 1is the graph of y � �2x � 1.
1
1O
y
x
x + y 5x + y = 5
y
xO
1
1
y = x
+ 4
y x + 4
1
1
O
y
x
y = x
+ 2y x + 2
1
1
O
y
x
y – x12
y = – x12
12x 2. No. For �5 � x � 5, �x� � �y� � 5 has two y-values.
Therefore, it is not a function.3. Yes. We know that �cx� � �c� �x�. For c � 0, �c� � c,
so the graph of y � �cx� is the same as the graphof y � c�x�.
Developing Skills4.
5.
6.
7.
8. y
xO
1–1–1
y = 2 x
y
xO
1
1–1–1
y = x + 3
y
xO
1–1–1
y = x – 1
y
xO
1
1–1–1
y = x + 3
y
xO
1
1–1
y = x – 1
340
14271AKTE2.pgs 9/25/06 10:41 AM Page 340
9.
10.
11.
12.
13. y
x1–1
x + y = 4
y
x
y = x
1–1–1
1
y
xO
1–1–1
x + 2 y = 7
y
xO
1–1–1
x + y = 5
y
xO
1
1–1–1
y = 2 x + 1
14.
15.
16. Reflected across the x-axis and shifted 1 unitdown
17. Reflected across the x-axis, stretched verticallyby a factor of 2, and shifted 2 units up
18. Shifted 2 units to the left and shifted 3 units down19. Reflected across the x-axis, shifted 1.5 units to
the right, and shifted 4 units up
9-11 Graphs Involving ExponentialFunctions (pages 391–392)Writing About Mathematics
1. If b is negative, then when x is even, y is positivebut when x is odd, y is negative. This would causeits graph to oscillate back and forth betweenpositive and negative y-values.
2. a. When r is positive, the equation representsgrowth. When r is negative, the equationrepresents decay.
b. Yes. If r � 1, then the growth will be morethan 100%, and the original value will morethan double.
c. No. The principal value can never be less thanzero. (Example: The population of a town candecrease, it can even disappear but therecannot be fewer than 0 residents.)
3. Yes. Every y-value for y � 3x � 2 occurs 2 unitsearlier than on the graph of y � 3x. Therefore,the graph is y � 3x shifted 2 units to the left.For every value of x in y � 3x � 2, the y-valueis 2 more than 3x. This means that it is the graphof y � 3x shifted 2 units up.
y
xO
1
1–1
2 x + 4 y – 6 = 0
y
xO
1
1–1–1
y4
x2– + – = 1
341
14271AKTE2.pgs 9/25/06 10:41 AM Page 341
Developing Skills4.
5.
6.
7.
8. y
x–1
1
1
y = (–)x13
y
x2
y = (–) x25
–1 1
y
xO
1–1
1
y = 1.5x
y
x2
y = 4x
1–1
y
x3
y = 3x
1
9.
10. a. (0, 1)b. Upwardc. Downward
11. a. b.
c. They are the same graph.Applying Skills12. 89,580 13. $7,969.2414. $29,936.85 15. 6 mi16. $9.8817. a. No. The total pay would be $5.12.
b. Yes. The total pay would be $167,772.16.
Review Exercises (pages 394–398)1. a. {1, 2, 3, 4}
b. {1, 2, 3}c. No.There are two y-values for x � 1 and x � 3.
2.
Yes. The diagonals of the quadrilateral both havelength 10 and, since they lie on the x- and y-axes,are perpendicular. Therefore, the graph is asquare.
y
xO
1–1–1
x + y = 5
y
x1
1
–1
y = 2–x
y
x1
1
–1
y = (–)x12
y
1
1
xO
y = 1.25x
342
14271AKTE2.pgs 9/25/06 10:41 AM Page 342
3. For each x-value, there are an infinite numberof y-values that make the inequality y � 2x � 1true. Therefore, it is not a function.
4. �25.6. y-intercept � �6, x-intercept � 47. �28. a.
b. Domain � {all real numbers} Range � {all real numbers}
9. a.
b. Domain � {all real numbers} Range � {3}
10. a.
b. Domain � {all real numbers} Range � {all real numbers}
11. a.
b. Domain � {all real numbers} Range � {all real numbers}
–1 1
1O
y
x
x + 2y = 8
y
xO1
1–1
y = –x23
y
xO
1
1–1–1
y = 3
y
xO
1
1–1–1
y = –x + 2
32
12. a.
b. Domain � {all real numbers} Range � {all real numbers}
13. a.
b. Domain � {all real numbers} Range � {all real numbers}
14. 2 15. 1 16. �217. 2 18. 19. Undefined20. 3 21. None 22. 023. 4 24. (2) 25. (2)26. (1) 27. (2) 28. (4)29. (2) 30. (4) 31. (1)32. (2) 33. (2)34. a–b.
c. Trapezoidd. DA � 10, BC � 6e. 5f. 40 sq units
35. y
xO
1
1–1–1
y = � x – 2 �
y
x
(–3, 3) C B (3, 3)
(–5, –2) D A (5, –2)
O1
1–1–1
212
y
xO
1
–1
2x –
y =
4
2x – y 4
y
xO
1
1–1–1
y – x > 2y –
x = 2
343
14271AKTE2.pgs 9/25/06 10:41 AM Page 343
36.
37.
38.
39. a. y � 2x � 350b. $2.00 per day that the child is not picked up
by 5:00 P.M.c. $356
40. a.
b. 9.6 galc. 300 mid. (9.6, 300)e. Yes. See graphf. 6.4 gal
y
x
500
400
300
200
100
5 10 15 20
Gasoline (gallons)
Mile
s dr
iven (9.6, 300)
O
y
x
3
1–1
y = 2.5x
y
xO
1
1–1–1
x + 2 y = 6
y
xO
1
1–1–1
y = x – 241. a. D � 20
b. 0.625 ftc. No
42. a. F �b.
c. See graphd. Yes. Most points are on or very near the graph
of the equation.43. 2,654 deer
Exploration (page 398)STEP 1.
STEP 2. See graphSTEP 3. m � 1, tan �BAC � 1STEP 4. m�BAC � 45STEP 5. Graphs will vary; slope � tan �BACSTEP 6.
STEP 7. m � �1, tan �BAC � �1STEP 8. Graphs will vary; slope � tan �BACSTEP 9. The slope of a line is equal to the tangent
of the angle whose vertex is the x-interceptand that opens in the positive direction.The measure of this angle is the tan�1 ofthe slope.
y
x
B (b, b)
A (2, 0)C (b, 0)
y
x
B (b, b)
A (2, 0)
C (b, 0)
y
x
100
80
60
40
20
20 40 60 80 100 120 140 160O
Chirps per minute
Tem
pera
ture
(°F
)
F = – C + 4014
14c 1 40
A 12 B
s
344
14271AKTE2.pgs 9/25/06 10:41 AM Page 344
Cumulative Review (pages 399–400)Part I
1. 1 2. 2 3. 3 4. 3 5. 16. 1 7. 4 8. 3 9. 2 10. 4
Part II11. 20 ft by 28 ft; 2(5x � 7x) � 96, 24x � 96, x � 4,
5 4 � 20, 7 4 � 2812. (11, 2), (11, 9), (4, 9) or (11, 2), (11, �5), (4, �5)
or (�3, 2), (�3, �5), (4, �5) or (�3, 2), (�3, 9),(4, 9); in a square, adjacent sides are perpen-dicular and all sides are congruent. Therefore,point B is either 7 units to the right or to the leftof A with a y-coordinate of 2. Points C and Dshare the same x-coordinates as A and B but are7 units above or below A and B.
y
xO
A
1
1–1–1
Part III13. �2; y � �2x � b, 4 � �2(2) � b, b � 8,
y � �2(5) � 8, y � �214. 3 boxes; 4a � 6b � 42, a � b � 8, b � 8 � a, 4a �
6(8 � a) � 42, 4a � 48 � 6a � 42, �2a � �6, a � 3Part IV15. 6.0 in., 8.5 in., 10.4 in, 12 in.; area of whole
dartboard � 12.02 � 144 with r � 12.0; areaof dartboard � , which makesr � 10.4; area of dartboard � ,which makes r � 8.5; area of dartboard �
� 36, which makes r � 6.16. 12 ft; sin 22° � , x � 124.5
x
14(144p)
14
12(144p) 5 72p1
2
34(144p) 5 108p
34
345
10-1 Writing an Equation Given Slope andOne Point (pages 403–404)Writing About Mathematics
1. Yes. The y-intercept is a specific point on the line,so Jen is correct that this is the same informationas coordinates of one point and the slope.
2. a.b. (2) Substitution principle
(3) Product of the means equals the productof the extremesDistributive propertyAddition property of equality
Developing Skills3. y � 2x � 2 4. y � 2x � 105. y � �3x � 7 6. y 5 25
3x 2 5
41
7. 8.9. a. y � 2x � 1 10. a. 4x � y � 5
b. b.c. y � 3x � 2d. c. 3x � y � �7e. d.f.
e. 4x � y � 3f.
11. a. y � �x � 7b.c. y 5 1
3x 1 53
y 5 237x 2 57
x 1 4y 5 212
14x 1 y 5 23 or
x 1 3y 5 1y 5 32x
13x 1 y 5 1
3 ory 5 223x
y 5 213x 2 2
x 1 4y 5 14
14x 1 y 5 7
2 ory 5 212x 1 1
y 5 234xy 5 1
2x 1 2
Chapter 10.Writing and Solving Systems of Linear Functions
14271AKTE2.pgs 9/25/06 10:41 AM Page 345
Applying Skills12. a. y � x � 3
b. (4, 7); see graphc.d. (6, 6); see graphe.f. Yes
13. a. y � 65x � 12b. 207 mic. 4.2 hr
10-2 Writing an Equation Given Two Points(pages 406–407)Writing About Mathematics
1. Yes. The coordinates of any point on the linecan be substituted into the equation, and weare given that (4, 11) is a point on the line;11 � 3(4) � b, 11 � 12 � b, b � �1
2. (1) Definition of slope(2) Simplification
Product of the means equals the productof the extremesDistributive propertyAddition property of equality
Developing Skills3.4. y � 2x � 35. y � 2x � 26. y � x � 27.8.9. 3x � y � �1
10.11. a.
b. y � x � 2c. y � 3x � 4d. Yes. Since the slopes of and are
negative reciprocals, the sides of the triangleare perpendicular.
12. No. The quadilateral does not have parallel sides;slope of WX � �1, slope of XY � � , slope of YZ � � , slope of ZW �1
Applying Skills13. a. (2, 7), (5, 13)
b. y � 2x � 3c. {x � x is a whole number}d. {2x � 3 � x is a whole number}
14. a. (3, 123), (1, 65) 15. a. (100, 7), (210, 9.2)b. y � 29x � 36 b. y � 0.02x � 5c. $36 c. $5d. $29/hr d. $0.02/copy
4951
97
CAvh
ABvh
y 5 213x 1 6
53x 1 y 5 0 or 5x 1 3y 5 0
32x 2 y 5 1 or 3x 2 2y 5 2
43x 2 y 5 22
3 or 4x 2 3y 5 22
y 5 52x 1 5
y 5 12x 1 3
y 5 212x 1 9
10-3 Writing an Equation Given theIntercepts (pages 409–410)Writing About Mathematics
1. a. gives the y-intercept 4, but there isno x-intercept since y � 4 is parallel to thex-axis.
b. gives the x-intercept 4, but there isno y-intercept since x � 4 is parallel to they-axis.
2. No. Division by 0 is undefined. The x- andy-intercepts are 0.
Developing Skills3. x-intercept � 5, y-intercept � 54. x-intercept � 6, y-intercept � �25. x-intercept � 10, y-intercept � �26. x-intercept � 1, y-intercept � 27. x-intercept � , y-intercept � 28. x-intercept � 7, y-intercept �9. x-intercept � , y-intercept � �1
10. x-intercept does not exist, y-intercept � �4
11.
12. a.b. �x �y � 1c.
Applying Skills13. a.
b. 514. a. x � 0, y � 0, and one of the following:
b. Yes. The right angle can be in any of thefour quadrants, causing the x- and y-interceptsto have coordinates (7, 0) and (0, 7), (�7, 0)and (0, 7), (�7, 0) and (0, �7), or (7, 0) and(0, �7).
10-4 Using a Graph to Solve a System ofLinear Equations (pages 415–416)Writing About Mathematics
1. No. The lines 2x � y � 7 and 2x � 5 � y areparallel but don’t coincide; that is, they form asystem of inconsistent equations.
2. No. All solutions to y � x � 4 are also solutionsto 2y � 8 � 2x since the second equation canbe shown to be equivalent to y � x � 4 dividedby 2. The lines form a system of dependentequations.
2x7 2
y7 5 1
2x7 1
y7 5 1,
x7 2
y7 5 1,
x7 1
y7 5 1,
x4 1
y3 5 1
3x4 1 3x
5 5 1
x2 2
2y3 5 1
2ba
13
72
83
x4 5 1
y4 5 1
346
y
x
1
1O
A(D)
B
C
14271AKTE2.pgs 9/25/06 10:41 AM Page 346
Developing Skills3. (�1, 3)
4. (0, 3)
5. (4, �3)
6. (3, 2)
7. (1, 3)y
x
1
1O
(1, 3)
y
x
1
1O
(3, 2)
y
x
1
1O
(4, –3)
y
x
1
1O
(0, 3)
y
x
1
1O
(–1, 3)
8. (0, 6)
9. (2, 6)
10. (3, 2)
11. (0, �2)
12. (1, 3)y
x
1
1O
(1, 3)
y
x
1
1O
(0, –2)
y
x
1
1O
(3, 2)
y
x
1
1O
(2, 6)
y
x
1
1O
(0, 6)
347
14271AKTE2.pgs 9/25/06 10:41 AM Page 347
13. (1, 6)
14. (0, �1)
15. (3, �2)
16. (5, �2)
17. (3.5, �2)y
x1O
–1(3.5, –2)
y
x
1
–1O
(5, –2)
y
x1–1
O
(3, –2)
y
x
1
–1O
(0, –1)
y
x
1
1O
(1, 6)
18. (3, 2)
19. (0, 0)
20. (�5, 3)
21. (�2, �2)
22. (�1, 0)y
x–1
O1
(–1, 0)
y
x
1
O 1
(–2, –2)
y
x
1
O
(–5, 3)
y
x
1
1O(0, 0)
y
x
1
1O
(3, 2)
348
14271AKTE2.pgs 9/25/06 10:41 AM Page 348
23. a.
b. Inconsistent24. a.
b. Consistent and dependent25. a.
b. Consistent and independent26. a.
b. Consistent and dependent27. a.
b. Inconsistent
y
x
y
x1O
1
y
x1O
–1
(–2, –3)
y
x1O1
y
x–1
O–1
28. a.
b. Consistent and independentApplying Skills29. a. x � y � 8; x � y � 2
b. 5, 3
30. a. x � y � 5; y � x � 7b. �1, 6
31. a. 2x � 2y � 12; y � 2xb. Length � 4m, width � 2m
y
x
(2, 4)
1O1
y
x
(–1, 6)
1O1
y
x
(5, 3)
1O1
y
x
(2, 1)
1O1
349
14271AKTE2.pgs 9/25/06 10:41 AM Page 349
32. a. 2x � 2y � 14; y � x � 3b. Length � 5 cm, width � 2 cm
33. a. y � 50b. y � 0.2x � 30c.
d. U-Drive-Ite. Safe Travelf. 100 mi
10-5 Using Addition to Solve a System ofLinear Equations (pages 421–422)Writing About Mathematics
1. Yes. Equation [A] would contain 14x and equation[B] would contain �14x, which as additive inversessum to 0 thereby eliminating the variable x.
2. Yes. Equation [A] would contain �6y and equation[B] would contain 6y, which as additive inversessum to 0 thereby eliminating the variable y.
Developing Skills3. (8, 4) 4. (a, b) � (9, 4)5. (5, 1) 6. (c, d) � (12, �1)7. (a, b) � (8, 4) 8. (a, b) �
9. (m, n) � 10. (3, 2)
11. (�3, 2) 12. (6, 5)13. (10, �6) 14. (�15, 12)15. (r, s) � (2, 1) 16. (�3, 1)17. (�7, 9) 18. (a, b) �19. (6, 5) 20. (r, s) � (6, 0)21. (6, 4) 22. (�8, 3)23. (24, 8) 24. (a, b) � (8, 12)25. (c, d) � (9, 4) 26. (a, b) � (6, 4)
A0, 252 B
A212, 3 B
A 12, 1 B
100
90
80
70
60
50
40
30
20
10
0 10 20 10090807060504030 110 120 130 140
Distance (miles)
Cos
t (do
llars
)
y
x
(100, 50)
y
x
(2, 5)
1O1
Applying Skills27. $200 in savings, $300 in a CD28. $150 at 3%, $250 at 5%29. $500 at 3%, $100 at 6%30. Robin, 7; Greta, 1431. Kiwi, $0.85; zucchini, $0.6032. 3,000 gal of gasoline; 1,000 gal of kerosene
10-6 Using Substitution to Solve a Systemof Linear Equations (pages 424–426)Writing About Mathematics
1. Solving x � 2y � 2 for x does not require divisionsince the coefficient is 1.
2. There is no solution because the system ofequations is inconsistent.
Developing Skills3. (7, 7) 4. (7, 14)5. (a, b) � (2, �1) 6. (4, 5)7. (a, b) � (�8, �3) 8. (a, b) � (6, 5)9. (a, b) � (4, 3) 10. (5, 4)
11. (d, h) � (7, 1) 12. (6, 4)13. (�4, 3) 14. (5, �1)15. (r, s) � (3, �3) 16. (a, b) � (5, 2)17. (7, 5) 18.19. (6, 18) 20. (a, b) � (10, 9)21. (�3, 4) 22. (60, 240)23. (c, d) � (5, 1) 24. (4, 3)25. (a, b) � (7, 7) 26. (a, b) � (12, �3)Applying Skills27. a. y � 2x � 6 28. a. y � x � 1.16
b. 2x � y � 78 b. 3x � y � 11.48c. 21 cm, 21 cm, c. Film, $ 2.58;
36 cm batteries, $3.7429. a. y � 2x � 0.30 30. a. y � x � 12
b. x � y � 3.9 b. x � y � 54c. Hot dog, $2.50; c. Jessica, 21; Terri, 33
cola, $1.40
10-7 Using Systems of Equations to SolveProblems (pages 429–431)Writing About Mathematics
1. She would have solved x � y � 12 for one of thevariables since the coefficients of x and y areboth 1, causing division to be unnecessary.
2. a. Answers will vary. Example: Substitution isthe most efficient since the first equation caneasily be solved for x or y.
b. Answers will vary. Example: Graphing is the least efficient because the solution involves fractions.
A 25, 35 B
A4, 12 B
350
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Developing Skills3. 30, 6 4. 35.5, 38.55. 35, 69 6. 10.5, 35.57. 10, 33 8. 30.4, 45.49. 400, 500
Applying Skills10. Length � 17 cm, width � 8 cm11. Length � 14 ft, width � 5 ft12. 30°, 150° 13. 65°, 115°14. 30°, 60° 15. 32°, 58°16. 40°, 70°, 70°17. Pretzel, $0.75; soda, $0.5018. Gardener, $60; helper, $3019. Bat, $9; ball, $4.5020. Veal, $6.25; pork, $2.5021. Brown rice, $2.50/kg; basmati rice, $2.90/kg22. 60 advance tickets, 40 tickets at the door23. 120 rackets24. Four $20 bills, two $10 bills25. Six 39-cent stamps, nine 24-cent stamps26. Squash, $0.49; eggplant, $0.6927. Wilma, $31,500; Roger, $35,50028. $400 at 5%, $1,000 at 8%29. $2,600 at 4%, $1,400 at 6%30. $9,000 at 8%, $12,000 at 6%31. 5 lb of $3/lb candy, 15 lb of $2/lb candy
10-8 Graphing the Solution Set of a Systemof Inequalities (pages 434–435)Writing About Mathematics
1. x � y � 4, y � 2x � 3; by graphing the opposite ofeach of the given inequalities (excluding the planedivider), the solution is the unshaded region of theoriginal system.
2. The points on the line y � 2x � 3 that are abovethe line x � y � 4
3. The empty set; because the plane dividers y � 4xand 4x � y � 3 are parallel, the portions of thegraph above the upper one and below the lowerone will not intersect.
Developing Skills4.
x
y
O
S
5.
6.
7.
8.
9.
x
y
O
S
x
y
O
S
x
y
OS
x
y
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x
y
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10.
11.
12.
13.
14.
x
y
O
S
x
y
O
S
x
y
O
S
x
y
O
S
x
y
O
S15.
16.
17.
18.
19.
x
y
O
S
x
y
O
S
x
y
O
S
x
y
O
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x
y
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352
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20.
21.
22.
23.
24. y � x � 1 25. y � �2xx � 2 y � x � 3
x � 3Applying Skills26. a. y � 2x, x � 2, y � 12
b.
x
y
O
S
2
2
x
y
O
S
x
y
O
S
x
y
O
S
x
y
O
S
c. See graph.d. No, only the coordinates that are integers.
There can be no part of a student.e. Answers will vary. Example: (3, 8)
27. a. x � y � 8, x � 1, y � 2b.
c. See graph.d. Yes, mileage can be any positive rational
number.e. Answers will vary. Example: (2.5, 4)
28.
4 cakes, 6 pies; 4 cakes, 7 pies; 4 cakes, 8 pies;5 cakes, 6 pies; 5 cakes, 7 pies; 6 cakes, 6 pies
Review Exercises (pages 436–438)1. (4, �1)2. a. Answers will vary.
b. y � 2x3. y � 3x � 5 4.5. y � �2x � 1 6. y � �37. 8. y � x
9. x � y � 1 10.27x
3 2 3y2 5 1
y 5 223x
y 5 2112x 1 131
2
x
y
O
x
y
O
2
2
x
y
O
R
1
1
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11. 12.
13. 14.15. x � �3 16. 2x � y � 417. 4x � 3y � 118. (4, 2)
19. (3, �3)
20. (�4, 0)
21. (7, �4) 22.23. (c,d) � (4,�12) 24. (�3, 5)25. (r, s) � (10, �5) 26. (0, 7)27. (5, �5) 28. (a, b) � (4, �1)29. (t, u) � (3, 9) 30. (a, b) � (8, 6)31. (400, 600) 32. (t, u) � (2, 4)33. a. (4.5, 1.5) b.
x
y
O
A21, 12 B
x
y
O
(–4, 0)
x
y
O
(3, –3)
x
y
O
(4, 2)
2x2 2
y1 5 1
2x 3 1
4y9 5 1
4x3 2
7y5 5 1x
5 2 y2 5 1 34.
35.
36.
37. x � y � 7, x � y � 18; (12.5, �5.5)38. 3n � 2p � 2.80, 2n � 5p � 2.60; pencil, $0.20;
notebook, $0.8039. x � y � 90, y � 2x � 15; 35°, 55°40. v � 3b, v � 2b � 180; 36°, 36°, 108°41. y � 1.03x, y � 254.41; $247
Exploration (page 439)Robby left first. Jason arrived at school first. Jasonpassed Robby at about 8:22. Robby walked at a rateof 3 mph. Jason biked at a rate of 9 mph.
Cumulative Review (pages 439–441)Part I
1. 4 2. 3 3. 4 4. 3 5. 46. 1 7. 1 8. 4 9. 2 10. 4
Part II11. (2, 0); y � �3x � 6, 0 � �3x � 6, x � 2
12. 2.5%;Part III13. Muffin, $1.25; coffee, $0.80; 2m � c � 3.30,
3m � 2c � 5.35
6,355 2 6,2006,200 5 0.025
x
y
O
A
x
y
O
A
x
y
O
A
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14. 42°; 54.8 ft � 37.2 ft � 17.6 ft, tan �A � ,�A � 42°
Part IV15. a. y � 1.5x � 25
5045403530252015105
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Time (hours)
Cos
t (do
llars
) (10, 40)y = 1.5x + 25
y = 4x
x
y
A
B C
D17.6 ft
37.2 ft
15.8 ft
15.817.6
b. y � 4x; See graph.c. 10 hr; 4x � 1.5x � 25, 2.5x � 25, x � 10
16.
a. 504 cm2; area of ABCD � 24 42 � 1,008 cm2;area of �ADE � � 384 cm2; area of �BCE � � 120 cm2; area of �ABE � 1,008 � 384 � 120 � 504 cm2
b. 106 cm; AE � � 40 cm; BE �
� 26 cm; 42 cm � 40 cm �26 cm � 108 cm"242 1 102
"242 1 322
12(10 ? 24)
12(24 ? 32)
A D
B C
E
24 cm
32 cm
10 cm
355
Introductory Page (page 442)12 arrangements
11-1 Special Products and Factors (page 446)Writing About Mathematics
1. No. All pairs of positive integers have a commonfactor of 1.
2. 200 � 14 � 14.29; Since 14.29 � 14, he could havetried one more number.
Developing Skills3. Prime 4. Composite 5. Prime6. Composite 7. Prime 8. Composite9. Prime 10. Composite 11. Composite
12. Neither 13. 5 7 14. 2 32
15. 24 32 16. 7 11 17. 27
18. 24 52 19. 2 101 20. 3 4321. 2 5 59 22. 22 79 23. 1, 2, 13, 2624. 1, 2, 5, 10, 25, 50 25. 1, 2, 3, 4, 6, 9, 12, 18, 3626. 1, 2, 4, 8, 11, 22, 44, 88 27. 1, 2, 4, 5, 10, 20, 25, 50,
10028. 1, 2, 11, 22, 121, 242 29. 1, 3730. 1, 2, 31, 62 31. 1, 11, 23, 25332. 1, 2, 3, 6, 17, 34, 51, 102 33. 1, 2, 5, 7, 10, 14, 35, 7034. 1, 13, 169
35. a. 12xy b. 6y2 c. 3x2
d. 4y3 e. 2y2
36. a. 3c9d12 b. 18c7de c. 27c8d9
d. 9c5d8e e. c4d3
37. 5 38. 4 39. 740. 6 41. 25 42. 3643. 4 44. 2r 45. 2x46. 5x 47. 9xy2z 48. 6ac2
49. ab 50. xyz 51. 1
11-2 Common Monomial Factors (page 448)Writing About Mathematics
1. No. The product 3a(2a � 5a2) yields a binomial.The factored form is 3a(1 � 2a � 5a2).
2. a. Yes; 2 and 4 will always be factors for anypositive integral values of a and b.
b. No. If a and b have a common factor c, otherthan 1, then 3a � 5b can be written as the product of c and .
Developing Skills3. 2(a � b) 4. 3(x � y) 5. b(x � y)6. x(c � d) 7. 4(x � 2y) 8. 3(m � 2n)9. 6(2x � 3y) 10. 9(2c � 3d) 11. 8(x � 2)
12. 7(y � 1) 13. 6(1 � 3c) 14. y(y � 3)
A 3ac 1 5b
c B
Chapter 11. Special Products and Factors
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15. x(2x � 5) 16. a(x � 5b) 17. 3y2(y2 � 1)18. 5x(2 � 3x2) 19. 2x(1 � 2x2) 20. p(1 � rt)21. r(r � l) 22. r(r � 2h) 23. 3(a2 � 3)24. 4y(3y � 1) 25. 3ab(b � 2a) 26. 7r2s(3rs � 2)27. 3(x2 � 2x � 10) 28. c(c2 � c � 2)29. 3a(3b2 � 2b � 1)Applying Skills30. 2(l � w) 31. 32. a. 8s3
b. 2s3
c. 8s3 � 2s3
d. 2s3(4 � )
11-3 The Square of a Polynomial (pages 449–450)Writing About Mathematics
1. When squaring, the exponent of each variable ismultiplied by 2, so the resulting exponent is even.
2. No. When the square of a number has an evennumber of digits, the exponent of 10 is odd.For example, 502 � 2,500 � 2.5 � 103.
Developing Skills3. a4 4. b6 5. d10
6. r2s2 7. m4n4 8. x6y4
9. 9x4 10. 25y8 11. 81a2b2
12. 100x4y4 13. 144c2d6 14.
15. 16. 17.
18. 19. 0.64x2 20. 0.25y4
21. 0.0001x2y2 22. 0.0036a4b2
Applying Skills23. 16x2 24. 100y2 25.26. 2.25x2 27. 9x4 28. 16x4y6
11-4 Multiplying the Sum and Differenceof Two Terms (pages 451–452)Writing About Mathematics
1. (5a � 10)(a � 2) � 5(a � 2)(a � 2) � 5(a2 � 4)2. Yes; (x � 2)2(x � 2)2 � (x � 2)(x � 2)(x � 2)(x � 2)
� (x � 2)(x � 2)(x � 2)(x � 2)� (x2 � 4)(x2 � 4)� (x2 � 4)2
Developing Skills3. x2 � 64 4. y2 � 100 5. n2 � 816. 144 � a2 7. c2 � d2 8. 9x2 � 19. 64x2 � 9y2 10. x4 � 64 11. 9 � 25y6
12. 13. r2 � 0.25 14. 0.09 � m2
15. a4 � 625 16. x4 � 81 17. a4 � b4
Applying Skills18. x2 � 49 19. 4x2 � 9 20. c2 � d2
21. 4a2 � 9b2 22. 900 � 9 � 891
a2 2 14
49x2
16x4
25
x2
364964a4b425
49x2y2
916a2
12h(a 1 b)
23. 2,500 � 4 � 2,49624. (70 � 5)(70 � 5) � 4,900 � 25 � 4,87525. (20 � 1)(20 � 1) � 400 � 1 � 399
11-5 Factoring the Difference of TwoSquares (pages 453–454)Writing About Mathematics
1. 17 and 23; (400 � 9) � (20 � 3)(20 � 3)2. Yes; 5a2 � 45 � 5(a2 � 9) � 5(a � 3)(a � 3).
The two binomial factors are (a � 3) and (a � 3).
Developing Skills3. (a � 2)(a � 2) 4. (c � 10)(c � 10)5. (3 � x)(3 � x) 6. (12 � c)(12 � c)7. (4a � b)(4a � b) 8. (5m � n)(5m � n)9. (d � 2c)(d � 2c) 10. (r2 � 3)(r2 � 3)
11. (5 � s2)(5 � s2) 12. (10x � 9y)(10x � 9y)13. 14. (x � 0.8)(x � 0.8)15. (0.2 � 7r)(0.2 � 7r) 16. (0.4y � 3)(0.4y � 3)17. (0.9 � y)(0.9 � y) 18. (9m2 � 7)(9m2 � 7)Applying Skills19. (x � 2)(x � 2) 20. (y � 3)(y � 3)21. (t � 7)(t � 7) 22. (t2 � 8)(t2 � 8)23. (2x � y)(2x � y) 24. a. c2 � d2
b. (c � d)(c � d)25. a. (4x2 � y2) 26. a. x2 � y2
b. (2x � y)(2x � y) b. (x � y)(x � y)27. (5a � 2b)(5a � 2b) 28. (3x � 4y)(3x � 4y)
11-6 Multiplying Binomials (page 456)Writing About Mathematics
1. Subtract the the square of the last term (b2) fromthe square of the first term (a2x2).
2. The inner product 10y and the outer product 21xcannot be combined because they are not liketerms.
Developing Skills3. x2 � 8x � 15 4. 18 � 9d � d2
5. x2 �15x � 50 6. 24 � 11c � c2
7. x2 � 5x � 14 8. n2 � 17n � 609. 45 � 4t � t2 10. 3x2 � 17x � 10
11. 3c2 � 16c � 5 12. y2 � 16y � 6413. a2 � 8a � 16 14. 4x2 � 4x � 115. 9x2 � 12x � 4 16. 14x2 � x � 317. 6y2 � 13y � 6 18. 9x2 � 24x � 1619. 4x2 � 20x � 25 20. 12t2 � 13t � 1421. 25y2 � 40y � 16 22. 10t2 � 17t � 323. 4a2 � 8ab � 3b2 24. 15x2 � xy � 28y2
25. 10c2 � 19cd � 6d2 26. a2 � 2ab � b2
27. 2a2 � 3a � 2ab � 3b 28. 24t2 � 4t � 6tz � z29. 81y2 � 18yw � 3w2
Aw 1 18 B Aw 2 18 B
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Applying Skills30. a. x2 � 9x � 20 31. a. x2 � 12x � 36
b. 2x2 � x � 3 b. x2 � 4x � 4c. 4x2 � 4x � 1d. 9x2 � 12x � 4
32. a. 2x � 3b. x � 5c. 2x � 13d. 2x2 � 23x � 65
11-7 Factoring Trinomials (page 461)Writing About Mathematics
1. Yes; (x � d)(x � e) � x2 � dx � ex � de � x2 � (d � e)x � de
If c � de and b � d � e, then x2 � (d � e)x �de � x2 � bx � c.
2. There are no two positive integers whose productis c and whose sum is 1.
Developing Skills3. (a � 2)(a � 1) 4. (c � 5)(c � 1)5. (x � 7)(x � 1) 6. (x � 1)(x � 10)7. (y � 4)(y � 2) 8. (y � 4)(y � 2)9. (y � 8)(y � 1) 10. (y � 8)(y � 1)
11. (y � 4)(y � 2) 12. (y � 4)(y � 2)13. (y � 8)(y � 1) 14. (y � 8)(y � 1)15. (x � 3)(x � 8) 16. (a � 2)(a � 9)17. (z � 5)(z � 5) 18. (x � 2)(x � 3)19. (x � 6)(x � 4) 20. (x � 2)(x � 1)21. (x � 7)(x � 1) 22. (y � 5)(y � 1)23. (c � 7)(c � 5) 24. (x � 9)(x � 2)25. (z � 12)(z � 3) 26. (2x � 1)(x � 2)27. (3x � 4)(x � 2) 28. (4x � 1)(4x � 1)29. (2x � 3)(x � 1) 30. (2x � 1)(2x � 5)31. (5a � 2)(2a � 1) 32. (3a � b)(a � 2b)33. (4x � 3y)(x � 2y)Applying Skills34. (x � 9)(x � 1) 35. (x � 5)(x � 4)36. (3x � 5)(x � 3) 37. (x � 5)38. (9x � 1) 39. (2x � 3)
11-8 Factoring a Polynomial Completely(pages 463–464)Writing About Mathematics
1. No. In both terms, a2 can be factored out;4a2 � a2b2 � a2(2 � b)(2 � b)
2. No; (x � 1)(x � 1)(x � 1) � x3 � x2 � x � 1Applying Skills
3. 2(a � b)(a � b) 4. 4(x � 1)(x � 1)5. a(x� y)(x � y) 6. s(t � 3)(t � 3)7. 2(x � 4)(x � 4) 8. 3(x � 3y)(x � 3y)9. 2(3m � 2)(3m � 2) 10. 7(3c � 1)(3c � 1)
11. x(x � 2)(x � 2) 12. z(z � 1)(z � 1)13. 4(a � 3)(a � 3) 14. (x2 � 1)(x � 1)(x � 1)15. (y2 � 9)(y � 3)(y � 3) 16. (c � d)(c � d)17. 3(x � 1)(x � 1) 18. 4(r � 3)(r � 4)19. x(x � 5)(x � 2) 20. 2(2x � 1)(x � 2)21. d(d � 4)(d � 4) 22. 2a(x � 3)(x � 2)23. x2(4 � y2)(2 � y)(2 � y)24. (a � 1)(a � 1)(a � 3)(a � 3)25. (y � 3)(y � 3)(y � 2)(y � 2)26. 5(x2 � 1)(x2 � 1) 27. b(2a � 1)(a � 3)28. 4(2x � 1)(2x � 1) 29. 25(x � 2y)(x � 2y)Applying Skills30. a, (3a � 2b), (4a � b)31. a. 3
b. (m � 2)32. a. 10a
b. (a � 1)
Review Exercises (pages 465–466)1. No; (2x � 2)(2x � 6) � 2(x � 1)2(x � 3) �
4(x � 1)(x � 3)2. 2 53 3. 4a 4. 8a2bc2
5. 9g6 6. 16x8 7. 0.04c4y2
8. 9. x2 � 4x � 4510. y2 � 14y � 48 11. a2b2 � 1612. 3d2 � 5d � 2 13. 4w2 � 4w � 114. 2x2 � 11cx � 12c2 15. 3(2x � 9b)16. y(3y � 10) 17. (m � 9)(m � 9)18. (x � 4h)(x � 4h) 19. (x � 5)(x � 1)20. (y � 2)(y � 7) 21. (8b � 3)(8b � 3)22. (11 � k)(11 � k) 23. (x � 4)(x � 4)24. (a � 10)(a � 3) 25. (x � 10)(x � 6)26. 16(y � 1)(y � 1) 27. 2(x � 2b)(x � 8b)28. (x2 � 1)(x � 1)(x � 1) 29. 3x(x � 4)(x � 2)30. k2 � 225 31. 16e3z2 � 4ez3
32. (15a � 2)(4a � 3) 33. (4)34. Answers will vary. Example: x2 � 9 is the
difference of two perfect squares (x � 3)(x � 3);x2 � 2x � 1 is a binomial squared (x � 1)2;x2 � 2x � 1 has no integral factors; x3 � 5x2 � 6xis the product of three factors x(x � 3)(x � 2)
35. 6x2 � 13x � 6 36. 64m2 � 16m � 137. (3x � 5) 38. a. 121 persons
b. 11 persons
Exploration (page 466)a. Let a be any integer and b � 1.Then the smaller
number (a � 1) is 2 less than the greater number(a � 1), which means they are either consecutiveeven or odd integers.Their product, (a � 1)(a � 1),is a2 � 1.
14a6b10
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b. (1) 1 � 3 � 22, so 3 � 22 � 1 or 22 � 12
1 � 3 � 5 � 32, so 5 � 32 � (1 � 3) � 32 � 22
1 � 3 � 5 � 7 � 42, so 7 � 42 � (1 � 3 � 5) � 42 � 32
(2) 2n � 1 � n2 � [1 � 3 � 5 � . . . � (2n � 3)]We know that [1 � 3 � 5 � . . . � (2n � 3)] �(n � 1)2, so 2n � 1 � n2 � (n � 1)2.
Cumulative Review (pages 466–468)Part I
1. 4 2. 1 3. 3 4. 2 5. 46. 3 7. 3 8. 4 9. 4 10. 2
Part II11. 28°; Explanations will vary. Example: sin x � ,
x � 28°
12. x � 16; , 3x � 2(x � 8), x � 16xx 1 8 5 2
3
60 68
32
x
3268
Part III13. a. 2w � 2(w � 10) � 68 or 2l � 2(l �10) � 68
or w � l � 10, 2w � 2l � 68b. Length � 12 ft, width � 22 ft; Explanations
will vary according to part a.14. 0.2% increase; 7,000 � 5 � 7 � 28 � 12 � 7,014;
Part IV15. 270 adults, 350 children; Explanations will
vary: a � c � 620, 5a � 3c � 2,400; 5(620 � c) �3c � 2,400; 3,100 � 5c � 3c � 2,400; c � 350,a � 270
16. a. y � �2x � 5b.
c. No; �2(�1) � 5, 4 � 5, 3 9d. 2; 1 � �2k � 5, 2k � 4, k � 2
3 5?
3 5?
–1 1
1 O x
y
7,014 2 7,0007,000 5
147,000 5 0.002
358
12-1 Radicals and the Rational Numbers(pages 475–476)Writing About Mathematics
1. � �3; on the other hand, does notexist in the set of real numbers because thesquare of any real number is nonnegative.
2. The product of an odd number of negativefactors is negative, so (�5)3 � �125. The productof an even number of factors is always positive,so (�5)4 � 625 and not �625.
Developing Skills3. 4 4. �8 5. �106. �13 7. 20 8. �259. 10. 11. �
12. 0.8 13. �1.2 14. �0.315. �0.02 16. 1 17. 318. 2 19. �2 20. 521. 0.6 22. �3 23. 3.224. �6.8 25. 1.3 26. �1527. 8 28. �4 29. 1030. 5.7 31. �0.5 32. �2.433. 8 34. 35. 0.736. 3 37. 97 38. 92
12
5923
412
"292"9
39. � 40. � 41. �42. � 43. � 44. �45. � 46. � 47. �
48. x � �2 49. y � � 50. x � �0.751. x � �4 52. y � �6 53. x � �554. y � �3 55. x � 2Applying Skills56. 10 in. 57. 13 cm 58. 25 m59. 39 ft 60. a. 6 ft 61. a. 14 yd
b. 24 ft b. 56 yd62. a. 11 cm 63. a. 15 m 64. 4x
b. 44 cm b. 60 m65. 101 � 102 � 12
102 � 102 � 12 � 12 or 72 � 72 � 22
103 � 102 � 12 � 12 � 12 or 92 � 32 � 32 � 22
or 72 � 72 � 22 � 12
104 � 102 � 22
105 � 102 � 22 � 12 or 82 � 52 � 42
106 � 92 � 52
107 � 92 � 52 � 12 or 72 � 72 � 32
108 � 102 � 22 � 22 or 62 � 62 � 62
109 � 102 � 32
110 � 102 � 32 � 12 or 92 � 52 � 22 or 72 � 62 � 52
29
Chapter 12. Operations with Radicals
14271AKTE2.pgs 9/25/06 10:41 AM Page 358
12-2 Radicals and the Irrational Numbers(pages 481–482)Writing About Mathematics
1. a. 999.9995 b. Yesc. Rational; 999.9995 is an exact square root
since (999.9995)2 � 999,999.2. No; , which his rational.
Developing Skills3. 2 and 3 4. 3 and 4 5. 6 and 76. �2 and �1 7. �4 and �3 8. 7 and 89. 8 and 9 10. �12 and �11 11. 11 and 12
12. 6 and 7 13. �1, , 2 14. 3, 4,15. �4, , �3 16. , 0,17. , 5, 18. , ,19. Rational 20. Irrational 21. Rational22. Irrational 23. Irrational 24. Rational25. Irrational 26. Rational 27. Rational28. Irrational 29. Rational 30. Irrational31. Rational 32. Rational 33. Irrational
In 34�48, part a, answers will vary depending on thenumber of digits in the calculator display.34. a. 1.414213562 35. a. 1.732050808
b. 1.4142 b. 1.732136. a. 4.582575695 37. a. 6.244997998
b. 4.5826 b. 6.245038. a. 8.94427191 39. a. 9.486832981
b. 8.9443 b. 9.486840. a. 10.39230485 41. a. 4.847679857
b. 10.3923 b. 4.847742. a. 9.391485505 43. a. �10.73312629
b. 9.3915 b. �10.733144. a. 5.344155686 45. a. 8.200609733
b. 5.3442 b. 8.200646. a. 66.2495283 47. a. 11.12205017
b. 66.2495 b. 11.122148. a. 11.59870682 49. a. 9.65
b. 11.5987 b. 9.64650. a. 3.86 51. a. 5.66 52. a. 24.07
b. 3.864 b. 5.657 b. 24.07153. a. 3.73 54. a. 17.27 55. a. 7.40
b. 3.732 b. 17.268 b. 7.40256. a. 9.20
b. 9.20057. Answers will vary; 7.615773106 � n � 7.61642961Applying Skills58. 4.2 cm 59. 5.4 cm 60. 9.8 cm61. 11.8 cm 62. 14.2 cm 63. 17.0 cm64. 13.83 65. 8.76 66. 8.9467. 38.72
2"112"192"23"30"21"72"72"15
"17"3
#1850 5 # 9
25 5 35
12-3 Finding the Principal Square Rootof a Monomial (pages 483–484)Writing About Mathematics
1. Yes; � �x� for all values of x, and since x � 0,�x � 0.
2. Yes;Developing Skills
3. 2a 4. 7z 5.6. 0.9w 7. 3c 8. 6y2
9. cd 10. 2xy 11. 12a2b12. 0.6m 13. 0.7ab 14. 8.4bx5
Applying Skills15. a. 7c 16. a. 8x 17. a. 10xy
b. 28c b. 32x b. 40xy18. a. 12ab 19. 41x
b. 48ab
12-4 Simplifying a Square-Root Radical(pages 486–487)Writing About Mathematics
1. No;
2. a. can be simplified further because 12 hasa perfect square as a factor.
b.c.
Developing Skills
3. 4. 5.
6. 7. 8.
9. 10. 11.
12. 13. 14.
15. 16. 17.
18. 19. 20.
21. 22. 23. (3)24. (3) 25. (3) 26. (4)
In 27�30, parts a and c, answers will vary dependingon the number of digits in the calculator display.27. a. 17.32050808 28. a. 13.41640786
b. b.c. 17.32050808 c. 13.41640786d. Yes d. Yes
29. a. 33.9411255 30. a. 5.291502622b. b.c. 33.9411255 c. 5.291502622d. Yes d. Yes
31. a. No; � � 5, but � �3 � 4 � 7.
b. No
"16"9"25"9 1 16
2"724"2
6"510"3
9y"3x6r"s
7x2"xx"3x24"x
3"1012"24
3"6
65"51
2"26"5
3"6"310"6
4"58"37"2
3"73"32"3
8"34"12 5 4"4 ? 3 5 4"4 ? "3 5 8"3
4"12
13"27 5 1
3"9 ? 3 5 13"9 ? "3 5 "3
45r
xa2 ? x
a2 5 x
a2 1 a2 5 xa
"x2
359
14271AKTE2.pgs 9/25/06 10:41 AM Page 359
32. a. No; � � 12, but �
� 13 � 5 � 8.b. No
12-5 Addition and Subtraction of Radicals(pages 490–491)Writing About Mathematics
1. The two operations are alike in that only liketerms can be added, and sometimes bothradicals and fractions can be simplified to getlike terms. The operations differ in thatfractions can always be written with a commondenominator and added, but there is not alwaysa common radical.
2. No. Using the distributive property,and not 3.
Developing Skills
3. 4. 5.6. 0 7. 8.9. 10. 11.
12. 13. 14.15. 16. 17.18. or
19. 20. 21.
22. 23. 24. 2x �25. (1) 26. (4) 27. (4)28. a. 29. a. 30. yd
b. 26.833 b. 17.321
12-6 Multiplication of Square-RootRadicals (page 493)Writing About Mathematics
1. Irrational; since a and b have no factors other than themselves and 1, and a b, cannot be a perfect square.
2. No. While � �2a�, if a is irrational, so is 2a.For instance, if a � , then � 2.
Developing Skills3. 3 4. 7 5. a6. 2x 7. 12 8. 2889. 10. 11.
12. 13. 14.15. �12a 16. 3y 17. 218. y 19. t 20. 5421. 10x 22. 9a 23.24. 25. 5x 26. 4t27. a. 120 28. a. 48 29. a.
b. Rational b. Rational b. Irrational108"2
3a"b3x"5
10"1036"270"69"210"32"7
"4a2
"4a2
"ab
110"210"312"5
9"23a"3"3x
5"b3"7a56"3
2163 "2251
3"20.5"22.3"22"33"58"36"28"y12"x4"5 1 3"222"74"36"215"39"5
"5 5 (3 2 1)"5 5 2"53"5 2
"25"169"144"169 2 25 30. a. 31. a. 32. a. 1
b. Irrational b. Rational b. Rational33. a. 34. a.
b. Irrational b. Irrational35. Answers will vary.
a.b.
Applying Skills36. 2 37. 1238. 72 39. 7540. a.
b.41. a. 12
b.
12-7 Division of Square-Root Radicals (pages 495–496)Writing About Mathematics
1. No. While is not equivalent
to . Ross took the square root of twice, or
the fourth root of .2. Rational; its square root, , is a rational number.
Developing Skills3. 6 4. 5 5.6. 7. 16 8.9. 10. 11. 3
12. 13. 14. 315. 8 16. 25 17.18.19. Irrational 20. Rational 21. Irrational22. Irrational 23. Rational 24. Irrational25. Irrational 26. Irrational 27.
28. 29. 30.
31. 32. 33. abc
34.Applying Skills35. 2 36. 37.38. or 3.5
Review Exercises (pages 497–498)1. a is a multiple of b and a and b have the
same sign.2. Irrational; since a is a perfect square, is
rational, but is irrational, so the product is irrational."ab 5 "a ? "b
"b"a
72
4"32"5
5ab "a or 5a"a
b
12"xy4"2
27"2 or 2"2
7"512"3 or "3
2
67
x2z"xyz or 12x#
xyz
bc2"b
14
73
42"25"22"3"7"7
14
#1681
#1681
23
#1681 5
"16"81
5 49, 49
2"12 ? "3 5 12
2"3 ? "2 5 2"6 5 4.8989794862"6
"5 ? "5 5 5, 2"2 ? 3"8 5 24"2 ? "5 5 "10, "5 ? "8 5 2"10
3"19012549 "7
20920"2
360
14271AKTE2.pgs 9/25/06 10:41 AM Page 360
3. 35 4. , 3, 5.6. �7 7. �3 8. �1.19. 20y2 10. 11.
12. 13. 14. 0.1m8
15. 16. 0.5a4b5 17. y � �918. m � �0.3 19. x � � 20. k � �21. a. 17.76
b. Rational; the radicand 315.4176 is a perfectsquare, and therefore rational, since (17.76)2 �315.4176.
22. 23. 0 24.25. 26. 32 27. 4528. 29. 7 30.31. 1 32. 33.34. (3) 35. (2)
In 36–39, part a, answers will vary depending on thenumber of digits in the calculator display.36. a. 13.92838828 37. a. 2.5198421
b. 13.93 b. 2.52038. a. �0.8366600265 39. a. �1.933182045
b. �0.8367 b. �1.93340. a. 5.292 m
b. 21.166 mc. The length in part a is a rounded value, which
adds almost 0.0005 m to the actual length;the perimeter was found by multiplying theunrounded length of a side by 4 and thenrounding.
41. 2x2 � � 3 42. 21.7x5
Exploration (pages 498–499)
STEP 1. By the Pythagorean theorem, the
hypotenuse .
STEP 4. After locating on the number line,
repeat step 2, drawing a rectangle whose dimensions
are by 1 to locate on the number line:
("n 2 1)2 1 12 5 ("n)2
"n"n 2 1
"n 2 1
c 5 "12 1 12 5 "2
O
1
1 √2 √3√4
√52 3x
x"3
8"10"28"314"10
2"36"2"2
6"210"2xy2#1
3xy4b"3b"7
9"26"5
35"182"2 Cumulative Review (pages 499–501)
Part I1. 2 2. 4 3. 14. 4 5. 3 6. 27. 4 8. 2 9. 2
10. 3Part II11. km; l tank � 48 L, � 600 km, 1 tank �
600 � 800 km; 48 L � 800 km, 1 L � km12. 4.43 ft; tan 12.5° � , 20.0(tan 12.5°) � x,
x � 4.43Part III13.
a. 150; area of ABCD � ,area of ABCE � � 102;area of AED � area of ABCD � area ofABCE � 252 � 102 � 150
b. 32; AD � ,
AE � ; perimeter of
AED � DE � AD � AE � 4 � 13 � 15 � 3214. Length � 12 ft, width � 6 ft; l � 2w, l � 1.5(w � 2),
l � 1.5w � 3; 2w � 1.5w � 3, 0.5w � 3, w � 6, l � 12Part IV15. ; side � �
� �
,16. a. y � �2x � 8
b.
c. (4, 0); 0 � �2x � 8, 2x � 8, x � 4d. 4
x
1
A
BO
1
y
13"2 1 13"2 1 10"2 5 36"213"2
"2 ? 169"50 1 288 5 "2(25 1 144)
#A5"2B2 1 A12"2B236"2
"122 1 92 5 "225 5 15
"122 1 52 5 "169 5 13
12(12)(13 1 4)
12(12)(13 1 8) 5 252
A
D
E
C
5
4
4 4
4
12
13
B
15
x20.0
1623
43
34 tank162
3
361
14271AKTE2.pgs 9/25/06 10:41 AM Page 361
362
13-1 Solving Quadratic Equations (pages 507–508)Writing About Mathematics
1. Yes. The equation can be rewritten as x2 � 9 � 0.This can be factored into (x � 3)(x � 3) � 0,which yields solutions x � 3 or x � �3.
2. The third factor was a constant, �8. There is nopossible way to set �8 � 0.
Developing Skills3. x � 1 or x � 2 4. z � 1 or z � 45. x � 4 6. r � 5 or r � 77. c � �1 or c � �5 8. m � �9 or m � �19. x � �1 10. y � �8 or y � �3
11. x � �1 or x � 5 12. x � �3 or x � 213. x � �5 or x � 3 14. t � �8 or t � 915. x � �3 or x � 4 16. x � �7 or x � 717. z � �2 or z � 2 18. m � �8 or m � 819. x � �2 or x � 2 20. d � 0 or d � 221. s � 0 or s � 1 22. x � �3 or x � 023. z � �8 or z � 0 24. x � �2 or x � 325. y � �4 or y � 7 26. c � 3 or c � 527. r � �2 or r � 2 28. x � �11 or x � 1129. y � 0 or y � 6 30. s � �4 or s � 031. y � �2 or y � 10 32. x � 4 or x � 533. x � �5 or x � 6 34. x � �9 or x � 635. x � �2 or x � 36. x � �5 or x � 737. y � �1 or y � 4 38. y � �8 or x � 539. x � �6 or x � 4 40. y � �6 or y � 341. x � �6 or x � 6 42. x � �243. x � �3 or x � �1 44. x � 1 or x � 7Applying Skills45. t � 1 sec and t � 2 sec46. t � 4 sec47. a. w(2w � 12) � 320 or 2w2 � 12w � 320
b. Length � 32 ft, width � 10 ft48. a.
b. 100 ft49. Length � 24 ft, width � 10 ft50. 10 cm, 24 cm
13-2 The Graph of a Quadratic Function(pages 518–521)Writing About Mathematics
1. The axis of symmetry of the parabola is at x � 4.Her graph shows only half of the parabola. Afterx � 4, the graph turns upward.
12(b 1 b 1 20)(b 1 20) 5 9,000
212
2. Answers will vary. Example: Since 4 is theturning point, it should be in the middle of thex-values. Answers may include x � 0 to x � 8or x � �4 to x � 12.
Developing Skills3. a.
b. x � 0c. (0, 0)
4. a.
b. x � 0c. (0, 0)
5. a.
b. x � 0c. (0, 1)
x
y
1
–1 1O
xy
–1–1 1
O
x
y
1
–1 1O
Chapter 13. Quadratic Relations and Functions
14271AKTE2.pgs 9/25/06 10:41 AM Page 362
6. a.
b. x � 0 c. (0, �1)7. a.
b. x � 0 c. (0, 4)8. a.
b. x � 1 c. (1, �1)9. a.
b. x � 1 c. (1, 1)
x
y
1
–1 1O
x
y
1
–1 1
O
–1
x
y
1
–1 1
O
–1
x
y
1
–1 1
O
10. a.
b. x � 3 c. (3, �1)11. a.
b. x � 2 c. (2, �1)12. a.
b. x � 1 c. (1, 0)13. a.
b. x � �1 c. (�1, 4)
x
y
1
1O
–1–1
x
y
1
1O–1
x
y
1
–11
O–1
x
y
1
–11
O
363
14271AKTE2.pgs 9/25/06 10:41 AM Page 363
14. a.
b. x � 2 c. (2, 1)15. a. x � 3 b. (3, �10)
c.
16. a. x � 1 b. (1, 7)c.
17. a. x � �4 b. (�4, �4)c.
x
y
1O
–1–1
x
y
1O–1
7
x
y
1
O
–1
x
y1
1
O
–1–1
18. a. x � �2 b. (�2, �1)c.
19. a. x � 1.5 b. (1.5, 4.75)c.
20. a. x � �0.5 b. (�0.5, 4.75)c.
21. a. y � �(x � 3)2
b. y � x2 � 9c. y � �6(x � 4)2 � 1
22. a. (�1, 0); x � �1b. y � (x � 1)2
23. a. (�3, 4); x � �3b. y � (�x � 3)2 � 4
24. a. (2, 5); x � 2b. y � (x � 2)2 � 5
25. a. (3, �4); x � 3b. y � �(x � 3)2 � 4
26. (2)
27
x
y
1
O
–1
4
x
y
1O
–1
4
x
y
1O
–1–1 1
364
14271AKTE2.pgs 9/25/06 10:41 AM Page 364
Applying Skills27. a. x � 4 b. y � x(x � 4) � x2 � 4x
c.
d. No. Lengths and areas cannot take onnegative values.
28. a. 2x � 6 b. y � x(2x � 6) � x2 � 3xc.
d. No. Lengths and areas cannot take onnegative values.
29. a. 10 � x b. y � x(10 � x) � 10x � x2
c.
d. length: 5 cm, width: 5 cm e. 25 cm2
1
1
O
y
x
x
y
–1 1
O
–1
12
x
y
1
1
O
–1–1
f. Answers will vary: 4 � 6 � 24, 3 � 7 � 21,2 � 8 � 16, 1 � 9 � 9
30. a. x 0 1 2 3 4
y 3 51 67 51 3
b.
c. 3 ft d. (1) 67 ft(2) 2 sec
13-3 Finding Roots From a Graph (pages 524–525)Writing About Mathematics
1. No. The vertex is above the x-axis and the graphopens upwards so it never intersects the x-axis.Thus, there are no real x-values that make y � 0.
2. Yes. Although the vertex is above the x-axis, thegraph opens downward, so there are two pointswhere it intersects the x-axis. The two rootsare �3 and 1.
Developing Skills3. a.
b. {�5, �1} c. (x � 5)(x � 1)4. a.
b. {�1} c. (x � 1)(x � 1)
x
y
1
O–1 1
x
y
1O
–2–1
h
t
100
80
60
40
20
00 1 2 3 4 5 6
365
14271AKTE2.pgs 9/25/06 10:41 AM Page 365
5. a.
b. {�1, 3} c. (x � 1)(x � 3)6. a.
b. {�1, 2} c. (x � 1)(x � 2)7. a.
b. {1} c. (x � 1)(x � 1)8. a.
b. {1, 2} c. (x � 1)(x � 2)
x
y
–1
O
–1
x
y
1
O–1 1
x
y
1O
–11
x
y
1O
–1 1
9. a.
b. No real solutionsc. Cannot be factored
10. a.
b. {�1, 6}c. (x � 1)(x � 6)
11. (x � 6) and (x � 8)12. {�6, 3}13. (2)14. x � �2 and x � 315. 2
13-4 Graphic Solution of a Quadratic-LinearSystem (pages 528–529)Writing About Mathematics
1. The empty set. If the graphs of the equationsdo not intersect, then there is no solution tothe systems.
2. x2 cannot be a negative number.Developing Skills
3. {(�3, 0), (1, 0)}4. {(�2, �3), (0, �3)}5. {(�1, �4)}
1O
1
y
x
x
y
1O
–1 1
366
14271AKTE2.pgs 9/25/06 10:41 AM Page 366
6. {(�4, 5), (2, 5)}7. c � �48. a.
b.
c. (0, �2) and (3, �5)d. (0, �2): y � (0)2 � 4(0) � 2 � �2
y � �(0) � 2 � �2 �(3, �5): y � (3)2 � 4(3) � 2 � �5
y � �(3) � 2 � �5 �9. {(�1, 1), (2, 4)}
x
y
O–1 1–1
x
y
1
1
O
x
y
10. {(�1, �1), (4, 4)}
11. {(2, 9), (�2, 1)}
12. {(�1, �5), (4, 0)}
13. {(2, 3), (5, 0)}
x
y
O1
–11
x
y
O1
–1 1
x
y
O1–1
x
y
O
1
1–1
367
14271AKTE2.pgs 9/25/06 10:41 AM Page 367
14. {(2, 3), (4, 3)}
15. {(�2, �3), (0, 1)}
16. {(�1, �4), (2, 2)}
17. a. h
x0 1 2 3
510152025303540
(–, 25)25
(–, 41)23
(–, 25)21
(1, 37) (2, 37)
(3, 5)(0, 5)
x
y
O1
–1–1
x
y
O1
1
x
y
O1
–1
b.
c. x � 0.5 sec and x � 2.5 sec18. a.
b.
c. 3.0 sec, height of 46 ft; 4.1 sec, height of 0 ft
13-5 Algebraic Solution of aQuadratic-Linear System (pages 532–533)Writing About Mathematics
1. There is no real number x whose square is anegative number.
2. Substituting 8 for x in to the first equation,82 � y2 � 49, we find that y2 must equal �15.However, there is no real number y whose squareis a negative number.
Developing Skills3. {(0, 0), (3, 3)} 4. {(0, 5), (1, 6)}5. {(1, 0), (4, 3)} 6. {(�2, 1), (1, 4)}7. {(�1, 8), (0, 9)} 8. {(�1, 1), (2, �2)}9. {(�3, �2), (1, 2)} 10. {(0.5, 2.5), (3, 5)}
11. {(�0.5, 2.5), (0, 3)} 12. { , (2, 1)}
13. {(�3, 1), } 14. { , (4, 0)}15. {(�5, 0), (3, 4)} 16. {(�8, �6), (6, 8)}17. {(�5, �5), (5, 5)} 18. {(�3.6, �5.2), (2, 6)}19. {(�4, �2), (2, 4)} 20. {(�1, 1)}
A 32, 54 BA 1
3, 199 B
A 13, 83 B
h
x0 1 2 3
510152025303540
368
14271AKTE2.pgs 9/25/06 10:41 AM Page 368
Applying Skills21. a. x � 2y � 1 or
b. x2 � 2y2 � y � 70c. {(7, 3)}; reject the solution as it is
outside the domaind. 7 � 7, 3 � 3, 3 � 1
22. a. 8 ftb. 12 ftc. Yes. The maximum height of the doorway
occurs 4 feet from the left end. If you movethe 6-foot-wide box through the middleof the doorway, the lowest clearances willbe at 1 and 7 feet from the left end. Using the formula, and
, so the 5-foot-tall box clears the doorway.
23. Length � 7 ft, width � 6 ft24. a. a2 � b2 � 29
b. 2bc. a � 2b � 1d. a � 5, b � 2; the solution is
outside the domaine. base � 4 ft, altitude � 5 ftf. (4 � 2 ) ftg. 10 sq ft
Review Exercises (pages 534–535)1. For each x-value (except x � 0), there are two
y-values: one positive and one negative.2. For each x-value, there is only one (positive)
y-value.3. a. {1, 2, 3, 4, 5} 4. a. {1}
b. {�1, 0, 1, 2, 3} b. {�1, 0, 1, 2, 3}c. Yes c. No
5. a. {�3, �2, �1, 0, 1, 2, 3} 6. a. {0, 1, 2, 3, 4}b. {0, 1, 4, 9} b. {1}c. Yes c. Yes
7. a. x � 3b.
c. (3, �3)
x
y
O1
–1–1
"29
a 5 2235 , b 5 214
5
234(7)2 1 6(7) 5 51
4
234(1)2 1 6(1) 5 51
4
A2203 , 223
6 B
y 5 x 2 12
d. Minimume. y � �3
8. a. x � 2b.
c. (2, �5)d. Minimume. y � �5
9. a. x � �1b.
c. (�1, 7)d. Maximume. y � 7
10. a. x � 3b.
c. (3, 8)d. Maximume. y � 8
x
y
O
1
1
x
y
O
1–1–1
1
x
y
O
1–1–1
369
14271AKTE2.pgs 9/25/06 10:41 AM Page 369
11. {(�4, 10), (3, 3)}
12. {(�2, �7), (3, �2)}
13. {(�1, �1), (3, 3)}
x
y
O
1
–11
x
y
O
1–1
x
y
O1
1–1–1
14. {(�1, 5), (4, 0)}
15. {(�1, 4), (1, 4)}
16. {(1, 1), (2, 0)}
17. {(�1, 4), (4, 11)}18. {(2, 5), (4, 9)}19. {(�1, �5), (3, �1)}20. {(1, 0), (6, 5)}21. {(�2, �6), (2, 6)}22. {(�2, �1), (2, 1)}23. y � �x2 � 5x � 3.2524. 6, 725. 18 ft � 8 ft
x
y
O–1
1
x
y
O
–1–1
1
1
x
y
O
–1–1 1
370
14271AKTE2.pgs 9/25/06 10:41 AM Page 370
Exploration (page 535)22 � 4 � 22 112 � 121 � 112
32 � 9 � 32 122 � 144 � 24 32
42 � 16 � 24 132 � 169 � 132
52 � 25 � 52 142 � 196 � 22 72
62 � 36 � 22 32 152 � 225 � 32 52
72 � 49 � 72 162 � 256 � 28
82 � 64 � 26 172 � 289 � 172
92 � 81 � 34 182 � 324 � 22 34
102 � 100 � 22 52 192 � 361 � 192
202 � 400 � 24 52
In the prime factorization of the squares, theexponents of the prime factors are always even.
If n � a3 � b2 � c4, n is not a perfect squarebecause the exponent of a is odd. The square root of n is irrational because .
Cumulative Review (pages 536–538)Part I
1. 2 2. 3 3. 4 4. 3 5. 16. 1 7. 3 8. 2 9. 4 10. 4
Part II11. 33 students; 242 (math) � 208 (science) � 183
(math and science) � 267 (math or science),300 (all students) � 267 (math or science) � 33(neither math nor science)
12. 16 cm, 19 cm, 19 cm; b � (b � 3) � (b � 3) � 54,3b � 6 � 54, b � 16
Part III13. 1.3%; 5p 2 15.5
5p < 0.013
"n 5 abc2"a
14. {(0, 1), (2, 3)}; x � 1 � �x2 � 3x � 1, x2 � 2x � 0,x(x � 2) � 0; y � 0 � 1 � 1, y � 2 � 1 � 3
Part IV15. 63°; �y-intercept� � 4, �x-intercept� � 2,
tan a � � 2, a � 63°
16. a. x � 4, y � 3, x � y � 10b.
c. Answers will vary. Examples: (4, 3), (5, 4), (6, 4)
1
1
O
y
x
x
y
–1
1
a
42
371
14-1 The Meaning of an Algebraic Fraction(pages 540–541)Writing About Mathematics
1. No; is not defined for x � 0.
2. No; is not defined for b � 0 and b � �1,
while is not defined only for b � �1.
Developing Skills3. 0 4. 05. 0 6. 57. 2 8. �29. 10.
11. 2, �2 12. �2, 7Applying Skills13. 14. 15.
16. 17.3x 1 2y
4m60
10x 1 20y
980p
c5
212
12
b2 2 bb 1 1
b 2 11 1 1b
xx
14-2 Reducing Fractions to Lowest Terms(pages 544–545)Writing About Mathematics
1. Kevin did not divide the whole numerator andthe whole denominator by a common factor.Since a and 4 are not common factors of thebinomials (a � 4) and (a � 8), they cannot befactored out. This fraction cannot be reduced.
2. The two fractions must be equivalent for allvalues of a 8, but a � 4 is the only value forwhich the equality is true.
Developing Skills3. (x 0) 4. (y 0)
5. (d 0) 6. (r 0)
7. (c 0, b 0) 8. (b 0, y 0)
9. (x 0, y 0) 10. (a 0, b 0, c 0)12
59
a2b
ac
910
2c3d
3y4
13
Chapter 14. Algebraic Fractions and Equations and Inequalities Involving Fractions
14271AKTE2.pgs 9/25/06 10:41 AM Page 371
11. 3x (x 0) 12. (x 0)
13. (a 0) 14. (x 0, y 0)
15. (a 0, c 0) 16. (x 0, y 0)
17. (a 0, b 0) 18. (x 0, y 0)
19. 20.
21. (x 0) 22. m � 5 (m 0)
23. (x 0) 24. (a 0)
25. (a 0, b 0) 26. (x 0, y 0)
27. (b 0) 28. (x �2)
29. (d �2) 30. (y �x)
31. 32. (r 3s)
33. 4 (a �b) 34. (x �3)
35. (x 1) 36. �1 (x 1)
37. (b 3, b �3) 38. (s r, s �r)
39. (a 4) 40. (y x)
41. (b 3, b �3) 42. (r 3)
43. (x 4, x �4) 44. (x �2)
45. (y 1) 46. (x 1, x 3)
47. (x �3, x 5) 48. (a 3, a �3)
49. (a 1, a 6)
50. (x �5, x �3)
51. (r �3, r 5)
52. or (x 3, x �4)
53. (x 3)
54. (x �5y, x 4y)
55. a. (1) 7 (2) 10 (3) 20(4) 2 (5) �4 (6) �10
b. Each reduced fraction is equal to the numericalvalue substituted for x.
c. No. When x � 5, the denominator x � 5 � 0,and the fraction is not defined.
d. x (x 5)e. 38,756f. No. The original expression is undefined only
at x � 5 while the new expression is undefinedat both x � 0 and x � 5.
14-3 Multiplying Fractions (pages 547–548)Writing About Mathematics
1. The denominator is �x2 � 3x � 2 because .
2. No. The product is undefined for z � 0, and itis also undefined for the values of x and y for
x2 2 3x 1 22x2 1 3x 2 2
5 x2 2 3x 1 22(x2 2 3x 1 2)
5 21
x 2 3yx 1 5y
2x 2 1x 2 3
2x 1 12x 2 3
12 2 xx 2 3
r 1 1r 1 3
2(x 2 5)x 1 3
aa 2 1
a 1 2a 1 3
x 1 5x 1 3
xx 2 1
3y 2 1
x 2 1x 1 2
x 1 3x 2 4
r 1 232 2b
b 1 3
2x 1 y
324 1 a2
2s 1 r
21b 1 3
x 1 15
x 2 33
2r 2 3sAa 2 0, a 2
13b Ba
3a 2 b
yy 1 x
dd 1 2
xx 1 2
6b 1 103b2
2x 1 3y4
4a 2 ba
a 2 2a
a 1 b3x
x 2 7x
4y 2 63
3x 1 64
19xy22
3
2x923ab
2c
y3x
34a
15x2
which its factors are undefined; the first factoris undefined for z � 0, x � �z, and the secondfactor is undefined for x � 0, x � z.
Developing Skills3. 4. 20 (y 0)5. 10x 6. 5d (d 0)7. 8. (m 0, n 0)9. (x 0, y 0) 10. (x 0, y 0)
11. (m 0) 12. (r 0, s 0)13. 2m (m 0, n 0) 14. (a 0, b 0, c 0)
15. 16. (a 0)
17. (x 0, y 0) 18. (b 0)
19. (b � 1)2 (a 0, b 0) 20. (x 0)21. (r 1) 22. (s �2)23. (x 0, x �3) 24. (x �1, x 1)
25. 2(a � 3) (a 3) 26. (x 2, x �2)27. (a 0, a b, a �b, b 0)
28. (a 2, a �2, b 0)29. (a �1, a 8) 30. (x �1, y 0)
31. (c 0, y �1)
32. (a 3, a �2)
33.34. (x �3, x �2, x 2)
35. 2 (y �9, y 9)36. (x 2, x �2, x 0)
37. (x 0, x 2)38. (x �1, x 1, x 2)39. �1 (x 9, x �9)40. �5(d � 5) (d 2, d �2, d �5)41. (a 6, a �6)
42.
14-4 Dividing Fractions (page 550)Writing About Mathematics
1. If x � 2, is not defined. If x � 3, the recipro-cal of , which the first fraction is multipliedby, is not defined.
2. No; so nocancellation can be done. Ruth did not write thereciprocal of the divisor before multiplying.
Developing Skills3. 4. (b 0)
5. (x 0, y 0) 6. (x 0)13
16yx
35b
a6
32(x 2 4) 4 x 2 4
5 5 32(x 2 4) ? 5
x 2 4
x 2 35
2x 2 2
23
2(a 1 6)2
36 1 a2
xx 1 1
212
1x(x 2 2)
103(x 2 2)
x 1 52x 2 3 (x 2
32, x 2 23
2, x 2 5)
3(2a 2 3)5(a 2 3)
y 2 3c
x 1 53y
52
24b2(a 2 2)
a 1 2
a 2 ba 1 b
7(x 1 1)x 1 2
13(x 2 1)
4x
2s3
r5
(x 1 1)(x 2 1)2
5x
b2(3a 2 1)3
y(x 2 y)5x
a2(a 1 3)90
x 1 1212
6a2bc
2s
4m3
yx
65
8mn
5x2
9
5a9
372
14271AKTE2.pgs 9/25/06 10:42 AM Page 372
7. (x 0, y 0) 8. (b 0, c 0, d 0)
9. (x 0, y 0) 10. (a 0, b 0, c 0)
11. (x 0) 12. (y 0)
13. (a 0, b 0)
14. 2(x � 1) (x 1)15. 2x(x � 4) (x 0, x 1)16.
17. (b 0, b 2, b �2)
18. (a 0, a b, a �b)
19.
20. (x 2, x �2, x 0)
21. (x 4y, x �4y, x �2y)
22. (x 2)23. 2(3 � y) (y �5, y �3)24. (x �1, x 1, x �2)25. x (x y, x �y) 26. (a 3, a �3)
27. (a b, a �b)
28. 0, 1, �1 29. 730. 1 (a 0, y 0, z 0)
14-5 Adding or Subtracting AlgebraicFractions (pages 554–555)Writing About Mathematics
1. No. A factor can only be cancelled out when it isa factor of each term of the numerator and eachterm of the denominator. Since 2 is not a factorof 11, it cannot be cancelled out.
2. Yes;
� or � 2 �
Developing Skills3. (c 0) 4. (t 0) 5. (c 0)
6. 1 (x �1) 7. 1 Qx R 8. 1 Qd R
9. (x �1, x 1) 10. (r 3, r �2)
11. 12. 13.
14. 15. 16.
17. 18. (x 0) 19. (x 0)
20. (b 0) 21. (d 0) 22.
23. 24. (b 0)
25. (y 0) 26. (c 0)
27. (x �1) 28. (x �y)3x 1 5y
x 1 y3x 1 8x 1 1
3c2 2 8c 1 12c2
12y2 2 17y 2 12
12y2
b 2 810b
7y 2 620
3a 2 56
5d2 1 75d
3a8b
2 38x
154x
2a 1 b14
a12
31x20
9ab20
22y15
x6
5x6
4r 2 3
1x 2 1
21223
4
65c
5r 2 2st
52c
21(a 2 x)a 2 x 5 2 1 1 5 3
2 2 x 2 aa 2 x
3a 2 3xa 2 x 5
3(a 2 x)a 2 x 5 3
2 2 x 2 aa 2 x 5
2(a 2 x) 2 (x 2a )a 2 x 5
2a 2 2x 2 x 1 aa 2 x
2(a 1 b)(a 2 b)
212
12
213
35
x 2 228x
Ay 2 1, y 212 B3
4(y 2 1)
14(a 1 b)
b(b 2 3)2(b 2 2)
Aa 2 232 B2a 2 3
2
4b2(a2 2 1)
a2
9(y 1 3)10y
32x(x 1 1)27
ab4c
y4
x2
acd4b
235 29. (x 3) 30. (y �1)
31. Qa R 32. (x 2)
33. (x 1) 34. (y 3, y �3)
35. (y 4, y �4)
36. (x 6, x �6)
37. (y 3, y �4)
38. (a �1) 39. (x �3)
40. (x �2)
41. (x y, x �4y)
42. (a 1, a �2, a �3)
43. (a 5, a 2, a �3)
44. 45. 46.
47. 48.
49. a. hr 50. hr 51. hr
b. hr
c. hr
14-6 Solving Equations with FractionalCoefficients (pages 559–561)Writing About Mathematics
1. No.Abby subtracted incorrectly; 3x � 0.2x � 2.8x,so x � �0.3.
2. Heidi’s method will lead to a correct solution,but she did not eliminate all of the decimals asshe had intended; 10(0.2x � 0.84) � 10(3x), 2x �8.4 � 30x, x � �0.3; 100(0.2x � 0.84) � 100(3x),20x � 84 � 300x, x � �0.3; 1,000(0.2x � 0.84) �1,000(3x), 200x � 840 � 3,000x; x � �0.3
Developing Skills3. x � 21 4. t � 108 5. x � 256. x � 16 7. m � 29 8. r � �139. y � 6 10. x � or 1.5 11. m � 30
12. x � 4 13. y � 9 14. x � 115. x � 21 16. r � 12 17. t � 918. a � 24 19. y � 1 20. y � 1221. t � 15 22. m � 3 23. y � 33024. x � 10.4 25. c � 20 26. y � 1527. x � 20 28. x � 395 29. x � �2030. x � 48 31. x � 31 32. x � 1,50033. x � 600 34. x � 700 35. x � 4036. a � 37. a � 38. 3039. 100 40. 30 41. 6042. 13, 14 43. 19, 21 44. 12, 12, 1845. 6, 18 46. 21, 35 47. 30, 6048. 60, 90
23607
12011
32
x 1 618
x 1 1030
9x 1 1540
5a 2 2a(a 2 1)
x45
2(x 2 5)15
2x 1 512
6(x 2 2)7
7(x 2 1)6
9x5
2a2 2 2a 2 2(a 1 3)(a 2 5)(a 2 2)
9a2 2 7a 1 5(a 2 1)(a 1 3)(a 1 2)
x2 2 18x 1 xy 1 3y 2 2y2
3(x 1 4y)(x 2 y)
2x2 1 3x 2 7x 1 2
x2 2 3x 2 15x 1 3
a2 1 2a 1 2a 1 1
2y2 1 11y 2 303(y 2 3)(y 1 4)
2x 1 243(x2 2 36)
25y 1 26
y2 2 16
23y 2 4
y2 2 92 5x
8(x 2 1)
296(x 2 2)
13
175(3a 2 1)
334(y 1 1)
172(x 2 3)
373
14271AKTE2.pgs 9/25/06 10:42 AM Page 373
Applying Skills49. Plot 1 � 30 ft, plot 2 � 40 ft, plot 3 � 10 ft,
plot 4 � 20 ft50. Sam is 6, his father is 3651. Robert is 24, his father is 4852. 52 students 53. 20 passengers54. $16 55. 50 seedlings56. 4 nickels, 12 dimes 57. 6 cans58. 12 nickels, 17 dimes 59. 3 dimes, 10 quarters60. 80 student tickets, 96 full-price tickets61. No; let x � number of dimes. Then 0.10x �
0.25(2x) � 4.50, 0.60x � 4.50, x � 0.75, but itis not possible to have a fractional numberof dimes.
62. Yes; 15 coins of each type total $6.00.63. $1,500 at 7%, $1,900 at 8%64. $1,750 at 10%, $250 at 11%
14-7 Solving Inequalities with FractionalCoefficients (pages 563–565)Writing About Mathematics
1. When multiplying both sides by a negative num-ber, the inequality must be reversed, so x � 6.
2. Positive integersDeveloping SkillsIn 3–23, answers will also require number lines withdomains equal to the real numbers.
3. x � 9 4. y � 15 5. c � 6 6. x � 5 7. y � 12 8. y � �19. t � �40 10. x � �6 11. x � 5
12. y � 13. x � 12 14. y � 615. d � 1 16. c � 17. m � 318. x � 18 19. x � 2 20. y � 921. r � 24 22. t � 17 23. a � 324. 17 25. 38 26. 13327. 113 28. 25, 10 29. 24, 20Applying Skills30. 51 calls 31. $17.99 32. 8 cans33. 4 nickels 34. Rhoda, 18; Alice, 2735. Mary, 16; Bill, 19 36. $1,20037. a. $120,000 b. $108,000 c. $105,00038. 15 books
14-8 Solving Fractional Equations (pages 568–569)Writing About Mathematics
1. No, the solution is undefined for r � 5 since itwould make the denominators equal to 0.
2. Pru’s answer of �5 must be rejected since it leadsto a denominator of 0 in the given equation. Theonly answer is 10, as found by Pam.
213
2123
Developing SkillsIn 3–6, responses will vary.
3. The solution x � 0 must be rejected because thefraction is undefined for x � 0.
4. The solution a � �1 must be rejected becausethe fraction is undefined for a � �1.
5. The solution x � 0 must be rejected because thefractions are undefined at x � 0.
6. The solution x � 1 must be rejected because thefractions are undefined for x � 1.
7. x � 2 8. y � 5 9. x � 310. x � 30 11. x � 2 12. y � 313. y � or 0.5 14. x � 3 15. y � 816. x � 6 17. y � 3 18. a � 419. b � 12 20. x � 8 21. x � 122. x � 2 23. a � 2 24. x � �525. x � 3 26. x � 4 27. x � 428. 29. z � 1 30. or 0.231. y � 3 32. a � �10 33. m � 234. a � 6 35. x � �2 36. or 0.537. or �0.5 38. x � { } or �; no solution39. x � �1 or x � 4 40. x � �2 or x � 341. b � �1 or b � 2 42. b � �2 or or 1.543. b � 4 or b � �3 44. x � �3 or x � 645. x � �1 46. x � (x 0, k 0)47. x � (x 0, k 0) 48. x � (x 0, c 0)49. x � de (x 0, x e)
50. Yes; and is undefined only forthe excluded values of a and c.
Applying Skills51. 4 52. 53. 3 54.
55. 56. 57. 18
58. Emily’s garden is 6 ft � 2 ft, Sarah’s garden is18 ft � 6 ft or Emily’s garden is 12 ft � 8 ft,Sarah’s garden is 18 ft � 12 ft
Review Exercises (pages 570–572)1. An algebraic fraction is the quotient of any two
algebraic expressions. A fractional expressionis a quotient of two polynomials. In other words,a fractional expression is a special case of analgebraic fraction.
2. 3. y � 44. 3x(2x � 3)(2x � 3) 5. (b 0, g 0)
6. (d 0) 7. x2 � 12
8. (y 0) 9. (x 0)10. x � 1 (x 0, y 0) 11. 12c (c 0)12. (a 0, b 0) 13. 2m
35
6b
2x3
2y 2 32
2d
23
x10
1520
1220
17
13
x 5A a
c B A c2
a Bc 5 1
a 1 bc
t6k
tk
b 5 32
y 5 212
y 5 12
r 5 15a 5 21
3
12
374
14271AKTE2.pgs 9/25/06 10:42 AM Page 374
14. (k 0) 15.
16. (x 0, y 0, z 0)
17. (x 0) 18. (x 0, x 5)
19. 2 (a �b) 20.
21. (x 5) 22.
23. Qa 0, R
24. 2b 25. 2 26. k � 1527. x � 11 28. y � 12 29. m � 730. t � 6 31. a � 5 or a � �432. r � (h 0) 33. r � (r 0)
34. r � (n 0, r 0) 35. $1,92036. 25 37. 90 points38. 300 mi at 50 mph, 360 mi at 60 mph39. a. n � (a 0)
b. (a 0)
c. (a 0)
d. $0.75; 16 cans for $12.00, 20 cans for $15.0040. $0.50 for coffee, $0.75 for a bagel41. 5 dimes
Exploration (page 572)(1) 0.5, 0.25, 0.2, 0.125, 0.1, 0.0625, 0.05, 0.04, 0.02,
0.01(2) They are all terminating decimals.(3) 2, 22, 5, 23, 2 � 5, 24, 22 � 5, 52, 2 � 52, 22 � 52
(4) The factors of the denominators are either 2or 5.
(5)
(6) They are all repeating decimals.(7) 3, 2 � 3, 32, 11, 22 � 3, 3 � 5, 2 � 32, 2 � 11, 23 � 3,
2 � 3 � 5(8) The denominators have at least one prime factor
other than 2 or 5.
0.0416, 0.030.3, 0.16, 0.1, 0.09, 0.083, 0.06, 0.05, 0.045,
n2 5$15.00
a
n1 5$12.00
a
Ta
an
c2p
S2ph
a 213, a 2 21
33
1 1 3a
5c 1 324
x 1 53(x 2 5)
13 2 y20
12
14x 1 136x
5z 2 2xxyz
7ax12
10k
(9) The ratio of two integers can be written as a termi-nating decimal if, when the fraction is in simplestform, the denominator has only 2 or 5 as prime fac-tors.The ratio of two integers can be written as arepeating decimal if, when the fraction is in simplestform, the denominator has a prime factor otherthan 2 or 5.
Cumulative Review (pages 573–574)Part I
1. 2 2. 2 3. 1 4. 3 5. 46. 3 7. 4 8. 2 9. 2 10. 3
Part II11. 4%;
12. 44 ft/sec;
Part III13. Soda: $0.80, fries: $1.00; 2s � f � 2.6, s � 2f � 2.8;
f � 2.6 � 2s, s � 2(2.6 � 2s) � 2.8, s � 5.2 � 4s �2.8, 3s � 2.4, s � 0.8, f � 1
14. a.
b. {(�1, 3), (3, 3)}Part IV15. 68.9 cm; sin 32° � , x � 68.916. Length: 15ft, width: 8 ft; lw � 120, l � 2w � 1;
(2w �1)w � 120, 2w2 � w � 120 � 0,(2w � 15)(w � 8) � 0, w � 8, l � 15 (Rejectw � �7.5)
36.5x
x
y
O–1
–1
44 ftsec
10 furlongs2.5 min ?
18 mi
1 furlong ? 5,280 ft
1 mi ? 1 min60 sec 5
3,640 2 3,5003,500 5 0.04
375
15-1 Empirical Probability (pages 581–583)Writing About Mathematics
1. No. The one out of four probability is based onresults for a very large group and is an average;it does not mean that for a particular group offour people, one will definitely have an accident,nor does it mean that only one will have anaccident. It is possible that there may be threeaccidents for some groups of four and none forothers.
2. No. Some books are more popular than othersand these books would have a higher likelihoodof being checked out. Others, like referencebooks, cannot be checked out at all, so wouldhave a probability of 0.
Developing Skills3. 4. 5.
6. 7. 8.
9. Answers will vary.10. a. 1
4
14
426 5 2
1325
110
14
16
Chapter 15. Probability
14271AKTE2.pgs 9/25/06 10:42 AM Page 375
b. Tom: � .300; Ann: 79, 300, � .263;Eddie: 102, 400, � .255; Cathy: 126, 500,
� .252c. Yesd. Answers will vary but should confirm the
result.11–15. Answers will vary.16. a. 1 b. 1 c. 0 d. 0Applying Skills17. 18. 19.
15-2 Theoretical Probability (pages 588–590)Writing About Mathematics
1. The regions are not the same size so the arrowdoes not have an equally likely chance of landingin each of the three regions.
2. No. The populations of the states are different sothe probability of the location of the next birth isnot equally likely for each state.
Developing Skills3. a. {H, T} b. c.
4. a. {3} b.
5. a. {2, 4, 6} b.
6. a. {1, 2} b.
7. a. {1, 3, 5} b.
8. a. {4, 5, 6} b.
9. a. {3, 4, 5, 6} b.
10. a. {3} b.
11. a. {2, 4} b.
12. a. {1, 2} b.
13. a. {1, 3, 5} b.
14. a. {4, 5} b.
15. a. {3, 4, 5} b.
16. a. b. c.
d. e. f.
g. h. i.
j.
17. a. b. c.
18. a. b. c.
d. e. f.
19.
20. a. b. c.
d. 411
38
36 5 1
225
24 5 1
2
216 5 1
8312 5 1
44
10 5 25
210 5 1
515
25
13
12
14
1252 5 3
13
252 5 1
26252 5 1
264
52 5 113
1352 5 1
4152
2652 5 1
2
1352 5 1
4452 5 1
131
52
35
25
35
25
25
15
46 5 2
3
36 5 1
2
36 5 1
2
26 5 1
3
36 5 1
2
16
12
12
2445 5 8
1515885 5 1
59790
1,000 5 79100
126500
102400
79300
60200
Applying Skills
21. a. b.
22. a. b.
23. a. b. c.
d. e. f.
g. h. i.
j.
15-3 Evaluating Simple Probabilities(pages 594–596)Writing About MathematicsIn 1 and 2, answers will vary.
1. Example: The probability of landing on a 7 whenrolling a die
2. Example: The probability of getting a numbergreater than 0 when rolling a die
Developing Skills3. a. { }, {H}, {T}, {H, T}
b. P(neither H or T) � 0P(H) �P(T) �P(either H or T) � 1
4. 5. 6.
7. 8. 9.
10. 0 11. 1
12. a. b. c.
d. 1 e. 0 f. 0
13. a. b. c.
d. 0 e. f. 1
g. 0 h. 1 i. 0
14. a. 0 b. 1 c.
d. 0 e. 1 f. 0
15. a. b. c. 0
d. 0 e. 1 f.
g. 0 h. 0 i. 0
16. a. {E1, E2} b.
17. a. {S1, S2, S3, S4} b.
18. a. {I, A, E} b. 38
411
25
1352 5 1
4
1352 5 1
4452 5 1
13
46 5 2
3
46 5 2
3
26 5 1
336 5 1
216
38
23
57
67
27
47
47
37
17
12
12
38
58
38
78
68 5 3
438
58
38
48 5 1
238
4840 5 1
2105
840 5 1168
1430 5 7
151630 5 8
15
376
14271AKTE2.pgs 9/25/06 10:42 AM Page 376
19. a. {E1, E2, I, E3} b.
20. a. {S, P, R, Y} b. 1
Applying Skills21. a. .6 b. .4 c. 1 d. 022. a. 10% b. 50% c. 40% d. 100%
e. 0%
23. a. b. c. d. 0
e. f. 0 g. 0 h.
i. j. 1 k. 0 l. 1
24. 8 25. 9 girls, 12 boys26. 18 caramels, 12 nut clusters 27. 28 rides
15-4 The Probability of (A and B) (pages 598–599)Writing About Mathematics
1. Answers will vary. Example: The probability ofrolling an even number and a 2 on a die.
2. Set A is a subset of set B.
Developing Skills
3. a. b. c.
d. 0 e. f. 0
4. a. b. c.
d. e. f. 0
g. h. i.
Applying Skills
5. a. b. c.
d. 1 e.
6. a. 21 b. (1)
(2)
(3)
7. a. b. c.
d. e. f.
g. h. i. 0
j. 212 5 1
6
512
212 5 1
6
212 5 1
6312 5 1
48
12 5 23
412 5 1
3512
712
2130 5 7
10
2830 5 14
15
2330
34
14
14
24 5 1
2
652 5 3
26252 5 1
261
52
152
252 5 1
26
152
252 5 1
261
52
36 5 1
2
16
16
26 5 1
3
35
15
25
15
25
35
47
15-5 The Probability of (A or B) (pages 603–605)Writing About Mathematics
1. No. By definition P(A or B) � P(A) � P(B) �P(A and B). However, 0 � P(A and B) � P(B).Therefore, P(A) � P(B) � P(A and B) � P(A) �P(B) � P(B) � P(A).
2. P(A or B) � P(A); since B is a subset of A, thenP(B) � P(A and B) so P(A or B) � P(A) �P(B) � P(A and B) � P(A).
Developing Skills3. a. Yes; b. Yes; c. Yes;
d. Yes; e. No;
4. a. Yes; b. Yes;
c. Yes; d. Yes;
e. No; f. Yes;
g. No; 1 h. No;
5. a. Yes; b. Yes;
c. Yes; d. No;
e. No; f. Yes;
g. No; h. Yes;
i. Yes;
6. a. Yes; b. Yes;
c. Yes; d. Yes;
e. Yes; 1 f. Yes;7. (2) 8. (2) 9. (1)
10. (2) 11. (2) 12. (3)
Applying Skills13. a. 2 14. a. .85 15.
b. 10 b. 289c. 4
15-6 The Probability of (Not A) (pages 607–609)Writing About Mathematics
1. P(not A) � 0; the event (not A) is an impossibility.Since P(not A) � 1 � P(A), if P(A) � 1, thenP(not A) � 1 � 1 � 0.
2. P(not A or not B) � 1. Since A and B are disjoint,the set (not A) contains every element of the univer-sal set, including the elements of B but not includingthe elements of A. Similarly, the set (not B) containsevery element of the universal set, including theelements of A but not including the elements of B.Therefore, the set (not A or not B) is equal to theuniversal set, and thus, P(not A or not B) � 1.
14
1016 5 5
8
1416 5 7
88
16 5 12
916
816 5 1
2
1652 5 4
13
3952 5 3
41652 5 4
13
1252 5 3
132852 5 7
13
1652 5 4
132652 5 1
2
852 5 2
138
52 5 213
46 5 2
3
26 5 1
336 5 1
2
36 5 1
246 5 2
3
46 5 2
326 5 1
3
35
35
45
45
25
377
14271AKTE2.pgs 9/25/06 10:42 AM Page 377
Developing Skills
3. a. b. c.
d. e. f.
g. 1 h. 0 i.
4. a. b. c.
d. e. f.
g. h.
5. a. P(P) � , P(I) � , P(C) � , P(N) �
, P(K) � , P(G) �
b. � 1
6. 7. .907
8. a. b. c.
d. 1 e. 0 f. 0
9. a. b. c.
d. 1 e.
10. a. b. c. 0
d. e. f.
g. h. i. 1
11. a. b. c.
d. e. f.
g. h. i. 0
j. k. l.
Applying Skills
12. a. (1) (2) (3)
b. � 1
c. (1) (2) (3)
13. a. b.
14. a. .2 b. .3 c. .5
d. .7 e. .8 f. .7
g. .5 h. 0
15. 89
710
310
712
812 5 2
39
12 5 34
312 1 4
12 1 512 5 12
12
512
412 5 1
33
12 5 14
5052 5 25
262852 5 7
132
52 5 126
5152
852 5 2
13
4852 5 12
133952 5 3
41652 5 4
13
152
1352 5 1
44
52 5 113
411
711
411
311
311
211
111
34
14
14
34
47
47
37
67
110 1 3
10 1 210 1 2
10 1 110 1 1
10
110
110
210 5 1
5
210 5 1
5310
110
4452 5 11
135152
5052 5 25
264852 5 12
134052 5 10
13
1252 5 3
133952 5 3
41352 5 1
4
46 5 2
3
46 5 2
326 5 1
336 5 1
2
36 5 1
256
16
16. a. b. c.
d. e. f.
g. 0 h. i. 0
j. k. l. 1
15-7 The Counting Principle, SampleSpaces, and Probability (pages 613–616)Writing About Mathematics
1. No. There are four equally likely outcomes:two heads, two tails, heads for nickel andtails for quarter, tails for nickel and headsfor quarter.
2. Yes; P(green 2) � P(red 3) � ,which is the same as P(green 3) � P(red 3) �
.
Developing Skills3. a. {(H, H),(H, T), (T, H), (T, T)}
b. 2 outcomes for the quarter � 2 outcomesfor the dime � 4 outcomes
c. 1d. 2
4. a. 6 � 6 � 36b.
5. a. 4b. 3c. 12d.
R
B
BWGRWGRBGRBW
W
G
123456
1 2 3 4 5 6
16 3 1
6 5 136
16 3 1
6 5 136
310
510 5 1
2
710
910
710
610 5 3
5
610 5 3
5410 5 2
51
10
378
14271AKTE2.pgs 9/25/06 10:42 AM Page 378
6. a. 4b. 4c. 16d.
7. a. 8. a. 9.
b. b.
c. c.
d.
10. a. 11. a. b.
b. c. d.
e. f.
g. h.
12. ; for each toss, there are exactly two outcomes:heads or tails.
Applying Skills13. a. 10 outfits 14. 80 ways 15. 56 teams
b. 40 outfitsc. 36 outfits
16. 42 meals 17. a. 40 mealsb. 24 mealsc. 12 meals
18. a.
b. {(T, T, T), (T, T, F), (T, F, T), (T, F, F), (F, T, T),(F, T, F), (F, F, T), (F, F, F)}
19. a. 4 ways 20. 180 versionsb. 64 waysc. 1,024 waysd. 4n ways
21. a. 676,000 license plates 22.b. 1,757,600 license platesc. 6,760,000 license plates
860 5 2
15
T
F
T
F
TFTF
T
F
TFTF
12
1464 5 7
324964
6364
164
1464 5 7
324964
18
6364
164
18
212 5 1
6
412 5 1
3112
312 5 1
416
14
112
12
23. a. 24. a. 25. a. .09 b. .49
b. b. c. .21 d. .21
c. e. .51 f. .5826. 25 27. 4 days28. a. (1) (2) (3)
b. c. 5,000 coupons
d. 1,994,897 coupons e.
15-8 Probability with Two or More Activities(pages 623–627)Writing About Mathematics
1. The probability of winning first prize is lessthan the probability of winning second prizebecause the sample space has decreased by 1;P(first prize) � , P(second prize) � ,and .
2. For the dice, there are 36 possible outcomesand 5 have a sum of 8 {(2, 6), (3, 5), (4, 4), (5, 3),(6, 2)}, so P(sum of 8) � . For the tiles, thereare 30 possible outcomes and 4 have a sum of8 {(2, 6), (3, 5), (5, 3), (6, 2)}, so P(sum of 8) �
. Since , the probability is greater with the dice.
Developing Skills3. a. 4. a. 5. a.
b. b. b.
c. c.
d. d.
e.6. a. {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3),
(2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2),(4, 3), (4, 4)}
b.
c.
d. 616 5 3
8
416 5 1
4
116
26 5 1
3
2042 5 10
212049
2242 5 11
212949
2042 5 10
212549
57
242 5 1
21449
27
536 . 2
15430 5 2
15
536
1n , 1
n 2 1
1n 2 1
1n
1,999,8972,000,000
1032,000,000
120,000
11,000,000
12,000,000
12n
312 5 1
4116
212 5 1
612
379
R
B
W
G
RBWGRBWGRBWGRBWG
1
2
3
4
1234123412341234
14271AKTE2.pgs 9/25/06 10:42 AM Page 379
7. a. (1) b. (1)
(2) (2)
(3) (3)
(4) (4)
(5)
(6) 0
(7)
8. a. b.
9. a. b.
Applying Skills10. a. {(L1, L2), (L1, L3), (L1, G1), (L1, G2), (L2, L1),
(L2, L3), (L2, G1), (L2, G2), (L3, L1), (L3, L2),(L3, G1), (L3, G2), (G1, L1), (G1, L2), (G1, L3),(G1, G2), (G2, L1), (G2, L2), (G2, L3), (G2, G1)}
b. (1) (2) c.
(3) (4)11. a. {(G1, G1), (G1, G2), (G1, G3), (G1, B1), (G1, B2),
(G2, G1), (G2, G2), (G2, G3), (G2, B1), (G2, B2),(G3, G1), (G3, G2), (G3, G3), (G3, B1), (G3, B2),(B1, G1), (B1, G2), (B1, G3), (B1, B1), (B1, B2),(B2, G1), (B2, G2), (B2, G3), (B2, B1), (B2, B2)}
1820 5 9
108
20 5 25
68 5 3
4220 5 1
106
20 5 310
L1
L2
L3
G1
G2
L2
L3
G1
G2
L1
L1
L1
L1
L2
L2
L2
L3
L3
L3
G1
G1
G1
G2
G2
G2
12
14
113
12
911
60132 5 5
11
78132 5 13
226
132 5 122
126132 5 21
2227
132 5 944
54132 5 9
2227
132 5 944
6132 5 1
2272
132 5 611
b. (1)
(2)
(3)
(4)
c.
12. a. b.
c. d.
e. f.
g. h.
i.13. a. {(H, Q), (H, D), (H, N), (Q, H), (Q,D),
(Q, N), (D, H), (D, Q), (D, N), (N, H),(N, Q), (N, D)}
b. (1) (2)
(3) (4)
(5) (6) 412 5 1
32
12 5 16
212 5 1
66
12 5 12
412 5 1
3212 5 1
6
451
1692,652 5 13
204451
22,652 5 1
1,3262,4962,652 5 16
17
1322,652 5 11
221650
2,652 5 25102
1562,652 5 1
1712
2,652 5 1221
615 5 2
5
1625
525 5 1
5
425
925
G1
G1
G2
G3
B1
B2
G1
G2
G3
B1
B2
G1
G2
G3
B1
B2
G1
G2
G3
B1
B2
G1
G2
G3
B1
B2
G2
G3
B1
B2
380
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15-10 Permutations with Repetitions(pages 638–639)Writing About Mathematics
1. a. {TAR, TRA, RAT, RTA, ART, ATR}b. There are 3 unique letters, each used once in
every word.2. a. {TOT, TTO, OTT}
b. There are 2 unique letters, one of which isrepeated twice in every word.
Developing Skills3. 60 4. 20 5. 306. 30 7. a. 720 8. a. 120
b. 120 b. 60c. c.
9. a. 360 10. a. 180 11. a. 60b. 60 b. 60 b. 30c. c. c.
12. a. 20 13. a. 60 14. a. 120b. 10 b. 0 b. 60c. c. 0 c.
15. 630 16. 420 17. 1,68018. 3,360 19. 3,780 20. 151,20021. 34,650 22. 30,240 23. a. 120
b. 6c.
24. a. 5 25. a. 10 26. a. 5b. 1 b. 3 b. 0c. c. c. 0
27. 4 28. 21 29. 5630. 360 31. 495Applying Skills32. 120 33. 8434. 9,189,180 35. 21,034,470,60036. 1,251,677,700 37. 1,396,755,36038. Answers will vary.
15-11 Combinations (pages 644–646)Writing About Mathematics
1. A permutation is used when order is important.For example, listing the different ways to choosea president, a vice president, and a treasurer.A combination is used when order is not impor-tant. For example, choosing a committee of 3.
2. r � 13. We know that 26Cr is an increasing func-tion of r. Since 26Cr � 26C26 � r, it will have a maxi-mum value. A maximum value will occur when r � n � r. When r � 13, n � r � 26 � 13 � 13.Therefore, 26C13 � 10,400,600 is the maximumvalue.
310
15
120
12
12
12
13
16
12
16
381
14. a. (1) 15. a. b.
(2) c. d.
(3) e. f.
(4) g.
b. 10
16. a. 17. a. 18. a.
b. b. b.
c.
19. a. 20. a.
b. b.
c.
15-9 Permutations (pages 634–635)Writing About Mathematics
1. n! � n [(n � 1) (n � 2) 2 1] and (n � 1) (n � 2) 2 1 � (n � 1)!, so n! � n(n � 1)!
2. 9P9 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1 and
9P8 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2, so they areequal
Developing Skills3. 24 4. 720 5. 5,0406. 8 7. 120 8. 69. 40,320 10. 336 11. 120
12. 90 13. 380 14. 7,92015. 5,040 16. 64,77017. 985,084,775,273,88018. 24 19. 60 20. 12021. 210 22. 19P7, 60P5, 24P7, 45P6
Applying Skills23. a. 24
b. {EMIT, EMTI, EIMT, EITM, ETMI, ETIM,MITE, MIET, MTIE, MTEI, MEIT, METI,ITEM, ITME, IMET, IMTE, IETM, IEMT,TEIM, TEMI, TIEM, TIME, TMEI, TMIE}
24. 120 25. 24 26. 40,32027. a. 30 28. a. 210
b. 870 b. 504c. 24,360 c. 990
d. n(n � 1)(n � 2)29. a. 6 30. 755,160 31. a. 650
b. 15,600 b. 676
In 32–35, part b, answers will vary depending on thenumber of digits in the calculator display.32. a. 60! 33. a. 26!
b. 8.320987113 � 1081 b. 4.032914611 � 1026
34. a. 35. a.b. 3.07762104 � 1021 b. 3.100796899 � 1012
40!32!
25!7!
47
22150 5 11
7538
12150 5 2
2518
30100 5 3
10
820 5 2
575200 5 3
86090 5 2
3
1236 5 1
3145400 5 29
8090200 5 9
20
3042 5 5
796
182 5 4891
46 5 2
31242 5 2
786
182 5 4391
3042 5 5
72442 5 4
730
182 5 1591
1242 5 2
7642 5 1
756
182 5 413
14271AKTE2.pgs 9/25/06 10:42 AM Page 381
Developing Skills3. 105 4. 220 5. 2106. 25 7. 1 8. 19. 9 10. 19,900 11. 35
12. 126 13. 1 14. 124,25115. 20 16. 84 17. 120
18. a. 3 b. 6 c. 10 d. 21e. 28 f. nC2 �
19. a. (1)
(2)
(3)
b. (1)
(2)
(3)c. Yes, the formulas are equivalent.
Applying Skills20. a. 2,002 21. 2,300 22. a. 1,001
b. 4,368 b. 286c. 715 c. 66
23. a. 1,540 24. a. 126 25. a. 2,598,960b. 120 b. 108 b. 24c. 540 c. 9 c. 48d. 660 d. 0 (there d. 1,287e. 66 are only 3 e. 1,320
car-repair f. 0manuals)
26. 176 27. 299
15-12 Permutations, Combinations, andProbability (pages 648–651)Writing About Mathematics
1. No. The probability that Olivia will be chosen is
equal to . Therefore, the probability that
she will not be chosen is equal to .2. Jenna was correct. Since order is important (we
are looking for alphabetical order), we need touse permutations.
Applying Skills3. a. 30 b. c. 15
d. (1) (2)
(3) (4)
4. a. 35 b. 12 c.
d. 1 e. 34
1235
115
815
115
615 5 2
5
1230 5 2
5
1 2 38 5 58
7C
2
8C35 3
8
11(10)(9)c(5)7! 5 330
11!(11 2 7)!7! 5 330
11P
77! 5
11!(11 2 7)!
7! 5 330
8(7)(6)3! 5 56
8!(8 2 3)!3! 5 56
8P
33! 5
8!(8 2 3)!
3! 5 56
n!2!(n 2 2)! 5
n(n 2 1)2
5. a. 84
b. (1) (2)
(3) (4)
6. a. (1) 35 (2) 7
(3) 63 (4) 210
b. (1) (2) (3)
(4) 0 (5)
7. a. b.
c. d.
e. f.
g. h.
8. a. b. c.
d. e. 1 f.
9. a. b.
c. d.
10. a. 210 b. 24 c.
11. a. 135 b. 720 c. 72 d.
12. a. 42
b. (1) 10 (2) 6 (3)
13. a. b. c.
d. e. f.
g. h.
14. a. 3,003 b. 60
c. (1) (2)
(3) (4)
(5) 0 (6)
15. a. 5,040 b. c.
16. a. 10 groups of people seated in 60 differentorders
b. c.
Review Exercises (pages 653–656)1. No. For Aaron, P(black) � where b is the num-
ber of black jellybeans in a dish of n jellybeans.However, Jake’s selection is not random. Jakematches Aaron’s selection. Therefore, if Aaronchooses a black jellybean, then Jake is certain to
bn
310
25
5765,040 5 4
35720
5,040 5 17
1263,003 5 6
143
63,003 5 2
1,001180
3,003 5 601,001
13,003
603,003 5 20
1,001
156
156
2156 5 3
8656 5 3
284056 5 5
7
3056 5 15
281056 5 5
281
56
610 5 3
5
310
14
34
624 5 1
4
1224 5 1
21824 5 3
4
1335
1835
1235
135
435
451
1251 5 4
17
162,652 5 4
66316
1,326 5 8663
781,326 5 1
176
1,326 5 1221
41,326 5 2
6631
1,326
27
63210 5 3
107
210 5 130
35210 5 1
6
24 5 1
21620 5 4
5
1220 5 3
5420 5 1
5
382
14271AKTE2.pgs 9/25/06 10:42 AM Page 382
choose a black jellybean, so the probability isequal to 1. If Aaron does not choose a blackjellybean, then Jake will definitely not choose ablack jellybean, so the probability is equal to 0.
2. No. The ratio of black jellybeans to totaljellybeans is , so there could be 16 outof 25, 24 out of 75, and so on.
3. (n � r � 1) 4. .5 5. .26. 40,320 7. 120 8. 1329. 1 10. 220 11. 780
12. 0 13. a. 120 14. 210
b. (1)
(2)
(3) 0
(4) 1
15. 24 16. 56 17. 65%
18. a. 210 b.
19. a. 126 b. (1) c.
(2) d. 1
(3)
20. a. b. c.
d. e.
21. a. b. c.
d. e. 0 f.
22. a. b. 0 c.
d. e. 1 f.
23. 8 girls, 16 boys
24. a. 21 b. (1) c. 20
(2) d.
(3)
(4) 0
25.
26. a. 120 b. 72 27.
c. d.
28. 29. 30. 1
31. 0 32. 33.
34. 35. 8 girls, 12 boys
36. 17 sophomores, 10 juniors, 18 seniors
37. a. 38. 60%
b. 1248 5 1
4
642,450 5 32
1,225
25
42,504850,668
15
36 5 1
226 5 1
3
35
72120 5 3
5
25
28 5 1
4
1821 5 6
7
220 5 1
101221 5 4
7
621
120
1484 5 1
6
535 5 1
7110
5152
3952 5 3
4
2852 5 7
131652 5 4
13852 5 2
13
351 5 1
1778
1,326 5 117
161,326 5 8
6636
1,326 5 1221
162,652 5 4
663
15126 5 5
42
60126 5 10
21
26 5 1
345
126 5 514
57
36 5 1
2
16
825
Exploration (pages 656–657)a. About 59 or 60 disksb. Answers will vary.c. Answers will vary. Example:
(1) Catch and tag a number of fish. Note thenumber tagged.
(2) Return the fish to the pond.(3) Catch some fish from the pond. Record the
number caught and the number of these thatare tagged.
(4) Return the fish to the pond.(5) Repeat steps (3) and (4) at least 10 times.Let x � the number of fish in the pond
n � the number of fish that were taggedc � the total number of fish caught and
recordedt � the total number of tagged fish
recordedUse the proportion
Cumulative Review (pages 657–659)Part I
1. 1 2. 3 3. 44. 1 5. 3 6. 17. 2 8. 4 9. 2
10. 4Part II11. 58°; tan �DAB � , �DAB � 58°
12. 84 cents/lb;Part III13. 72 students; Japanese only � 50 (J) � 13
(S and J) � 10 (E and J) � 5 (E and S and J) �22, Spanish only � 100 (S) � 13 (S and J) � 45(E and S) � 5 (E and S and J) � 37, Japanese orSpanish � 22 � 37 � 13 � 72
14. 140; � 1407!3! 3 3!
5(98 cents) 1 7(74 cents)(5 1 7) lb 5 84
centslb
y
xA B
CD
1
1
85
tc 5 n
x
383
14271AKTE2.pgs 9/25/06 10:42 AM Page 383
Part IV15. 60%; 0.80 � 0.75 � 0.616. a. 2w2 � 4w � 720; l � 2w � 4
b. Length � 36 m, width � 20 m; 2w2 � 4w �720, w2 � 2w � 360, w2 � 2w � 360 � 0,(w � 20)(w � 18) � 0, w � 20 [reject �18],l � 2(20) � 4 � 36
384
16-1 Collecting Data (pages 665–667)Writing About MathematicsIn 1 and 2, answers will vary.
1. Even in a census, some groups may be under-counted; for example, the homeless or peoplewho do not wish to respond to the census form.
2. No. Only highly-motivated people will returntheir questionnaires: either those who wanted thefree coupon or those who had strong opinionsabout the product. Voluntary responses usuallydo not represent the group.
Developing Skills3. Qualitative 4. Qualitative 5. Quantitative6. Qualitative 7. Quantitative 8. Qualitative9. Quantitative 10. Quantitative
11. a. Biasedb. Basketball players are taller than average.
12. a. Biasedb. Seniors are generally be taller than under-
classmen.13. a. Biased
b. Students of a single age and gender are gen-erally similar in size to one another but non-representative of other groups of students.
14. a. Biasedb. Girls are usually shorter than boys; half the
student population would be unrepresented.15. a. Unbiased16. a. Biased
b. Boys are usually taller than girls; half thestudent population would be unrepresented.
17. a. Biasedb. Three is too small a sample space.
18. a. Unbiased
In 19–24, responses will vary.19. No. Students at the gym for a sports event would
most likely want additional pep rallies for sportsevents.
20. Yes. Students at the library21. Yes22. No. Since cheerleaders would be involved in
the pep rallies, they would be more likely wantadditional pep rallies.
23. No. Students at the prom committee would mostlikely want more dances.
24. Yes
25. (2), (3), (5)26. Collecting data, organizing data, and drawing
conclusions from the data27. The shorter the person, the faster the time; how-
ever, we visually associate the bigger pictureswith better results, which in this case is false.Also, the area of the figures and the scale usedexaggerate the change in time.
28. The sample sizes of the groups are too small tomake a valid conclusion Also, the variable ofinterest is difficult to define.
Hands-On ActivityResults will vary
16-2 Organizing Data (pages 672–674)Writing About Mathematics
1. The stem-and-leaf diagram shows theindividual data items and is more visual; in agrouped table, the individual items cannot bedetermined.
2. The stem could be the hundreds digits from 0 to6, with the tens and ones digits as the leaves. So,2 would be 0 � 02 and 654 would be 6 � 54.
Developing Skills
3. a. Interval Tally Frequency
180–189 6
170–179 10
160–169 12
150–159 6
140–149 2
b. (1) 8 students(2) 28 students(3) 160–169(4) 140–149
c. Stem Leaf
18 0 0 0 1 3 317 0 1 1 1 3 4 5 5 7 816 0 2 2 2 3 3 4 4 6 8 8 915 0 5 7 8 8 814 7 9
d. 36 cme. 8 students
Key: 16 � 2 � 162 cm
Chapter 16. Statistics
14271AKTE2.pgs 9/25/06 10:42 AM Page 384
4. a. Interval Tally Frequency
50–59 5
60–69 3
70–79 12
80–89 14
90–99 7
100–109 9
b. (1) 30 batteries(2) 20 batteries(3) 80–89(4) 60–69
c. Stem Leaf
5 2 3 4 7 86 2 3 77 0 0 2 2 2 3 3 5 5 6 8 98 0 0 1 2 2 4 4 4 5 5 5 6 7 89 0 1 1 2 5 6 7
10 0 1 1 3 2 4 5 6 8
d. 56 hr e.
5. a. Interval Tally Frequency
35–39 1
30–34 2
25–29 5
20–24 5
15–19 10
10–14 4
5–9 6
0–4 5
b. Interval Tally Frequency
32–39 2
24–31 6
16–23 14
8–15 7
0–7 9
c. Stem Leaf
3 1 2 62 0 0 0 1 2 5 5 5 7 81 0 1 2 3 5 6 7 7 7 8 8 9 9 90 0 3 4 4 5 6 6 7 8 9
d. 36 hr/wke. 2
38 5 119
Key: 3 � 1 � 31 hr/wk
850 5 4
25
Key: 5 � 0 � 50 hr
6. a. Interval Tally Frequency
91–100 7
81–90 10
71–80 7
61–70 2
51–60 2
41–50 2
b. Interval Tally Frequency
89–100 8
77–88 9
65–76 8
53–64 3
41–52 2c. 81–90 d. 77–88e. Yes. The scores from 81–88 are common to
both regions.7. a. The intervals are of different lengths.
b. The last two intervals are half the length ofthe first three intervals.
c. The intervals 9–16 and 17–24 overlap.d. 0 is not included in any interval but was
represented in the data.Hands-On ActivityResults will vary.
16-3 The Histogram (pages 679–680)Writing About Mathematics
1. Answers will vary. If the stem-and-leaf diagramwere to be rotated 90° counterclockwise, it wouldhave the same shape as the histogram.The longestlist of data corresponds to the tallest bar. However,individual data items can be read from the stem-and-leaf diagram but not from the histogram.
2. The other intervals would be 21–25, 26–30, 31–35,36–40, and 41–45. No, since the old and newintervals overlap, it is not possible to determinethe frequencies of the new intervals.
Developing Skills3. 10
8
6
4
2
0 81–90
Interval
Freq
uenc
y
51–60 61–70 71–80 91–100
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4.
5.
6. a. 150b. 10–12c. 20%d. 68
Applying Skills
7. a. Interval Tally Frequency
35–37 4
32–34 6
29–31 7
26–28 1
23–25 2
b.
c. 29–31d. 10 gamese. 10%
Freq
uenc
y
Interval
8
6
4
2
023–25 26–28 29–31 32–34 35–37
Freq
uenc
y
Interval
42
36
30
24
18
12
6
01–3 4–6 7–9 10–12 13–15 16–18
Freq
uenc
y
Interval
14
12
10
8
6
4
2
05–9 30–3425–29 20–2415–19 10–14
8. a. Interval Tally Frequency
37.0–40.9 8
33.0–36.9 9
29.0–32.9 8
25.0–28.9 4
21.0–24.9 1
b.
c. 5 students d.
Hands-On ActivityResults will vary.
16-4 The Mean, the Median, and the Mode(pages 686–690)Writing About Mathematics
1. No. Rene is not giving each test the sameweight in the calculation. The correct mean is(67 � 79 � 91) � 3 � 79.
2. Yes.When n is odd, the median of a set of n num-bers is the middle value, that is, the amount ofnumbers less than the median equals the amountof numbers greater than the median. Excluding the median, each half is .The median is one
number more, .
Developing Skills3. a. 7 b. 24 c. d. 0.714. a. 3 b. 7 c. 4 d. 80
e. 3.2 f. 2 g. 22.5 h. 6.55. 5 6. 50.57. a. 2 b. 2, 8 c. 8
d. No mode e. 2, 8, 9 f. 1g. 2 h. No mode i. 2, 7j. 19
515
n 2 12 1 1 5 n 2 1
2 1 22 5 n 1 12
n 2 12
530 5 1
6Fr
eque
ncy
Interval
10
8
6
4
2
0
21.0
–24
.925
.0 –
28.9
29.0
–32
.933
.0 – 36
.937
.0 –
40.9
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8. a. 9 9. a. 2b. 8 b. 4 or 5c. No mode c. Any number except for 2, 4,d. 8, 9 or 5
10. (3) 11. (2) 12. (4)13. (3) 14. (3) 15. (3)16. (2) 17. a. 7 18. 80.25
b. 1119. 31, 32, 33 20. 18, 20, 22 21. 16, 33, 44Applying Skills22. 82 23. 86 24.25. 95 26. 10027. a. 82
b. 100c. 63d. Not possible; Al would need a score of 115.
28. a. 165 kg 29. 1.3 in. 30. 174 cmb. 56.25 kg
31. 13.5 32. 15 33. $134. 70 35. a. (1) $475
(2) $447.50(3) $445, $450
b. The mean because it makesthe “average” salary appearhigher.
c. The median or mode becausethey make the “average”salary appear lower.
36. a. (1) 2 mi(2) 1 mi(3) 1 mi
b. 3c. The mean is greater than all but two of the
distances.37. a. (1) 1 in.
(2) in.(3) in.
b. Answers will vary. Example: The carpenter’s average-size nail is inch because he uses it most often.
Hands-On ActivityResults will vary.
16-5 Measures of Central Tendency andGrouped Data (pages 695–697)Writing About Mathematics
1. The 25th and 26th data values must be equal. Letx represent the 25th data value and y representthe 26th. Then their mean, (x � y) � 2, is either xor y. If the mean is x, (x � y) � 2 � x, x � y � 2x,
34
34
34
2x 1 3y5
y � x. If the mean is y, (x � y) � 2 � y, x � y �2y, x � y.
2. The set has an even number of data values, andthe two middle values are not equal. Unless themiddle values are equal, their mean will not beone of the values.
Developing Skills3. a. 16 4. a. 21
b. 7 b. 18c. 7 c. 19d. 6 d. 20
5. a. 20 6. a. 26b. 21.75 b. 35–44c. 21 c. 45–54d. 20
7. a. 71 8. a. 28b. 22–27 b. 76–100c. 28–33 c. 26–50
Applying Skills
9. a. Interval Frequency
20 0
19 1
18 2
17 4
16 3
15 2
14 1
13 1
12 1
b. 16c. 17d. 16
10. a. (1) 4 min(2) 4 min(3) 4 min
b. The mean, median, and mode are equal.11. a. (1) 25 (2) 40
(3) 40 (4) 38b. The mode shows the suit size sold most
frequently.
12. a. Interval Frequency
91–100 9
81–90 6
71–80 3
61–70 0
51–60 2
b. 91–100 c. 81–90
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13. a. Interval Frequency
180–199 4
160–179 10
140–159 7
120–139 9
100–119 5
b.
c. 140–159 d. 160–179
16-6 Quartiles, Percentiles, andCumulative Frequency (pages 707–710)Writing About Mathematics
1. a. Yes.The individual scores can be read from thestem-and-leaf diagram and the number if scoresat or below a given score can be determined.
b. No. Since the individual scores are not shown,the number of scores at or below a given scorecannot be determined.
2. a. 6th b. 18thDeveloping Skills
3. a. Min � 12, Q1 � 21, med � 30, Q3 � 37, max � 44b.
4. a. Min � 67, Q1 � 74.5, med � 79, Q3 � 87,max � 92
b.
5. a. Min � 0, Q1 � 1, med � 2, Q3 � 3, max � 9b.
0 1 2 3 4 5 6 7 8 9 10
60 65 70 75 80 85 90 95 100
9 12 15 18 21 24 27 30 33 36 39 42 45
Freq
uenc
y
Weight (pounds)
10
8
6
4
2
0
110
– 119
120
– 139
140
– 159
160
– 179
180 –
199
6. a. Min � 3.6, Q1 � 4.1, med � 4.4, Q3 � 4.85,max � 5.0
b.
7. a.
b. 11–20c. 31–40d. 41–50
8. a.
b. 5–9c. 10–14d. 15–19
9. a.
b. 5–8c. 9–12d. 9–12
Interval
Cum
ulat
ive
freq
uenc
y 20
16
12
8
4
1–4 1–8 1–12 1–16 1–20
Interval
Cum
ulat
ive
freq
uenc
y 24
20
16
12
8
4
5–9 5–14 5–19 5–24 5–29
Interval
Cum
ulat
ive
freq
uenc
y 24
20
16
12
8
4
1–10 1–20 1–30 1–40 1–50
3 3.5 4 4.5 5 5.5 6
388
14271AKTE2.pgs 9/25/06 10:42 AM Page 388
10. a.
b. 11–15 c. 20%11. a.
b. 18–22 c. 23–27d. 35% e. 13–17
Applying Skills12. a. 70% b. 280 students c. 1–120
d. 1–30 e. 35% or
13. a. Number Cumulative of Letters Frequency Frequency
1 4 4
2 14 18
3 20 38
4 20 58
5 3 61
6 18 79
7 5 84
8 2 86
9 1 87
10 1 88
b. 4c. 3, 6
720
Interval
Cum
ulat
ive
freq
uenc
y
40
35
30
25
20
15
10
5
03–7 3–12 3–17 3–22 3–27 3–32 3–37
Interval
Cum
ulat
ive
freq
uenc
y 20
16
12
8
4
01–5 1–10 1–15 1–20 1–25
d.
e.
f. 92.6% or 93%14. 187 students
15. a. Height Cumulative (inches) Frequency Frequency
77 2 22
76 2 20
75 7 18
74 5 11
73 3 6
72 2 3
71 1 1
b.
c. 73 in. d. 75 in.16. 510 children 17. (4) 18. (1)
16-7 Bivariate Statistics (pages 721–724)Writing About Mathematics
1. Answers will vary. Example: As the pollution in apond increases, the population of fish decreases.
2. The line of best fit can be used to make predictionsabout one of the variables given the other variable.
Height (inches)
Cum
ulat
ive
freq
uenc
y 24
20
16
12
8
4
71–7
7
71–7
1 71
–72
71–7
371
–74
71–7
571
–76
Freq
uenc
y
Number of letters
9080706050403020100
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
389
14271AKTE2.pgs 9/25/06 10:42 AM Page 389
Applying Skills3. a.
b. Positive c. No d. 9.5 gale. 304 mi f. See graph g. 96 mi
4. a.
b. Positive c. Yes d. 7.5 mine. $0.81 f. y � 0.108x; see graphg. y � 0.109x � 0.008h. $1.51 i. $1.52
5. a.
b. Negative c. Yes d. $1.18e. 60 heads f. See graphg. As the price increases, the number of heads of
lettuce decreases.
Cos
t in
dolla
rs
Minutes
4.00
3.00
2.00
1.00
010 20 30 40
y = 0.108x
400
350
300
250
200
150
100
50
02 4 6 8 10 12 14
Gallons of gas
Mile
s dr
iven
y = 32
x (9.5, 304)
6. a.
b. Positive c. Yesd. y � 4.16x � 125; see graphe. y � 4.52x � 151f. approx. 133 lb g. approx. 78 in.
7. a.
b. Positive c. Yesd. y � 16.4x � 54.3; see graphe. y � 15.5x � 47 f. 224.5 million
8. a.
Years after 1990
Cel
lula
r ph
ones
(in
mill
ions
)
160
140
120
100
80
60
40
20
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Height (in.)
Wei
ght (
lb)
240
230
220
210
200
190
180
170
160
150
140130
60 80757065
390
Age nominated
Yea
rs a
s a
Supr
eme
Cou
rt ju
dge
40 45 50 55 60 65 70
35
30
25
20
15
10
5
0
Hea
ds o
f let
tuce
Price (in dollars)
100
90
80
70
60
50
40
30
20
10
0 0.25 3.25 3.002.25 2.001.751.50 1.251.000.750.50
14271AKTE2.pgs 9/25/06 10:42 AM Page 390
b. No linear correlation c. Nod. Answers will vary. See graphe. y � �0.40x � 36.6f. None of these
9. a.
b. Positivec. y � or y � 0.182x � 0.148; see graphd. y � 0.14x � 0.38 e. approx. 2 cups
Review Exercises (pages 725–728)1. Yes. If the set contains N integers and N is even,
then there are pairs each with a sum of (N � 1).The average of the middle two numbers is .
The mean is . If
N is odd, there are pairs each with a sum of
(N � 1) plus one number equal to , which
is also the median.The sum of the numbers is
,
so the mean is .2. a. When N is odd, or when the two middle
numbers in the set are equal.b. When N is even and when the two middle
numbers in the set are not equal.3. (1) a. (2) a.
b. 3.5 b. 3c. 3 c. 1
(3) a. (4) a.b. 3 b. 3c. 3 c. 2, 3
4. 5y � 85. a. Stem Leaf
5 4 96 0 3 5 57 2 5 7 8 8 8 88 0 1 79 1
Key: 5 � 4 � 54
46742
3
32732
3
N2 1 N2N 5 N 1 1
2
N 2 12 (N 1 1) 1 N 1 1
2 5 N2 2 1 1 N 1 12 5 N2 1 N
2
N 1 12
N 2 12
N2 (N 1 1)
N 5N(N 1 1)
2N 5N 1 1
2
N112
N2
211x 1 13
88
2.00
1.75
1.50
1.25
1.00
0.750.50
0.25
0 1 2 3 4 5 6 7 8Number of potatoes
Cup
s of
dre
ssin
gb.
c.
d.
6. a. (1) 67 kg (2) 70 kg (3) 72 kgb. (1) 62 kg (2) 65 kg (3) 67 kg
7. 93 8. a. 80° b. 81° c. 83°
9. a. Interval Tally Frequency
0–5 3
6–11 9
12–17 6
18–23 0
24–29 2
b.
c. 6–11 d. 6–11
Freq
uenc
y
Number of hours
109876543210
0–5 6–11 12–17 18–23 24–29
50 55 60 65 70 75 80 85 90 95
Cum
ulat
ive
freq
uenc
yInterval
181614121086420
50–59 50–69 50–79 50–89 50–99
Freq
uenc
y
Interval
876543210
50–59 60–69 70–79 80–8990–99
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14271AKTE2.pgs 9/25/06 10:42 AM Page 391
10. a. Univariateb. 80.4c. 80d. 70
e. Cumulative Score Frequency Frequency
60 1 1
70 9 10
80 8 18
90 2 20
100 5 25
f.
g. 76%h.
11. a. Stem Leaf
2 71 77 86 92 973 01 03 03 21 36 70 82 794 04 26 32 35 42 44 49 57 72 86 895 20 23 25
b. 404 votesc. Q1 � 303, Q3 � 457d.
12. a. 15b. 50th percentilec. 7 students
13. a. Qualitative b. Quantitativec. Quantitative d. Quantitativee. Quantitative f. Quantitativeg. Qualitative h. Qualitative
14. The sample sizes for the current and former alco-holics are too small.This is not an experiment, andtherefore, it is not possible to establish cause-and-effect.
15. a. Bivariate
200 250 300 350 400 450 500 550 600
Key: 2 � 71 � 271 votes
825
Cum
ulat
ive
freq
uenc
y
Score
25
20
15
10
5
060–60 90–70 60–80 60–90 60–100
b.
c. Yes; positived. Yes; the cost increases as the number of
oranges increases.e. Answers will vary. Example: y � 0.232x � 0.455;
see graphf. $1.38
16. The bar representing the net income for 2002 isalmost as tall as the bar for 2003, giving theimpression that the income of the company hasbeen steadily increasing each year. However,in 2002 XYZ Company lost $4,000 and in 2003it earned a profit of $5,123.
Exploration (page 728)a. In 2004, the ratio of students who scored at or
below 1370 to students who scored at or above1370 was less than in 2000.
b. Other students also increased their grade pointaverages, causing the ratio of students scoring ator below 3.4 to students scoring at or above 3.4one semester to equal the ratio of students scoringat or below 3.8 to students scoring at or above 3.8the next semester.
Cumulative Review (pages 729–730)Part I
1. 4 2. 2 3. 2 4. 1 5. 36. 4 7. 3 8. 3 9. 3 10. 1
Part II11. 15 persons; w � m � 3, ,
5m � 15 � 6m � 9, m � 6, w � 912. 19°; sin x � , x � 19°Part III13. 10.5 m, 17.5 m, 21 m; 3x � 5x � 6x � 49.0,
14x � 49.0, x � 3.5, 3x � 10.5, 5x � 17.5, 6x � 2114. a. 2 hr, 1 hr, 0.5 hr, 0.25 hr; 8x � 4x � 2x � x �
3.75, 15x � 3.75, x � 0.25, 2x � 0.5, 4x � 1,8x � 2
b. ; 3.75 � 4 � 1516
1516 hr
1237
m 1 3m 1 (m 1 3) 5 m 1 3
2m 1 3 5 35
2
1.5
1
0.5
00 1 2 3 4 5
Weight (lb)
Cos
t ($)
392
14271AKTE2.pgs 9/25/06 10:42 AM Page 392
Part IV15. a. y � �x2 � 8x; length � � 8 � x, area �
x(8 � x) � �x2 � 8xb. y
xO
1
–1
16 2 2x2
c. 16 sq ft; x � , y � �(4)2 � 8(4) �
�16 � 32 � 16
16. a. 8:20 A.M.; � 2h, 8h � 2 � 2h, 6h � 2,
h � ,
b. ; 2 mph hr � 23 mi1
323 mi
13 hr ? 60 min
1 hr 5 20 min13
8 Ah214 B
282(21) 5 4
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