A-26 Answers to Odd-Numbered Problems

Chapter 5

Exercise Set 5.1

Vocabulary Review 1. rational 3. function 5. quotient

Problems 1. x − 4 ____ 6 3. (a 2 + 9)(a + 3) 5. 2y + 3 _____ y + 1 7. x − 2 ____ x − 1 9. x − 3 ____ x + 2 11. x

2 − x + 1 _______ x − 1 13. −4a ___ 3 15. b − 1 ____ b + 1

17. 7x − 3 _____ 7x + 5 19. 4x + 3 _____ 4x − 3 21. x + 5 _____ 2x − 7 23. a2 − ab + b2

_________ a − b 25. 2x − 2 _____ x 27. x + 3 ____ y − 4 29. x + 2 31. x 2 + 2x + 4 ________ x + 2 33. 4x 2 + 6x + 9 _________ 2x + 3

35. 1 ________ x 2 – 2x + 4 37. −1 39. −(y + 6) 41. − 3a + 1 _____ 3a − 1 43. 3 45. x + a 47. x + a + 3 49. {x | x ≠ 1} 51. {x | x ≠ 2} 53. {t | t ≠ 4, t ≠ −4} 55. g(0) = −3, g(−3) = 0, g(3) = 3, g(−1) = −1, g(1) = undefi ned 57. h(0) = −3, h(−3) = 3, h(3) = 0, h(−1) is undefi ned, h(1) = −1 59. a. 4 b. 4 61. a. 5 b. 5 63. a. x + a b. 2x + h 65. a. x + a b. 2x + h 67. a. x + a − 3 b. 2x + h − 3 69. a. 2x + 2a − 3 b. 4x + 2h + 3 71.

Exercise Set 5.2

Vocabulary Review 1. product, numerators

Problems 1. 1 _ 6 3. 9 _ 4 5. 1 _ 2 7. 15y ___ x 2 9. b _ a 11. 2y 5

___ z 3 13. x + 3 ____ x + 2 15. y + 1 17. 3(x + 4) ______ x − 2 19. y 2

_____ xy + 1 21. x 2 + 9 _____ x 2 − 9

23. 1 _ 4 25. 1 27. (a − 2)(a + 2) __________ a − 5 29. 9t 2 − 6t + 4 ________ 4t 2 − 2t + 1 31. x + 3 ____ x + 4 33. a − b ____ 5 35. 5c − 1 _____ 3c − 2 37. 5a − b _____________ 9a 2 + 15ab + 25b 2 39. 2

41. x(x − 1)(x + 1) 43. (a + 4b)(a − 3b) ___________ (a − 4b)(a + 5b) 45. 2y − 1

_____ 2y − 3 47. (y − 2)(y + 1) __________ (y + 2)(y − 1) 49. x − 1 ____ x + 1 51. x − 2 ____ x + 3 53. w(y − 1)

______ w − x

55. (m + 2)(x + y) __________ (2x + y)2 57. (1 − 2d)(d + c)

___________ d − c 59. (r − s)(r + s ) _________ (r 2 + s 2)2 61. 3x 63. 2(x + 5) 65. x − 2 67. −(y − 4) or 4 − y

69. (a − 5)(a + 1) 71. a. 5 __ 21 b. 5x + 3 __________ 25x 2 + 15x + 9 c. 5x − 3 __________ 25x 2 + 15x + 9 d. 5x + 3 _____ 5x − 3 73. a. (x + 2)2

______ (x − 1)2 b. (x − 3)2 c. 1 ______ (x − 3)2

75. 77. $305 79.

81. Venus: 7,110 miles; Saturn: 75,050 miles 83. 2 _ 3 85. 47 ___ 105 87. x − 7 89. −4x + 10 91. (x + 1)(x − 1)

93. 2(x + 5) 95. (a − b)(a2 + ab + b2) 97. (2x + 3)(2x – 3) 99. x4 − x2 y2 + y4

_________ x2 + y2 101. (a − 1)(a + 5) __________ 3a2 − 2a + 1 103. −(p2 − pq + q2)

Find the Mistake 1. Multiplying x 2 – 4 _ x – 2

∙ x 2 + 2x – 8 _ x 2 + 6x + 8

we get x – 2. 3. To divide the rational expression 5x 2 + 21x – 20 _ 15x 2 – 17x + 4

by x 2 – 25 _ 3x 2 – 4x + 1

we

multiply by the reciprocal of the expression that follows the division symbol.

Exercise Set 5.3

Vocabulary Review 1. common 3. diff erent, equivalent, numerators, lowestProblems 1. 1 3. −1 5. 1 ____ x + y 7. 1 9. a

2 + 2a − 3 ________ a3 11. 1 13. 5 _ 4 15. 1 _ 3 17. 41 __ 24 19. 19 ___ 144 21. 31 __ 24

23. 2 _ 3 25. a. 1 __ 16 b. 9 _ 4 c. 13 __ 24 d. 5x + 15 ______ (x − 3)2 e. x + 3 ____ 5 f. x − 2 ____ x − 3 27. f(x) − g(x) = 1 _ 2 29. f(x) − g(x) = 1 _ 5 31. x + 3 ______ 2(x + 1)

33. a − b _________ a2 + ab + b2 35. 2y − 3 _________ 4y 2 + 6y + 9 37. 2(2x − 3)

__________ (x − 3)(x − 2) 39. 1 _____ 2t − 7 41. 4 __________ (a − 3)(a + 1) 43. −4x 2 _____________________ (2x + 1)(2x − 1)(4x 2 + 2x + 1)

45. 2 ___________ (2x + 3)(4x + 3) 47. a __________ (a + 4)(a + 5) 49. x + 1 __________ (x − 2)(x + 3) 51. x − 1 __________ (x + 1)(x + 2) 53. 1 __________ (x + 2)(x + 1) 55. 1 __________ (x + 2)(x + 3)

Weeks Weight (lb)

x W(x)

0 200 1 194.3 4 184 12 173.3 24 168

73. a. C(n) = 1200 + 540n b. _

C (n) = 1200 + 540n _________ n c. $660 d. $620 e. Decreasing 75. 2 _ 3 77. 20x 2y 2 79. 72x 4y 5 81. (x + 2)(x − 2) 83. x 2(x − y) 85. 2(y + 1)(y − 1) 87. a. 2 b. −4 c. Undefi ned d. 2 e. 1 f. −6 g. 4 h. −3 Find the Mistake 1. The rational expression 8x 2 + 2x – 1 _________ 16x 2 – 6x – 7 reduced to lowest terms is 4x – 1 _

8x – 7 .

3. Th e rational function f(x) = x 3 – 4x 2 + 4x + 16 ____________ x 2 – 6x + 8 is undefi ned when f(4) and f(2).

Number of Copies Price per Copy ($)

x p(x)

1 20.33 10 9.33 20 6.40 50 4.00 100 3.05

Time Concentration

t c(t)

5 10.40 10 2.70 15 1.24 20 0.73

Answers to Odd-Numbered Problems A-27

57. 4x + 5 _____ 2x + 1 59. 22 − 5t _____ 4 − t 61. 2x 2 + 3x − 4 _________ 2x + 3 63. 2x − 3 _____ 2x 65. 1 _ 2 67. 3 ____ x + 4 69. (2x + 1)(x + 5) ______________ (x − 2)(x + 1)(x + 3) 71. 51 __ 10

73. 1 hour, 20 minutes 75. x + 4 _ x = x 2 + 4 _____ x 77. 1 _ x + 1 ____ x + 1 = 2x + 1 ______ x(x + 1) 79. 6 _ 5 81. 1 _ 9 83. x + 2 85. 3 − x

87. (x + 2)(x − 2) 89. (x − 3)(x 2 + 3x + 9) 91. (x – 1) _____ (x + 3) 93. a

2 + ab + a − v ___________ u + v 95. 6x + 5 ___________ (4x − 1)(3x − 4) 97. y(y 2 + 1) ___________ (y + 1)2(y − 1)2

Find the Mistake 1. If we combine the rational expressions x + 5 _____ 3x + 9 and 4 ____ x 2 – 9 and reduce to the lowest terms, we get x – 1 ____ 3x – 9 . 3. If we subtract 5 _

2x 2 + x – 3 from 2 _

x 2 – 1 , we get –1 _________ (x + 1)(2x + 3) .

Exercise Set 5.4

Vocabulary Review complex, quotient, numerator, LCD, reciprocal, denominator

Problems 1. 9 _ 8 3. 2 __ 15 5. 119 ___ 20 7. 1 ____ x + 1 9. a + 1 ____ a − 1 11. y − x ____ y + x 13. 1 __________ (x + 5)(x − 2) 15. 1 _______ a2 − a + 1 17. x + 3 ____ x + 2

19. a + 3 ____ a − 2 21. x − 3 ____ x + 3 23. a − 1 ____ a + 1 25. − x _ 3 27. y 2 + 1

_____ 2y 29. −x 2 + x − 1 _________ x − 1 31. 5 _ 3 33. 2x − 1 _____ 2x + 3 35. − 1 ______ x(x + h)

37. 3c + 4a − 2b _________ 5 39. (t − 4)(t + 1) _________ (t + 6)(t − 3) 41. (5b − 1)(b + 5)

___________ 2(2b − 11) 43. − 3 ______ 2x + 14 45. 2m − 9 47. – 2 _ x 49. f(x) ___ g(x) = x − 3 ____ x

51. f(x) ___ g(x) = x + 4 ____ x + 2 53. a. −4 ___ ax b. −1 __________ (x + 1)(a + 1) c. − a + x ____ a 2x 2 55. a. As v approaches 0, the denominator approaches 1 b. v = fs __ h − s

57. xy − 2x 59. 3x − 18 61. ab 63. 2t 2 − 3s 65. (y + 5)(y − 5) 67. x(a + b) 69. 2 71. −2, 6

Exercise Set 5.5

Vocabulary Review 1. rational 3. function

Problems 1. − 35 __ 3 3. − 18 __ 5 5. 36 __ 11 7. 2 9. 5 11. 2 13. −3, 4 15. 1, − 4 _ 3 17. No solution 19. 5 21. − 1 _ 2 , 5 _ 3 23. 2 _ 3 25. −4 27. −6 29. −5 31. 53 __ 17 33. 2 35. No solution 37. 22 __ 3 39. 18 41. No solution43. a. − 9 _ 5 , 5 b. 9 _ 2 c. No solution 45. a. 1 _ 3 b. 3 c. 9 d. 4 e. 1 _ 3 , 3 47. a. 6 __________ (x − 4)(x + 3) b. x − 3 ____ x − 4 c. 5

49. x = ab ____ a − b 51. y = x − 3 ____ x − 1 53. y = 1 − x _____ 3x − 2

55. 57. 59.

61.

x

y

f(x) = 1x − 3

x

y

f(x) = 4x + 2

x

y

f(x) = 2x − 4

x

y

f(x) = 6x + 1

63. Vertical asymptote: x = 5 _ 3 ; Domain: { x | x ≠ 5 _ 3 }

65. Vertical asymptote: x = 3 _ 2 ; Domain: { x | x ≠ 3 _ 2 } 67. Vertical asymptote: x = ±3; Domain: {x | x ≠ ±3} 69. Vertical asymptote: x = 1, –3; Domain: {x | x ≠ 1, –3}71. Vertical asymptote: x = 1, 2; Domain: {x | x ≠ 1, 2} 73. Vertical asymptote: x = – 1 _ 2 , 3; Domain: { x | x ≠ – 1 _ 2 , 3 }

75. 24 __ 5 feet 77. 2,358 79. 12.3 81. 3 83. 9, −1 85. 60 87. − 2 _ 3 , − 5 _ 2 , 5 _ 2 89. −2, 2, 3 91. x = − a _ 5 93. v = 16t 2 + s ______ t 95. f = pg

____ g − p Find the Mistake 1. When we solve the rational equation 1 ____ x – 3 – 5 ____ x + 3 = x ____ x 2 – 9 , we get x = 18 __ 5 3. Th e graph for f(x) = 3 ____ x – 1 will come very close but not touch the line x = 1.

A-28 Answers to Odd-Numbered Problems

Exercise Set 5.6

Vocabulary Review 1. blueprint 3. table

Problems 1. 1 _ 5 and 3 _ 5 3. 3 or 1 _ 3 5. 3, 4 7. 3 9. a-b. 11. 6 mph 13. a.

c. Th ey are the same. 1.5 ____ 5 − x = 3 ____ 5 + x c. Th ey are the same. 150 _____ x + 15 = 120 ___ x d. Th e speed of the current is 5 _ 3 mph. d. Th e speed of train A is 75 mph, and for train B it is 60 mph

15. 540 mph 17. 54 mph 19. 16 hours 21. 15 hours 23. 5.25 minutes 25. 10 = 1 _ 3 [ ( x + 2 _ 3 x ) + 1 _ 3 ( x + 2 _ 3 x ) ] ; x = 27 __ 2 27. a. 30 grams b. 3.25 moles 29. 31. 33. 35.

37. 121,215 fans 39. 2 41. −2x 2 y 2 43. 185.12 45. x − 3 47. 4x 3 − 8x 2 49. 4x 3 − 6x − 20 51. −3x + 9 53. (x + a)(x − a) 55. (x − 7y)(x + y)

Exercise Set 5.7Vocabulary Review 1. average 3. factor 5. estimate

Problems 1. 2x 2 − 4x + 3 3. −2x 2 − 3x + 4 5. 2y 2 + 5 _ 2 − 3 ___ 2y 2 7. − 5 _ 2 x + 4 + 3 _ x 9. 4ab3 + 6a 2 b 11. −xy + 2y 2 + 3xy 2

13. x + 2 15. a − 3 17. 5x + 6y 19. x 2 +xy + y 2 21. (y 2 + 4)(y + 2) 23. (x + 2)(x + 5) 25. (2x + 3)(2x − 3) 27. x − 7 + 7 ____ x + 2 29. 2x + 5 + 2 _____ 3x − 4 31. y 2 − 3y − 13 33. x −3 35. 3y 2 + 6y + 8 + 37 _____ 2y − 4 37. a 3 + 2a 2 + 4a + 6 + 17 ____ a − 2 39. y 3 + 2y 2 + 4y + 8 41. p(x)

___ q(x) = 2x 2 − 5x + 1 + 4 ____ x + 1 43. p(x) ___ q(x) = x 2 − 2x + 1 45. (x + 3)(x + 2)(x + 1)

47. (x + 3)(x + 4)(x − 2) 49. yes 51. −7; same 53. a. (x − 2)(x 2 − x + 3) b. (x − 5)(x 3 − x + 1) 55. b. p(x) = (x − 4) (x + 2)(x + 3)

57. a.

b. _

C (x) = 2 _ x + 0.15 c.

d. It decreases.

59. a. T(100) = 11.95, T(400) = 32.95, T(500) = 39.95 b. 4.95 + .07m ________ m c. _

T (100) = 0.1195, _

T (400) = 0.082, _

T (500) = 0.0799 61. 1,487,340 people 63. 2 __ 3a 65. (x − 3)(x + 2) 67. 1 69. 3 − x ____ x + 3 71. No solution 73. 196 75. 4 77. 1.6 79. 3 81. 2,400Find the Mistake 1. Dividing the polynomial 15x 3 + 6x 2 + 9x by the monomial 3x is the same as multiplying by 1 __ 3x . 3. When using long division to divide 2x 3 – 9x 2 + 19x – 15 __

2x – 3 , we fi rst estimate 2x 3 ÷ 2x = x 2.

d r t

Upstream 1.5 5 − x 1.5 ____ 5 − x

Downstream 3 5 + x 3 ____ 5 + x

d r t

Train A 150 x + 15 150 _____ x + 15

Train B 120 x 120 ___ x

x

y

f(x) = x − 3x − 1

(0, 3)

(3, 0) x

y

f(x) = x + 3x − 1

(−3, 0)

(0, −3)

x

y

f(x) = x − 3x + 1

(3, 0)

(0, 23)

x

y

f(x) = x + 2x + 4

(0, ) (22, 0) 12

x 1 5 10 15 20

C(x) 2.15 2.75 3.50 4.25 5.00

e. f. y = C(x): domain = {x|1 ≤ x ≤ 20}; range = {y|2.15 ≤ y ≤ 5.00}

y = _

C (x): domain = {x|1 ≤ x ≤ 20}; range = {y|0.25 ≤ y ≤ 2.15}

y = C(x)

0 4 8 12 16 200

1

2

3

4

6

5

x

y

y = C(x)

Number of Checks

Cos

ts (

$)

Number of Checks

Cos

t ($

)

x 1 5 10 15 20

__

C (x) 2.15 0.55 0.35 0.28 0.25

Answers to Odd-Numbered Problems A-29

Exercise Set 5.8Vocabulary Review 1. d 3. c 5. f 7. g 9. i 11. k Problems 1. 30 3. −6 5. 40 7. 81 __ 5 9. 64 11. 108 13. 300 15. ±2 17. 1600 19. ±8 21. 50 __ 3 pounds 25. 12 pounds per square inch

29. 1504 ____ 15 square inches 31. 1.5 ohms 33. a. P = 0.21 √— L c. 3.15 35. 0.6M − 42 37. 16x3 − 40x2 + 33x − 9

39. 4x 2 − 3x 41. 6x 2 − 2x − 4 43. 11

Chapter 5 Review

1. x − y 2. x − 2 ____ x + 1 3. 201.9 feet/min 4. a. 3 b. 3 5. a. 2x + 2 +h b. x + a + 2 6. 3(x + 3) 7. 5(x + 3)8. 2x + 4 ________ x2 + 2x + 4 9. 93 ___ 140 10. 25 __ 24 11. 1 ____ a − 9 12. 3(4x − 3) ___________ (x − 1)(2x − 1) 13. 2x __________ (x + 1)(x + 3) 14. x + 1 __________ (x − 1)(x + 3) 15. 4a + 7 _____ 4a + 9

16. x + 4 ____ x + 3 17. 5 _ 8 18. − 5 _ 3 , 2 19. − 5 _ 3 , 3 20. 5, 6 21. 6 22. 11 mph 23. 30 mins 24. 2x2y − 5x3 + 7y3

25. 3x 2 + 2x + 4 + 8 _____ 3x − 5 26. 3 _ 2 27. 112

Chapter 5 Cumulative Review

1. − 125 ___ 27 2. 49 __ 64 3. −40 4. 93 5. 9 6. 12 7. 1 __ 11 8. 1 9. − 11 __ 5 10. 2a − 9b > 2a + 9b 11. 14112. −139 13. 94 14. 90 15. −27x − 20 16. 20x 17. 1 _________ (y + 3)(y +2) 18. 1 ____ y − x 19. 20t4 − 1 __ 20 20. x − 9 ____ x + 3

21. 3x5n − 2x2n 22. 2a4 + 2a3 + 2a2 + 3a + 3 − 5 ____ a − 1 23. x5 24. x 7y 2 25. x + 5 26. 1 _______ a2 + a + 1 27. y = − 3 _ 5 x + 7

28. y = −1 29. 22 ∙ 32 ∙ 11 30. (x − 5)(x + 8) 31. (x − 6 + y)(x + 6 + y) 32. ( y − 4 _ 5 x ) ( y 2 + 4 _ 5 xy + 16 __ 25 x 2 ) 33. 14 34. −5 35. −1 36. 3 _ 2 37. 31 __ 3 38. − 2 _ 3 , 6 39. (1, 3) 40. ( 57 __ 31 , 8 __ 31 ) 41. x ≥ −6 42. 2 < x < 643. 44. 45. 4 46. 10 47. 7 48. 3

Chapter 5 Test

1. x + y 2. 2x − 3 _ x − 1

3. 4.1 feet/second 4. a. 8 b. 8 5. a. 2x + h + 1 b. x + a + 1 6. 3(a + 2) 7. 4(a − 2)

8. x + 1 ____ 2 9. 27 _ 50

10. 17 _ 32

11. 1 _ a + 5 12. 2(4x + 1) __

(x + 1)(2x − 1) 13. x _

(x + 4)(x + 3) 14. x + 2

_ (x − 5)(x + 3) 15. 2a + 11 _ 2a + 13

16. x − 5 _ x + 2

17. − 10 _ 19 18. Ø 19. − 12

_ 67 20. 7, 3 21. −2 22. 7 mph 23. 10 hours 24. 2x2 + 11xy − 2y2

25. x 2 − 3x + 2 + 2 _ 4x + 3

26. 1 __ 14 27. 25 __ 4

23. a. T = 4P b.

c. 70 pounds per square inch

0 20 40 60 80 1000

50

100

150

200

250

300

p

T

Number of Checks

Tem

per

ature

(ke

lvin

)

Pressure (lbs/sq in)

Tem

per

ature

(ke

lvin

)

27. a. f = 80 __ d b.

c. An f-stop of 8

0 20 40 60 80 100 12002468

10121416

d

f

Aperture diameter (mm)

f-st

op

Aperture diameter (mm)

f-sto

p

x

y

−6x +

5y =

−15

x

y

3x + y <

−4