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7.E: Chapter 7 Review Exercises

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Simplify, Multiply, and Divide Rational Expressions

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

1. 5a+33a2

Answer

a23

2. b7b225

3. 5x2y28y

Answer

y0

4. x3x2x30

Simplify Rational Expressions

In the following exercises, simplify.

5. 1824

Answer

34

6. 9m418mn3

7. x2+7x+12x2+8x+16

Answer

x+3x+4

8. 7v3525v2

Multiply Rational Expressions

In the following exercises, multiply.

9. 58415

Answer

16

10. 3xy28y316y224x

11. 72x12x28x+32x2+10x+24x236

Answer

3x2

12. 6y22y109y2y26y+96y2+29y20

Divide Rational Expressions

In the following exercises, divide.

13. x24x12x2+8x+12÷x2363x

Answer

3x(x+6)(x+6)

14. y2164÷y3642y2+8y+32

15. 11+ww9÷121w29w

Answer

111+w

16. 3y212y634y+3÷(6y242y)

17. c2643c2+26c+16c24c3215c+10

Answer

5c+4

18. 8a2+16aa4a2+2a24a2+7a+10÷2a26aa+5

Multiply and Divide Rational Functions

19. Find R(x)=f(x)g(x) where f(x)=9x2+9xx23x4 and g(x)=x2163x2+12x.

Answer

R(x)=3

20. Find R(x)=f(x)g(x) where f(x)=27x23x21 and g(x)=9x2+54xx2x42.

Add and Subtract Rational Expressions

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

21. 715+815

Answer

1

22. 4a22a112a1

23. y2+10yy+5+25y+5

Answer

y+5

24. 7x2x29+21xx29

25. x2x73x+28x7

Answer

x+4

26. y2y+11121y+11

27. 4q2q+3q2+6q+53q2q6q2+6q+5

Answer

q3q+5

28. 5t+4t+3t2254t28t32t225

Add and Subtract Rational Expressions Whose Denominators Are Opposites

In the following exercises, add and subtract.

29. 18w6w1+3w216w

Answer

15w+26w1

30. a2+3aa243a84a2

31. 2b2+3b15b249b2+16b149b2

Answer

3b2b+7

32. 8y210y+72y5+2y2+7y+252y

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

33. 7a23a10,3aa2a20

Answer

(a+2)(a5)(a+4)

34. 6n24,2nn24n+4

35. 53p2+17p6,2m3p223p8

Answer

(3p+1)(p+6)(p+8)

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

36. 75a+32b

37. 2c2+9c+3

Answer

11c12(c2)(c+3)

38. 3xx29+5x2+6x+9

39. 2xx2+10x+24+3xx2+8x+16

Answer

5x2+26x(x+4)(x+4)(x+6)

40. 5qp2qp2+4qq21

41. 3yy+2y+2y+8

Answer

2(y2+10y2)(y+2)(y+8)

42. 3w15w2+w20w+24w

43. 7m+3m+25

Answer

2m7m+2

44. nn+3+2n3n9n29

45. 8aa2644a+8

Answer

4a8

46. 512x2y+720xy3

Add and Subtract Rational Functions

In the following exercises, find R(x)=f(x)+g(x) where f(x) and g(x) are given.

47. f(x)=2x2+12x11x2+3x10,g(x)=x+12x

Answer

R(x)=x+8x+5

48. f(x)=4x+31x2+x30,g(x)=5x+6

In the following exercises, find R(x)=f(x)g(x) where f(x) and g(x) are given.

49. f(x)=4xx2121,g(x)=2x11

Answer

R(x)=2x+11

50. f(x)=7x+6,g(x)=14xx236

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

51. 7xx+214x2x24

Answer

x22x

52. 25+5613+14

53. x3xx+51x+5+1x5

Answer

(x8)(x5)2

54. 2m+mnnm1n

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

55. 13+1814+112

Answer

118

56. 3a21b1a+1b2

57. 2z249+1z+79z+7+12z7

Answer

z521z+21

58. 3y24y322y8+1y+4

Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

59. 12+23=1x

Answer

x=67

60. 12m=8m2

61. 1b2+1b+2=3b24

Answer

b=32

62. 3q+82q2=1

63. v15v29v+18=4v3+2v6

Answer

no solution

64. z12+z+33z=1z

Solve Rational Equations that Involve Functions

65. For rational function, f(x)=x+2x26x+8,

  1. Find the domain of the function
  2. Solve f(x)=1
  3. Find the points on the graph at this function value.
Answer
  1. The domain is all real numbers except x2 and x4
  2. x=1,x=6
  3. (1,1),(6,1)

66. For rational function, f(x)=2xx2+7x+10,

  1. Solve f(x)=2
  2. Find the points on the graph at this function value.

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

67. Vl=hw for l

Answer

l=Vhw

68. 1x2y=5 for y

69. x=y+5z7 for z

Answer

z=y+5+7xx

70. P=kV for V

Solve Applications with Rational Equations

Solve Proportions

In the following exercises, solve.

71. x4=35

Answer

x=125

72. 3y=95

73. ss+20=37

Answer

s=15

74. t35=t+29

Solve Applications Using Proportions

In the following exercises, solve.

75. Rachael had a 21-ounce strawberry shake that has 739 calories. How many calories are there in a 32-ounce shake?

Answer

1161 calories

76. Leo went to Mexico over Christmas break and changed $525 dollars into Mexican pesos. At that time, the exchange rate had $1 US is equal to 16.25 Mexican pesos. How many Mexican pesos did he get for his trip?

Solve Similar Figure Applications

In the following exercises, solve.

77. ΔABC is similar to ΔXYZ. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

clipboard_e3ea5ba04cf9cf3eca601a80313183f6d.png

Answer

b=9;x=213

78. On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is 700 miles, find the distance from

  1. Paris to Rome
  2. Paris to Vienna

clipboard_e52ae790b0233f8e2e5c50b849fb21443.png

79. Francesca is 5.75 feet tall. Late one afternoon, her shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.

Answer

23 feet

80. The height of a lighthouse in Pensacola, Florida is 150 feet. Standing next to the statue, 5.5-foot-tall Natasha cast a 1.1-foot shadow. How long would the shadow of the lighthouse be?

Solve Uniform Motion Applications

In the following exercises, solve.

81. When making the 5-hour drive home from visiting her parents, Lolo ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles when it turned bad. How fast did she drive when the weather was bad?

Answer

45 mph

82. Mark is riding on a plane that can fly 490 miles with a tailwind of 20 mph in the same time that it can fly 350 miles against a tailwind of 20 mph. What is the speed of the plane?

83. Josue can ride his bicycle 8 mph faster than Arjun can ride his bike. It takes Luke 3 hours longer than Josue to ride 48 miles. How fast can John ride his bike?

Answer

16 mph

84. Curtis was training for a triathlon. He ran 8 kilometers and biked 32 kilometers in a total of 3 hours. His running speed was 8 kilometers per hour less than his biking speed. What was his running speed?

Solve Work Applications

In the following exercises, solve.

85. Brandy can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?

Answer

45 hour

86. Prem takes 3 hours to mow the lawn while her cousin, Barb, takes 2 hours. How long will it take them working together?

87. Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?

Answer

12 days

88. Marta and Deb work together writing a book that takes them 90 days. If Sue worked alone it would take her 120 days. How long would it take Deb to write the book alone?

Solve Direct Variation Problems

In the following exercises, solve.

89. If y varies directly as x when y=9 and x=3, find x when y=21.

Answer

7

90. If y varies inversely as x when y=20 and x=2 find y when x=4.

91. Vanessa is traveling to see her fiancé. The distance, d, varies directly with the speed, v, she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?

Answer

301 mph

92. If the cost of a pizza varies directly with its diameter, and if an 8” diameter pizza costs $12, how much would a 6” diameter pizza cost?

93. The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?

Answer

288 feet

Solve Inverse Variation Problems

In the following exercises, solve.

94. If m varies inversely with the square of n, when m=4 and n=6 find m when n=2.

95. The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for $6, how many tickets can Madelyn afford to buy if the price increased to $8?

Answer

97 tickets

96. On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an 11-inch string on a violin has a frequency of 360 cycles per second, what frequency does a 12-inch string have?

Solve Rational Inequalities

Solve Rational Inequalities

In the following exercises, solve each rational inequality and write the solution in interval notation.

97. x3x+40

Answer

(4,3]

98. 5xx2>1

99. 3x2x42

Answer

[6,4)

100. 1x24x12<0

101. 124x21x

Answer

(,2][4,)

102. 4x2<3x+1

Solve an Inequality with Rational Functions

In the following exercises, solve each rational function inequality and write the solution in interval notation

103. Given the function, R(x)=x5x2, find the values of x that make the function greater than or equal to 0.

Answer

(,2)[5,)

104. Given the function, R(x)=x+1x+3, find the values of x that make the function greater than or equal to 0.

105. The function C(x)=150x+100,000 represents the cost to produce x, number of items. Find

  1. The average cost function, c(x)
  2. How many items should be produced so that the average cost is less than $160.
Answer
  1. c(x)=150x+100000x
  2. More than 10,000 items must be produced to keep the average cost below $160 per item.

106. Tillman is starting his own business by selling tacos at the beach. Accounting for the cost of his food truck and ingredients for the tacos, the function C(x)=2x+6,000 represents the cost for Tillman to produce x, tacos. Find

  1. The average cost function, c(x) for Tillman’s Tacos
  2. How many tacos should Tillman produce so that the average cost is less than $4.

Practice Test

In the following exercises, simplify.

1. 4a2b12ab2

Answer

a3b

2. 6x18x29

In the following exercises, perform the indicated operation and simplify.

3. 4xx+2x2+5x+612x2

Answer

x+33x

4. 2y2y21÷y3y2+yy31

5. 6x2x+20x2815x2+11x7x281

Answer

x3x+9

6. 3a3a3+5aa2+3a4

7. 2n2+8n1n21n27n11n2

Answer

3n2n1

8. 10x2+16x78x3+2x2+3x138x

9. 1m1n1n+1m

Answer

nmm+n

In the following exercises, solve each equation.

10. 1x+34=58

11. 1z5+1z+5=1z225

Answer

z=12

12. z2z+834z8=3z216z168z2+2z64

In the following exercises, solve each rational inequality and write the solution in interval notation.

13. 6xx62

Answer

[3,6)

14. 2x+3x6>1

15. 12+12x25x

Answer

(,0)(0,4][6,)

In the following exercises, find R(x) given f(x)=x4x23x10 and g(x)=x5x22x8.

16. R(x)=f(x)g(x)

17. R(x)=f(x)g(x)

Answer

R(x)=1(x+2)(x+2)

18. R(x)=f(x)÷g(x)

19. Given the function, R(x)=22x2+x15, find the values of x that make the function less than or equal to 0.

Answer

(2,5]

In the following exercises, solve.

20. If y varies directly with x, and x=5 when y=30, find x when y=42.

21. If y varies inversely with the square of x and x=3 when y=9, find y when x=4.

Answer

y=8116

22. Matheus can ride his bike for 30 miles with the wind in the same amount of time that he can go 21 miles against the wind. If the wind’s speed is 6 mph, what is Matheus’ speed on his bike?

23. Oliver can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Oliver’s dad working alone to split the logs?

Answer

Oliver’s dad would take 445 hours to split the logs himself.

24. The volume of a gas in a container varies inversely with the pressure on the gas. If a container of nitrogen has a volume of 29.5 liters with 2000 psi, what is the volume if the tank has a 14.7 psi rating? Round to the nearest whole number.

25.

The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio. The diagram gives the map distances between these cities in inches.

clipboard_ed99a0bbb595e2e6dbfc06d21e311369b.png

The actual distance from Dayton to Cincinnati is 48 miles. What is the actual distance between Dayton and Columbus?

Answer

The distance between Dayton and Columbus is 64 miles.


This page titled 7.E: Chapter 7 Review Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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