15.E: Review Exercises

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. \(\dfrac{5 a+3}{3 a-2}\)

Answer

\(a \neq \dfrac{2}{3}\)

2. \(\dfrac{b-7}{b^{2}-25}\)

3. \(\dfrac{5 x^{2} y^{2}}{8 y}\)

Answer

\(y \neq 0\)

4. \(\dfrac{x-3}{x^{2}-x-30}\)

Simplify Rational Expressions

In the following exercises, simplify.

5. \(\dfrac{18}{24}\)

Answer

\(\dfrac{3}{4}\)

6. \(\dfrac{9 m^{4}}{18 m n^{3}}\)

7. \(\dfrac{x^{2}+7 x+12}{x^{2}+8 x+16}\)

Answer

\(\dfrac{x+3}{x+4}\)

8. \(\dfrac{7 v-35}{25-v^{2}}\)

Multiply Rational Expressions

In the following exercises, multiply.

9. \(\dfrac{5}{8} \cdot \dfrac{4}{15}\)

Answer

\(\dfrac{1}{6}\)

10. \(\dfrac{3 x y^{2}}{8 y^{3}} \cdot \dfrac{16 y^{2}}{24 x}\)

11. \(\dfrac{72 x-12 x^{2}}{8 x+32} \cdot \dfrac{x^{2}+10 x+24}{x^{2}-36}\)

Answer

\(\dfrac{-3 x}{2}\)

12. \(\dfrac{6 y^{2}-2 y-10}{9-y^{2}} \cdot \dfrac{y^{2}-6 y+9}{6 y^{2}+29 y-20}\)

Divide Rational Expressions

In the following exercises, divide.

13. \(\dfrac{x^{2}-4 x-12}{x^{2}+8 x+12} \div \dfrac{x^{2}-36}{3 x}\)

Answer

\(\dfrac{3 x}{(x+6)(x+6)}\)

14. \(\dfrac{y^{2}-16}{4} \div \dfrac{y^{3}-64}{2 y^{2}+8 y+32}\)

15. \(\dfrac{11+w}{w-9} \div \dfrac{121-w^{2}}{9-w}\)

Answer

\(\dfrac{1}{11+w}\)

16. \(\dfrac{3 y^{2}-12 y-63}{4 y+3} \div\left(6 y^{2}-42 y\right)\)

17. \(\dfrac{\dfrac{c^{2}-64}{3 c^{2}+26 c+16}}{\dfrac{c^{2}-4 c-32}{15 c+10}}\)

Answer

\(\dfrac{5}{c+4}\)

18. \(\dfrac{8 a^{2}+16 a}{a-4} \cdot \dfrac{a^{2}+2 a-24}{a^{2}+7 a+10} \div \dfrac{2 a^{2}-6 a}{a+5}\)

Multiply and Divide Rational Functions

19. Find \(R(x)=f(x) \cdot g(x)\) where \(f(x)=\dfrac{9 x^{2}+9 x}{x^{2}-3 x-4}\) and \(g(x)=\dfrac{x^{2}-16}{3 x^{2}+12 x}\).

Answer

\(R(x)=3\)

20. Find \(R(x)=\dfrac{f(x)}{g(x)}\) where \(f(x)=\dfrac{27 x^{2}}{3 x-21}\) and \(g(x)=\dfrac{9 x^{2}+54 x}{x^{2}-x-42} \).

Add and Subtract Rational Expressions

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

21. \(\dfrac{7}{15}+\dfrac{8}{15}\)

Answer

\(1\)

22. \(\dfrac{4 a^{2}}{2 a-1}-\dfrac{1}{2 a-1}\)

23. \(\dfrac{y^{2}+10 y}{y+5}+\dfrac{25}{y+5}\)

Answer

\(y+5\)

24. \(\dfrac{7 x^{2}}{x^{2}-9}+\dfrac{21 x}{x^{2}-9}\)

25. \(\dfrac{x^{2}}{x-7}-\dfrac{3 x+28}{x-7}\)

Answer

\(x+4\)

26. \(\dfrac{y^{2}}{y+11}-\dfrac{121}{y+11}\)

27. \(\dfrac{4 q^{2}-q+3}{q^{2}+6 q+5}-\dfrac{3 q^{2}-q-6}{q^{2}+6 q+5}\)

Answer

\(\dfrac{q-3}{q+5}\)

28. \(\dfrac{5 t+4 t+3}{t^{2}-25}-\dfrac{4 t^{2}-8 t-32}{t^{2}-25}\)

Add and Subtract Rational Expressions Whose Denominators Are Opposites

In the following exercises, add and subtract.

29. \(\dfrac{18 w}{6 w-1}+\dfrac{3 w-2}{1-6 w}\)

Answer

\(\dfrac{15 w+2}{6 w-1}\)

30. \(\dfrac{a^{2}+3 a}{a^{2}-4}-\dfrac{3 a-8}{4-a^{2}}\)

31. \(\dfrac{2 b^{2}+3 b-15}{b^{2}-49}-\dfrac{b^{2}+16 b-1}{49-b^{2}}\)

Answer

\(\dfrac{3 b-2}{b+7}\)

32. \(\dfrac{8 y^{2}-10 y+7}{2 y-5}+\dfrac{2 y^{2}+7 y+2}{5-2 y}\)

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

33. \(\dfrac{7}{a^{2}-3 a-10}, \dfrac{3 a}{a^{2}-a-20}\)

Answer

\((a+2)(a-5)(a+4)\)

34. \(\dfrac{6}{n^{2}-4}, \dfrac{2 n}{n^{2}-4 n+4}\)

35. \(\dfrac{5}{3 p^{2}+17 p-6}, \dfrac{2 m}{3 p^{2}-23 p-8}\)

Answer

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

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

36. \(\dfrac{7}{5 a}+\dfrac{3}{2 b}\)

37. \(\dfrac{2}{c-2}+\dfrac{9}{c+3}\)

Answer

\(\dfrac{11 c-12}{(c-2)(c+3)}\)

38. \(\dfrac{3 x}{x^{2}-9}+\dfrac{5}{x^{2}+6 x+9}\)

39. \(\dfrac{2 x}{x^{2}+10 x+24}+\dfrac{3 x}{x^{2}+8 x+16}\)

Answer

\(\dfrac{5 x^{2}+26 x}{(x+4)(x+4)(x+6)}\)

40. \(\dfrac{5 q}{p^{2} q-p^{2}}+\dfrac{4 q}{q^{2}-1}\)

41. \(\dfrac{3 y}{y+2}-\dfrac{y+2}{y+8}\)

Answer

\(\dfrac{2\left(y^{2}+10 y-2\right)}{(y+2)(y+8)}\)

42. \(\dfrac{-3 w-15}{w^{2}+w-20}-\dfrac{w+2}{4-w}\)

43. \(\dfrac{7 m+3}{m+2}-5\)

Answer

\(\dfrac{2 m-7}{m+2}\)

44. \(\dfrac{n}{n+3}+\dfrac{2}{n-3}-\dfrac{n-9}{n^{2}-9}\)

45. \(\dfrac{8 a}{a^{2}-64}-\dfrac{4}{a+8}\)

Answer

\(\dfrac{4}{a-8}\)

46. \(\dfrac{5}{12 x^{2} y}+\dfrac{7}{20 x y^{3}}\)

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. \(f(x)=\dfrac{2 x^{2}+12 x-11}{x^{2}+3 x-10}, g(x)=\dfrac{x+1}{2-x}\)

Answer

\(R(x)=\dfrac{x+8}{x+5}\)

48. \(f(x)=\dfrac{-4 x+31}{x^{2}+x-30}, g(x)=\dfrac{5}{x+6}\)

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

49. \(f(x)=\dfrac{4 x}{x^{2}-121}, g(x)=\dfrac{2}{x-11}\)

Answer

\(R(x)=\dfrac{2}{x+11}\)

50. \(f(x)=\dfrac{7}{x+6}, g(x)=\dfrac{14 x}{x^{2}-36}\)

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

51. \(\dfrac{\dfrac{7 x}{x+2}}{\dfrac{14 x^{2}}{x^{2}-4}}\)

Answer

\(\dfrac{x-2}{2 x}\)

52. \(\dfrac{\dfrac{2}{5}+\dfrac{5}{6}}{\dfrac{1}{3}+\dfrac{1}{4}}\)

53. \(\dfrac{x-\dfrac{3 x}{x+5}}{\dfrac{1}{x+5}+\dfrac{1}{x-5}}\)

Answer

\(\dfrac{(x-8)(x-5)}{2}\)

54. \(\dfrac{\dfrac{2}{m}+\dfrac{m}{n}}{\dfrac{n}{m}-\dfrac{1}{n}}\)

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

55. \(\dfrac{\dfrac{1}{3}+\dfrac{1}{8}}{\dfrac{1}{4}+\dfrac{1}{12}}\)

Answer

\(\dfrac{11}{8}\)

56. \(\dfrac{\dfrac{3}{a^{2}}-\dfrac{1}{b}}{\dfrac{1}{a}+\dfrac{1}{b^{2}}}\)

57. \(\dfrac{\dfrac{2}{z^{2}-49}+\dfrac{1}{z+7}}{\dfrac{9}{z+7}+\dfrac{12}{z-7}}\)

Answer

\(\dfrac{z-5}{21 z+21}\)

58. \(\dfrac{\dfrac{3}{y^{2}-4 y-32}}{\dfrac{2}{y-8}+\dfrac{1}{y+4}}\)

Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

59. \(\dfrac{1}{2}+\dfrac{2}{3}=\dfrac{1}{x}\)

Answer

\(x=\dfrac{6}{7}\)

60. \(1-\dfrac{2}{m}=\dfrac{8}{m^{2}}\)

61. \(\dfrac{1}{b-2}+\dfrac{1}{b+2}=\dfrac{3}{b^{2}-4}\)

Answer

\(b=\dfrac{3}{2}\)

62. \(\dfrac{3}{q+8}-\dfrac{2}{q-2}=1\)

63. \(\dfrac{v-15}{v^{2}-9 v+18}=\dfrac{4}{v-3}+\dfrac{2}{v-6}\)

Answer

no solution

64. \(\dfrac{z}{12}+\dfrac{z+3}{3 z}=\dfrac{1}{z}\)

Solve Rational Equations that Involve Functions

65. For rational function, \(f(x)=\dfrac{x+2}{x^{2}-6 x+8}\),

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 \(x \neq 2\) and \(x \neq 4\)
2. \(x=1, x=6\)
3. \((1,1),(6,1)\)

66. For rational function, \(f(x)=\dfrac{2-x}{x^{2}+7 x+10}\),

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. \(\dfrac{V}{l}=h w\) for \(l\)

Answer

\(l=\dfrac{V}{h w}\)

68. \(\dfrac{1}{x}-\dfrac{2}{y}=5\) for \(y\)

69. \(x=\dfrac{y+5}{z-7}\) for \(z\)

Answer

\(z=\dfrac{y+5+7 x}{x}\)

70. \(P=\dfrac{k}{V}\) for \(V\)

Solve Applications with Rational Equations

Solve Proportions

In the following exercises, solve.

71. \(\dfrac{x}{4}=\dfrac{3}{5}\)

Answer

\(x = \dfrac{12}{5}\)

72. \(\dfrac{3}{y}=\dfrac{9}{5}\)

73. \(\dfrac{s}{s+20}=\dfrac{3}{7}\)

Answer

\(s = 15\)

74. \(\dfrac{t-3}{5}=\dfrac{t+2}{9}\)

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. \(\Delta ABC\) is similar to \(\Delta XYZ\). The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

Answer

\(b=9 ; \; x=2 \dfrac{1}{3}\)

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

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

\(\dfrac{4}{5}\) 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. \(\dfrac{x-3}{x+4} \leq 0\)

Answer

\((-4,3]\)

98. \(\dfrac{5 x}{x-2}>1\)

99. \(\dfrac{3 x-2}{x-4} \leq 2\)

Answer

\([-6,4)\)

100. \(\dfrac{1}{x^{2}-4 x-12}<0\)

101. \(\dfrac{1}{2}-\dfrac{4}{x^{2}} \geq \dfrac{1}{x}\)

Answer

\((-\infty,-2] \cup[4, \infty)\)

102. \(\dfrac{4}{x-2}<\dfrac{3}{x+1}\)

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, \(R(x)=\dfrac{x-5}{x-2}\), find the values of \(x\) that make the function greater than or equal to 0.

Answer

\((-\infty, 2) \cup[5, \infty)\)

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

105. The function \(C(x)=150 x+100,000\) 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. \(c(x)=\dfrac{150 x+100000}{x}\)
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 \(C(x)=2 x+6,000\) 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. \(\dfrac{4 a^{2} b}{12 a b^{2}}\)

Answer

\(\dfrac{a}{3 b}\)

2. \(\dfrac{6 x-18}{x^{2}-9}\)

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

3. \(\dfrac{4 x}{x+2} \cdot \dfrac{x^{2}+5 x+6}{12 x^{2}}\)

Answer

\(\dfrac{x+3}{3 x}\)

4. \(\dfrac{2 y^{2}}{y^{2}-1} \div \dfrac{y^{3}-y^{2}+y}{y^{3}-1}\)

5. \(\dfrac{6 x^{2}-x+20}{x^{2}-81}-\dfrac{5 x^{2}+11 x-7}{x^{2}-81}\)

Answer

\(\dfrac{x-3}{x+9}\)

6. \(\dfrac{-3 a}{3 a-3}+\dfrac{5 a}{a^{2}+3 a-4}\)

7. \(\dfrac{2 n^{2}+8 n-1}{n^{2}-1}-\dfrac{n^{2}-7 n-1}{1-n^{2}}\)

Answer

\(\dfrac{3 n-2}{n-1}\)

8. \(\dfrac{10 x^{2}+16 x-7}{8 x-3}+\dfrac{2 x^{2}+3 x-1}{3-8 x}\)

9. \(\dfrac{\dfrac{1}{m}-\dfrac{1}{n}}{\dfrac{1}{n}+\dfrac{1}{m}}\)

Answer

\(\dfrac{n-m}{m+n}\)

In the following exercises, solve each equation.

10. \(\dfrac{1}{x}+\dfrac{3}{4}=\dfrac{5}{8}\)

11. \(\dfrac{1}{z-5}+\dfrac{1}{z+5}=\dfrac{1}{z^{2}-25}\)

Answer

\(z=\dfrac{1}{2}\)

12. \(\dfrac{z}{2 z+8}-\dfrac{3}{4 z-8}=\dfrac{3 z^{2}-16 z-16}{8 z^{2}+2 z-64}\)

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

13. \(\dfrac{6 x}{x-6} \leq 2\)

Answer

\([-3,6)\)

14. \(\dfrac{2 x+3}{x-6}>1\)

15. \(\dfrac{1}{2}+\dfrac{12}{x^{2}} \geq \dfrac{5}{x}\)

Answer

\((-\infty, 0) \cup(0,4] \cup[6, \infty)\)

In the following exercises, find \(R(x)\) given \(f(x)=\dfrac{x-4}{x^{2}-3 x-10}\) and \(g(x)=\dfrac{x-5}{x^{2}-2 x-8}\).

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

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

Answer

\(R(x)=\dfrac{1}{(x+2)(x+2)}\)

18. \(R(x)=f(x) \div g(x)\)

19. Given the function, \(R(x)=\dfrac{2}{2 x^{2}+x-15}\), 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

\(y=\dfrac{81}{16}\)

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 \(4 \dfrac{4}{5}\) 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.

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.