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7.9: Chapter 8 Review Exercises

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Chapter Review Exercises

Simplify 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.

Exercise 7.9.1

2a+13a2

Answer

a23

Exercise 7.9.2

b3b216

Exercise 7.9.3

3xy25y

Answer

y0

Exercise 7.9.4

u3u2u30

Evaluate Rational Expressions

In the following exercises, evaluate the rational expressions for the given values.

Exercise 7.9.5

4p1p2+5 when p=1

Answer

56

Exercise 7.9.6

q25q+3 when q=7

Exercise 7.9.7

y28y2y2 when y=1

Answer

72

Example 7.9.8

z2+24zz2 when z=3

Simplify Rational Expressions

In the following exercises, simplify.

Exercise 7.9.9

1024

Answer

512

Exercise 7.9.10

8m416mn3

Exercise 7.9.11

14a14a1

Answer

14

Exercise 7.9.12

b2+7b+12b2+8b+16

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

Exercise 7.9.13

c2c24c2

Answer

c+1c+2

Exercise 7.9.14

d1616d

Exercise 7.9.15

7v3525v2

Answer

75+v

Exercise 7.9.16

w23w2849w2

Multiply and Divide Rational Expressions

Multiply Rational Expressions

In the following exercises, multiply.

Exercise 7.9.17

38·215

Answer

120

Exercise 7.9.18

2xy28y3·16y24x

Exercise 7.9.19

3a2+21aa2+6a7·a1ab

Answer

3b

Exercise 7.9.20

5z25z2+40z+35·z213z

Divide Rational Expressions

In the following exercises, divide.

Exercise 7.9.21

t24t12t2+8t+12÷t2366t

Answer

6t(t+6)2

Exercise 7.9.22

r2164÷r3642r28r+32

Exercise 7.9.23

11+ww9÷121w29w

Answer

111+w

Exercise 7.9.24

3y212y634y+3÷(6y242y)

Exercise 7.9.25

c2643c2+26c+16c24c3215c+10

Answer

5c+4

Exercise 7.9.26

8m28mm4·m2+2m24m2+7m+10÷2m26mm+5

​​​​​Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Exercise 7.9.27

35+25

Answer

1

Exercise 7.9.28

4a22a112a1

Exercise 7.9.29

p2+10pp+5+25p+5

Answer

p+5

Exercise 7.9.30

3xx1+2x1

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Exercise 7.9.31

d2d+43d+28d+4

Answer

d7

Exercise 7.9.32

z2z+10100z+10

Exercise 7.9.33

4q2q+3q2+6q+53q2+q+6q2+6q+5

Answer

q3q+5

Exercise 7.9.34

5t+4t+3t2254t28t32t225

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

Exercise 7.9.35

18w6w1+3w216w

Answer

15w+26w1

Exercise 7.9.36

a2+3aa243a84a2

Exercise 7.9.37

2b2+3b15b249b2+16b149b2

Answer

3b2b+7

Exercise 7.9.38

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

Add and Subtract Rational Expressions With Unlike Denominators

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

Exercise 7.9.38

4m23m10,2mm2m20

Answer

(m+2)(m5)(m+4)

Exercise 7.9.39

6n24,2nn24n+4

Exercise 7.9.40

53p2+17p6,2m3p223p8

Answer

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

Find Equivalent Rational Expressions

In the following exercises, rewrite as equivalent rational expressions with the given denominator.

Exercise 7.9.41

Rewrite as equivalent rational expressions with denominator (m+2)(m5)(m+4)

4m23m10,2mm2m20.

Exercise 7.9.42

Rewrite as equivalent rational expressions with denominator (n2)(n2)(n+2)

6n24n+4,2nn24.

Answer

6n+12(n2)(n2)(n+2),2n24n(n2)(n2)(n+2)

Exercise 7.9.43

Rewrite as equivalent rational expressions with denominator (3p+1)(p+6)(p+8)

53p2+19p+6,7p3p2+25p+8

​​​​​​Add Rational Expressions with Different Denominators

In the following exercises, add.

Exercise 7.9.44

23+35

Answer

1915

Exercise 7.9.45

75a+32b

Exercise 7.9.46

2c2+9c+3

Answer

11c12(c2)(c+3)

Exercise 7.9.47

3dd29+5d2+6d+9

Exercise 7.9.48

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

Answer

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

Exercise 7.9.49

5qp2qp2+4qq21

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

Exercise 7.9.50

3vv+2v+2v+8

Answer

2(v2+10v2)(v+2)(v+8)

Exercise 7.9.51

3w15w2+w20w+24w

Exercise 7.9.52

7m+3m+25

Answer

2m7m+2

Exercise 7.9.53

nn+3+2n3n9n29

Exercise 7.9.54

8dd2644d+8

Answer

4d8

Exercise 7.9.55

512x2y+720xy3

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

Exercise 7.9.56

5aa+210a2a24

Answer

a22a

Exercise 7.9.57

25+5613+14

Exercise 7.9.58

x3xx+51x+5+1x5

Answer

(x8)(x5)2

Exercise 7.9.59

2m+mnnm1n

​​​​​​​Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Exercise 7.9.60

6+2q45q+4

Answer

(q2)(q+4)5(q4)

Exercise 7.9.61

3a21b1a+1b2

Exercise 7.9.62

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

Answer

z521z+21

Exercise 7.9.63

3y24y322y8+1y+4

Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

Exercise 7.9.64

12+23=1x

Answer

67

Exercise 7.9.65

12m=8m2

Exercise 7.9.66

1b2+1b+2=3b24

Answer

32

Exercise 7.9.67

3q+82q2=1

Exercise 7.9.68

v15v29v+18=4v3+2v6

Answer

no solution

Exercise 7.9.69

z12+z+33z=1z

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

Exercise 7.9.70

Vl=hw for l

Answer

l=Vhw

Exercise 7.9.71

1x2y=5 for y

Exercise 7.9.72

x=y+5z7 for z

Answer

z=y+5+7xx

Exercise 7.9.73

P=kV for V

​​​​​​Solve Proportion and Similar Figure Applications Similarity

Solve Proportions

In the following exercises, solve.

Exercise 7.9.74

x4=35

Answer

125

Exercise 7.9.75

3y=95

Exercise 7.9.76

ss+20=37

Answer

15

Exercise 7.9.77

t35=t+29

​​​​​​​In the following exercises, solve using proportions.

Exercise 7.9.78

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

Answer

1161 calories

Exercise 7.9.79

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.

Exercise 7.9.80

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

This image shows two triangles. The large triangle is labeled A B C. The length from A to B is labeled 8. The length from B to C is labeled 7. The length from C to A is labeled b. The smaller triangle is triangle x y z. The length from x to y is labeled 2 and two-thirds. The length from y to z is labeled x. The length from x to z is labeled 3.

Answer

b=9; x=213

Exercise 7.9.81

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. a. Paris to Rome
  2. b. Paris to Vienna

This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Paris”, the next vertex is labeled “Vienna”, and the next vertex is labeled “Rome”. The distance from Paris to Vienna is 7.7 centimeters. The distance from Vienna to Rome is 7 centimeters. The distance from Rome to Paris is 8.9 centimeters.

Exercise 7.9.82

Tony is 5.75 feet tall. Late one afternoon, his 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

Exercise 7.9.83

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

​​​​​​​Solve Uniform Motion and Work Applications Problems

Solve Uniform Motion Applications

In the following exercises, solve.

Exercise 7.9.84

When making the 5-hour drive home from visiting her parents, Lisa 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 in the bad weather. How fast did she drive when the weather was bad?

Answer

45 mph

Exercise 7.9.85

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?​​​​​​​

Exercise 7.9.86

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

Answer

16 mph

Exercise 7.9.87

Mark 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.

Exercise 7.9.88

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

Answer

45 hour

Exercise 7.9.89

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

Exercise 7.9.90

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

Exercise 7.9.91

Sue 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?

​​​​​​​Use Direct and Inverse Variation

Solve Direct Variation Problems

In the following exercises, solve.

Exercise 7.9.92

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

Answer

7

Exercise 7.9.93

If y varies directly as x, when y=20 and x=2, find y when x=4.

Exercise 7.9.94

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

Answer

36

Exercise 7.9.95

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?

Exercise 7.9.96

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?

Answer

$9

Exercise 7.9.97

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?

​​​​​​​Solve Inverse Variation Problems

In the following exercises, solve.

Exercise 7.9.98

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​​​​​​​

Exercise 7.9.99

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?​​​​​​​

Practice Test

In the following exercises, simplify.

Exercise 7.9.1

3a2b6ab2

Answer

a2b​​​​​​​

Exercise 7.9.2

5b25b225

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

Exercise 7.9.3

4xx+2·x2+5x+612x2

Answer

x+33x

Exercise 7.9.4

5y4y8·y2410

Exercise 7.9.5

4pq+5p

Answer

4+5qpq

Exercise 7.9.6

1z93z+9

Exercise 7.9.7

23+3525

Answer

1916

Exercise 7.9.8

1m1n1n+1m

In the following exercises, solve each equation.

Exercise 7.9.9

12+27=1x

Answer

x=1411

Exercise 7.9.10

5y6=3y+6

Exercise 7.9.11

1z5+1z+5=1z225

Answer

z=12

Exercise 7.9.12

t4=35

Exercise 7.9.13

2r2=3r1

Answer

r=4

In the following exercises, solve.

Exercise 7.9.14

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

Exercise 7.9.15

If y varies inversely with x and x=6 when y=20, find y when x=2.

Answer

y=60

Exercise 7.9.16

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

Exercise 7.9.17

The recommended erythromycin dosage for dogs, is 5 mg for every pound the dog weighs. If Daisy weighs 25 pounds, how many milligrams of erythromycin should her veterinarian prescribe?

Answer

125 mg

Exercise 7.9.18

Julia spent 4 hours Sunday afternoon exercising at the gym. She ran on the treadmill for 10 miles and then biked for 20 miles. Her biking speed was 5 mph faster than her running speed on the treadmill. What was her running speed?

Exercise 7.9.19

Kurt 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 Kurt’s speed on his bike?

Answer

14 mph

Exercise 7.9.20

Amanda jogs to the park 8 miles using one route and then returns via a 14-mile route. The return trip takes her 1 hour longer than her jog to the park. Find her jogging rate.

Exercise 7.9.21

An experienced window washer can wash all the windows in Mike’s house in 2 hours, while a new trainee can wash all the windows in 7 hours. How long would it take them working together?

Answer

159 hour

Exercise 7.9.22

Josh 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 Josh’s dad working alone to split the logs?

Exercise 7.9.23

The price that Tyler pays for gas varies directly with the number of gallons he buys. If 24 gallons cost him $59.76, what would 30 gallons cost?

Answer

$74.70

Exercise 7.9.24

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.

Exercise 7.9.25

The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio, as shown on the figure below, that gives the map distances between these cities in inches.

This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Dayton”, the next vertex is labeled “Columbus”, and the next vertex is labeled “Cincinnati”. The distance from Dayton to Columbus is 3.2 inches. The distance from Columbus to Cincinnati is 5.3 inches. The distance from Cincinnati to Dayton is 2.4 inches.

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

Answer

64 miles

​​​​​​​

This page titled 7.9: Chapter 8 Review Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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