Introductory and Intermediate Algebra for College Students 4th

Robert Blitzer

Chapter 7

Rational Expressions - all with Video Answers

Educators

Section 1

Rational Expressions and Their Simplification

Problem 1

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{5}{2 x}$$

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Problem 2

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{11}{3 x}$$

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Problem 3

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{x}{x-8}$$

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Problem 4

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{x}{x-6}$$

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Problem 5

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{13}{5 x-20}$$

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Problem 6

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{17}{6 x-30}$$

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Problem 7

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{x+3}{(x+9)(x-2)}$$

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Problem 8

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{4 x}{(3 x-17)(x+3)}$$

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Problem 9

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{4 x}{(3 x-17)(x+3)}$$

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Problem 10

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{8 x}{(4 x-19)(x+2)}$$

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Problem 11

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{x+5}{x^{2}+x-12}$$

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Problem 12

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{7 x-14}{x^{2}-9 x+20}$$

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Problem 13

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{x+5}{5}$$

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Problem 14

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{x+7}{7}$$

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Problem 15

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{y+3}{4 y^{2}+y-3}$$

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Problem 16

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{y+8}{6 y^{2}-y-2}$$

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Problem 17

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{y+5}{y^{2}-25}$$

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Problem 18

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{y+7}{y^{2}-49}$$

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Problem 19

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{5}{x^{2}+1}$$

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Problem 20

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state.
$$\frac{8}{x^{2}+4}$$

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Problem 21

In Exercises $21-76,$ simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{14 x^{2}}{7 x}$$

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Problem 22

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{9 x^{2}}{6 x}$$

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Problem 23

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{5 x-15}{25}$$

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Problem 24

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{7 x+21}{49}$$

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Problem 25

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{2 x-8}{4 x}$$

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Problem 26

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{3 x-9}{6 x}$$

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Problem 27

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{3}{3 x-9}$$

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Problem 28

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{12}{6 x-18}$$

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Problem 29

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{-15}{3 x-9}$$

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Problem 30

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{-21}{7 x-14}$$

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Problem 31

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{3 x+9}{x+3}$$

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Problem 32

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{5 x-10}{x-2}$$

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Problem 33

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x+5}{x^{2}-25}$$

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Problem 34

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x+4}{x^{2}-16}$$

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Problem 35

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{2 y-10}{3 y-15}$$

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Problem 36

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{6 y+18}{11 y+33}$$

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Problem 37

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x+1}{x^{2}-2 x-3}$$

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Problem 38

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x+2}{x^{2}-x-6}$$

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Problem 39

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{4 x-8}{x^{2}-4 x+4}$$

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Problem 40

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2}-12 x+36}{4 x-24}$$

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Problem 41

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{y^{2}-3 y+2}{y^{2}+7 y-18}$$

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Problem 42

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{y^{2}+5 y+4}{y^{2}-4 y-5}$$

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Problem 43

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{2 y^{2}-7 y+3}{2 y^{2}-5 y+2}$$

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Problem 44

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{3 y^{2}+4 y-4}{6 y^{2}-y-2}$$

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Problem 45

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{2 x+3}{2 x+5}$$

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Problem 46

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{3 x+7}{3 x+10}$$

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Problem 47

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2}+12 x+36}{x^{2}-36}$$

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Problem 48

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2}-14 x+49}{x^{2}-49}$$

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Problem 49

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{3}-2 x^{2}+x-2}{x-2}$$

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Problem 50

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{3}+4 x^{2}-3 x-12}{x+4}$$

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Problem 51

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{3}-8}{x-2}$$

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Problem 52

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{3}-125}{x^{2}-25}$$

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Problem 53

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{(x-4)^{2}}{x^{2}-16}$$

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Problem 54

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{(x+5)^{2}}{x^{2}-25}$$

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Problem 55

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x}{x+1}$$

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Problem 56

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x}{x+7}$$

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Problem 57

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x+4}{x^{2}+16}$$

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Problem 58

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x+5}{x^{2}+25}$$

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Problem 59

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x-5}{5-x}$$

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Problem 60

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x-7}{7-x}$$

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Problem 61

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{2 x-3}{3-2 x}$$

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Problem 62

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{5 x-4}{4-5 x}$$

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Problem 63

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x-5}{x+5}$$

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Problem 64

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x-7}{x+7}$$

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Problem 65

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{4 x-6}{3-2 x}$$

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Problem 66

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{9 x-15}{5-3 x}$$

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Problem 67

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{4-6 x}{3 x^{2}-2 x}$$

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Problem 68

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{9-15 x}{5 x^{2}-3 x}$$

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Problem 69

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2}-1}{1-x}$$

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Problem 70

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2}-4}{2-x}$$

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Problem 71

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{y^{2}-y-12}{4-y}$$

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Problem 72

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{y^{2}-7 y+12}{3-y}$$

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Problem 73

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2} y-x^{2}}{x^{3}-x^{3} y}$$

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Problem 74

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x y-2 x}{3 y-6}$$

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Problem 75

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2}+2 x y-3 y^{2}}{2 x^{2}+5 x y-3 y^{2}}$$

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Problem 76

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$\frac{x^{2}+3 x y-10 y^{2}}{3 x^{2}-7 x y+2 y^{2}}$$

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Problem 77

Simplify each rational expression.
$$\frac{x^{2}-9 x+18}{x^{3}-27}$$

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Problem 78

Simplify each rational expression.
$$\frac{x^{3}-8}{x^{2}+2 x-8}$$

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Problem 79

Simplify each rational expression.
$$\frac{9-y^{2}}{y^{2}-3(2 y-3)}$$

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Problem 80

Simplify each rational expression.
$$\frac{16-y^{2}}{y(y-8)+16}$$

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Problem 81

Simplify each rational expression.
$$\frac{x y+2 y+3 x+6}{x^{2}+5 x+6}$$

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Problem 82

Simplify each rational expression.
$$\frac{x y+4 y-7 x-28}{x^{2}+11 x+28}$$

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Problem 83

Simplify each rational expression.
$$\frac{8 x^{2}+4 x+2}{1-8 x^{3}}$$

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Problem 84

Simplify each rational expression.
$$\frac{x^{3}-3 x^{2}+9 x}{x^{3}+27}$$

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Problem 85

The rational expression
$$\frac{\frac{130 x}{100-x}}{ }$$ describes the cost, in millions of dollars, to inoculate $x$ percent of the population against a particular strain of flu.
a. Evaluate the expression for $x=40, x=80,$ and $x=90$ Describe the meaning of each evaluation in terms of percentage inoculated and cost.
b. For what value of $x$ is the expression undefined?
c. What happens to the cost as $x$ approaches $100 \% ?$ How can you interpret this observation?

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Problem 86

The rational expression
$\frac{60,000 x}{100-x}$
describes the cost, in dollars, to remove $x$ percent of the air pollutants in the smokestack emissions of a utility company that burns coal to generate electricity.
a. Evaluate the expression for $x=20, x=50,$ and $x=80$ Describe the meaning of each evaluation in terms of percentage of pollutants removed and cost.
b. For what value of $x$ is the expression undefined?
c. What happens to the cost as $x$ approaches $100 \% ?$ How can you interpret this observation?

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Problem 87

Determine the dosage of a drug prescribed for children. In this expression, $A=$ the child's age and $D=$ the adult dosage.
If the normal adult dosage of medication is 1000 milligrams, what dosage should an 8 -year-old child receive?

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Problem 88

Determine the dosage of a drug prescribed for children. In this expression, $A=$ the child's age and $D=$ the adult dosage.
If the normal adult dosage of medication is 1000 milligrams, what dosage should a 4-year-old child receive?

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Problem 89

A company that manufactures bicycles has costs given by the equation
$$C=\frac{100 x+100,000}{x}$$ in which $x$ is the number of bicycles manufactured and $C$ is the cost to manufacture each bicycle.
a. Find the cost per bicycle when manufacturing
500 bicycles.
b. Find the cost per bicycle when manufacturing 4000 bicycles.
c. Does the cost per bicycle increase or decrease as more bicycles are manufactured? Explain why this happens.

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Problem 90

A company that manufactures small canoes has costs given by the equation
$$C=\frac{20 x+20,000}{x}$$ in which $x$ is the number of canoes manufactured and $C$ is the cost to manufacture each canoe.
a. Find the cost per canoe when manufacturing 100 canoes.
b. Find the cost per canoe when manufacturing $10,000$ canoes.
c. Does the cost per canoe increase or decrease as more canoes are manufactured? Explain why this happens.

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Problem 91

A drug is injected into a patient and the concentration of the drug in the bloodstream is monitored. The drug's concentration,
$y,$ in milligrams per liter, after $x$ hours is modeled by
$$y=\frac{5 x}{x^{2}+1}$$
The graph of this equation, obtained with a graphing utility, is shown in the figure in a $[0,10,1]$ by $[0,3,1]$ viewing rectangle.
Use the equation to find the drug's concentration after 3 hours. Then identify the point on the equation's graph that conveys this information.
(THE IMAGES CANNOT COPY)

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Problem 92

A drug is injected into a patient and the concentration of the drug in the bloodstream is monitored. The drug's concentration,
$y,$ in milligrams per liter, after $x$ hours is modeled by
$$y=\frac{5 x}{x^{2}+1}$$
The graph of this equation, obtained with a graphing utility, is shown in the figure in a $[0,10,1]$ by $[0,3,1]$ viewing rectangle.
Use the graph of the equation to find after how many hours the drug reaches its maximum concentration. Then use the equation to find the drug's concentration at this time.
(THE IMAGES CANNOT COPY)

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Problem 93

What is a rational expression? Give an example with your explanation.

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Problem 94

Explain how to find the number or numbers, if any, for which a rational expression is undefined.

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Problem 95

Explain how to simplify a rational expression.

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Problem 96

Explain how to simplify a rational expression with opposite factors in the numerator and denominator.

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Problem 97

Use the graph shown for Exercises $91-92$ to write a description of the drug's concentration over time. In your description, try to convey as much information as possible that is displayed visually by the graph.

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Problem 98

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
Simplifying rational expressions is similar to reducing fractions.

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Problem 99

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
I cannot simplify rational expressions without knowing how to factor polynomials.

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Problem 100

The rational expressions
$$\frac{7}{14 x} \text { and } \frac{7}{14+x}$$ can both be simplified by dividing each numerator and each denominator by 7 .

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Problem 101

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
I evaluated $\frac{3 x-3}{4 x(x-1)}$ for $x=1$ and obtained 0

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Problem 102

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$\frac{3 x+1}{3}=x+1$$

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Problem 103

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$\frac{x^{2}+3}{3}=x^{2}+1$$

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Problem 104

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The expression $\frac{-3 y-6}{y+2}$ simplifies to the consecutive integer that follows $-4$

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Problem 105

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$\frac{3 x+7}{3 x+10}=\frac{8}{11}$$

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Problem 106

Write a rational expression that cannot be simplified.

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Problem 107

Write a rational expression that is undefined for $x=-4$

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Problem 108

Write a rational expression with $x^{2}-x-6$ in the numerator that can be simplified to $x-3$

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Problem 109

Use the $[\text { GRAPH }]$ or $[\text { TABLE }]$ feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility.
$$\frac{3 x+15}{x+5}=3, \quad x \neq-5$$

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Problem 110

Use the $[\text { GRAPH }]$ or $[\text { TABLE }]$ feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility.
$$\frac{2 x^{2}-x-1}{x-1}=2 x^{2}-1, x \neq 1$$

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Problem 111

Use the $[\text { GRAPH }]$ or $[\text { TABLE }]$ feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility.
$$\frac{x^{2}-x}{x}=x^{2}-1, x \neq 0$$

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Problem 112

Use a graphing utility to verify the graph in Figure 7.1 on page $496 .$ TRACE What do you observe?

KL

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Problem 113

Multiply: $\frac{5}{6} \cdot \frac{9}{25} .$ (Section 1.2, Example 5)

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Problem 114

Divide: $\frac{2}{3} \div 4 .$ (Section 1.2, Example 6)

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Problem 115

Solve by the addition method:
$\left\{\begin{array}{l}2 x-5 y=-2 \\ 3 x+4 y=20 . \text { (Section 4.3, Example 3) }\end{array}\right.$

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Problem 116

Will help you prepare for the material covered in the next section. In each exercise, perform the indicated operation.
$$\frac{2}{5} \cdot \frac{3}{7}$$

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Problem 117

Will help you prepare for the material covered in the next section. In each exercise, perform the indicated operation.
$$\frac{3}{4} \div \frac{1}{2}$$

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Problem 118

Will help you prepare for the material covered in the next section. In each exercise, perform the indicated operation.
$$\frac{5}{4} \div \frac{15}{8}$$

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