CHAPTER 7 Rational Equations and Functions -...

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7.1 Cars and Growing OldIntroduction to Rational Functions ● p. 353

7.2 Rational Expressions, Part ISimplifying, Adding, and SubtractingRational Expressions ● p. 361

7.3 Rational Expressions, Part IIMultiplying and Dividing RationalExpressions ● p. 365

7.4 SolutionsSolving Rational Equations and Inequalities ● p. 369

7.5 Holes and BreaksGraphing Rational Functions andDiscontinuities ● p. 383

7.6 Work, Mixture, and MoreApplications of Rational Equations andFunctions ● p. 399

Over 40 million used cars are sold in the United States every year. When buying expensive itemssuch as automobiles, people must often save money for many months or even years. You will userational functions to calculate how long it will take to save enough money to buy a used car.

7C HA PT E R

Rational Equations and Functions

Chapter 7 ● Rational Equations and Functions 351

7

352 Chapter 7 ● Rational Equations and Functions

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Lesson 7.1 ● Introduction to Rational Functions 353

Problem 1 Saving for a CarYou want to purchase a used car, but it costs $10,000. With your current job, youwill need to work for more than two years to save that much money. So, you startapplying for other jobs, hoping to make more money.

1. How long would it take to save $10,000 if you could save

a. $50 per week?

b. $100 per week?

c. $200 per week?

d. $250 per week?

e. $300 per week?

ObjectivesIn this lesson, you will:

● Graph rational functions.● Determine the asymptotes of rational

functions.● Determine the discontinuities of rational

functions.

Key Terms● rational function● discontinuous● discontinuities● asymptotes● inverse variation● constant of variation

7.1 Cars and Growing OldIntroduction to Rational Functions

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2. Complete the table. Then use the information in the table to construct a graph forthe situation.

73. Can this situation be modeled by a function? Explain your answer.

4. If this situation can be modeled by a function, what type of function will modelthis situation? How do you know?

Quantity Name

Unit

Expression

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5. Write an algebraic equation to model the situation.

6. What happens to the graph as x gets closer to zero? What happens to the graphas x gets very large?

7. Will the graph have any x- or y-intercepts? Explain.

Problem 2The situation in this problem is modeled with a rational function. A rational functionis a function that is written as the quotient of two polynomials. You have alreadyseen some specific types of rational functions. Linear, quadratic, cubic, and higher-order polynomial functions are types of rational functions. Rational functionscan be either continuous or discontinuous, and their graphs can have many differentshapes. A discontinuous function cannot be drawn without lifting your pencilbecause it has holes or jumps. These holes or jumps are called discontinuities.

Even though the Saving for a Car situation makes sense only for positive values(those values in the first quadrant), looking at the graph of the function for all realvalues of x will help you understand this rational function.

1. Graph the function from Problem 1 on the grid for �500 � x � 500.

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2. Answer the following questions about the graph in Question 1.

a. What are the domain and range of the function?

b. What are the x- and y-intercepts of the function?

c. What are the intervals of increase and decrease?

d. Describe the end behavior of the function.

e. What happens to the function as x gets very large? Very small?

f. Can x � 0? Explain.

g. Can y � 0? Explain.

h. Is the function continuous or discontinuous? If it is discontinuous, name anylocations of discontinuity.

In this problem, the x- and y-axes are called asymptotes. An asymptote is a lineapproached by a graph of a relation such that the distance between the graph andthe line approaches zero.

The function in the Saving for a Car problem is an example of an inverse variation. An

inverse variation is a function that can be written in the form . With inverse

variation, if one variable increases, the other variable decreases. The constant ofvariation, k, represents the constant product of the two variables.

In the Saving for a Car problem, the constant of variation represented the product ofthe number of weeks worked and the amount saved each week, or $10,000.

y �kx

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Problem 3One day, Dan Petersen was having a disagreement with his father. During theirdiscussion, Mr. Petersen said, “You should listen to me. I know more. I’ve livedlonger. You’re not even half my age.” That got Dan thinking about when he would behalf of his father’s age. Right now, Dan is 16 years old and his father is 36 years old.

1. Calculate Dan’s age as a percentage of his father’s age.

2. How old was Dan’s father when Dan was born?

3. When will Dan be half of his father’s age? How old will each be?

4. When will Dan be three quarters of his father’s age? How old will each be?

5. Will Dan ever be as old as his father? Explain.

6. If x represents Dan’s age, what expression represents his father’s age?

7. What expression represents the ratio of Dan’s age to his father’s age?

8. What is the domain and range for this problem situation?

9. What is the domain and range of the mathematical function, ?

10. Are your answers for Questions 8 and 9 the same? Explain.

f(x) �x

x � 20

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11. Complete the following table of values.

7

x

�70

�60

�50

�40

�30

�20

�10

0

xx � 20

x

10

15

20

30

40

50

60

70

xx � 20

12. Create a graph using the values from the table.

13. What are the x- and y-intercepts?

14. What are the intervals of increase and decrease?

15. Is the function continuous or discontinuous? If it is discontinuous, name anylocations of discontinuity.

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16. Describe the end behavior.

17. What are the asymptotes?

18. Explain why the function has the asymptotes it does. Your explanation may bein terms of either the mathematical function or of the problem situation.

19. Describe the behavior near the vertical asymptote.

20. Is the situation in this problem an example of direct variation, inverse variation,or neither? Explain your answer.

21. What type of function models this situation?

Be prepared to share your work with another pair, group, or the entire class.

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Lesson 7.2 ● Simplifying, Adding, and Subtracting Rational Expressions 361

Problem 1 Simplifying Rational ExpressionsAs with simplifying fractions, you can simplify rational expressions, expressionsformed by dividing one polynomial by another, by factoring the numerators anddenominators into their prime factors and dividing out any common factors.

1. Simplify the following fractions.

a.

b.

c.

d.

2. Simplify the following rational expressions. (List the restrictions, those values ofthe variables for which the expression would be undefined, before factoring andsimplifying.)

a.

b.

c.

d.�9wx2 yz4

51w6 y3 z

3xyz27xy

5x3 y25x2 y3

2x2

12x2 y

5751025

69115

240740

25125


● Simplify rational expressions.● Add rational expressions.● Subtract rational expressions.

Key Terms● rational expressions● restrictions

7.2 Rational Expressions, Part ISimplifying, Adding, and SubtractingRational Expressions

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3. Simplify the following rational expressions. (Make sure to list the restrictions forthe variable.)

a.

b.

c.

d.

e.

f.x3 � 5x2 � x � 5

x2 � 6x � 5

25x2 � 95x2 � 12x � 9

x3 � 7x2 � 18x3x2 � 9x

x2 � 5x � 63x � 9

3xy � 3yx2 � 1

2x2 � 8x � 2

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Lesson 7.2 ● Simplifying, Adding, and Subtracting Rational Expressions 363

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Problem 2 Adding and Subtracting RationalExpressions

Rational expressions are defined as one polynomial expression divided by another.When evaluated at specific values of variables, rational expressions can be simplified to fractions. So, adding and subtracting rational expressions is very similarto adding and subtracting fractions.

1. Calculate the following sums and differences.

a.

b.

c.

d.

e. How are parts (a)–(d) similar?

f. What steps did you follow to add and subtract these expressions?

2. Calculate the Least Common Denominator (LCD) for each of the following rational expressions. (Make sure to list the restrictions for the variable.)

a. LCD:

b. LCD:

c. LCD:

d. LCD: 2x

x2 � 5x � 6,

7x � 11x2 � 6x � 9

�11x

, 7

x � 4,

xx2 � 16

�23x � 2

, 4x

3x2 � 7x � 6

3x � 4

, 7x

x � 4

3x5�

23�

2y15�

2x � 3y5

3x2�

2y5�

2x4

12�

23�

16

16�

56�

16


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3. Calculate the following sums and differences. (Make sure to list the restrictionsfor the variable.)

a.

b.

c.

d.

4. Calculate the following sums and differences. (Make sure to list the restrictionsfor the variable and simplify the answers when possible.)

a.

b.

c.

d.

e.


x � 2x2 � 4x � 3

�x � 3

x2 � 5x � 4�

4x � 1

3x � 54x2 � 12x � 9

�4

2x � 3�

2x3

x � 6x2 � 5x � 4

�45x

4x � 3x2 � 25

�6

5x � 25

2x � 5x

�45x� 4

x � 7x2 � 3x � 2

�4

x2 � 7x � 10

5x � 6x2 � 9

�4

x � 3

�3x � 2

�7x

5x � 5

�4

x � 5

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Lesson 7.3 ● Multiplying and Dividing Rational Expressions 365

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Problem 1 Multiplying Rational ExpressionsAs with multiplying fractions, you may first calculate the products of the numerators anddenominators of rational expressions and then simplify or divide out common factorsbefore calculating the products. In either case, the final answer should be simplified.

1. Calculate the products of the numerators and denominators, and then simplify.(Make sure to list the restrictions for the variables.)

a.

b.

c.

d.

2. Simplify the rational expressions by first simplifying each of the expressionsbeing multiplied. Then multiply the two expressions. (Make sure to list therestrictions for the variables.)

a.

b.2xy15

�5x

16y3

1225

�516

2xy5

�15y8x2

2425

�1516

49

�65

215

�58


● Multiply rational expressions.● Divide rational expressions.

7.3 Rational Expressions, Part IIMultiplying and Dividing Rational Expressions


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c.

d.

e. What are the advantages or disadvantages of each method?

3. Using either method, calculate the following products. (Make sure to list therestrictions for the variables.)

a.

b.

c.

d.

e.

f.3x

x2y � xy�

x2 � 1x2 � 8x � 7

�xy5

7x � 73x2 �

x � 59x2 � 9

�x2 � 5x � 6

x3 � 6x2 � 5x

3x2 � 5xx2 � 3x � 28

�x2 � 49

7x2

x � 5x2 � 4x � 3

�x � 3

4x � 20

5x2

3x2 � 75�

3x � 154x2

3x5x � 15

�x � 39x2

3ab2

4c�

2c2

27ab

xy14z

�7xz2

5y

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Lesson 7.3 ● Multiplying and Dividing Rational Expressions 367

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Problem 2 Dividing Rational ExpressionsDividing rational expressions uses the same process as dividing fractions. As withdividing fractions, you multiply by the multiplicative inverse of the divisor.

1. Calculate the quotients of the following expressions.

a.

b.

c.

d.

2. Calculate the quotients or products of the following expressions. (Make sure tolist the restrictions for the variables.)

a.

b.

c.

d.

3. Calculate the quotients of the following expressions. (Make sure to list therestrictions for the variables.)

a.

b.7x2

3x2 � 27�

4x2

3x � 9

3xx2 � 9

�9x2

x � 3

9ab2

4c�

18c2

5ab

xy14z

�7xz2

5y

5z4xy

�7z2xy

xy2

5z�

3xy10z2

4xy5�

12y25x2

1415�

1225

59�

1027

215�

825


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c.

d.

e.

f.


4xx2y2 � xy

�x2 � 4

3x2 � 19x � 14�

x � 2xy

4x � 45x2 � x � 6

x2 � 16�

x2 � 7x � 6x3 � x2 � 12x

3x2 � 15xx2 � 3x � 40

�5x2

x2 � 64

x2 � 4x � 42x � 10

�2x � 44x � 8

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Lesson 7.4 ● Solving Rational Equations and Inequalities 369

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Problem 1A rational equation is an equation containing one or more rational expressions. Youhave already solved simple rational equations with a single variable as a denominator.

1. Solve each of the following rational equations.

a.

b.5x� 12 � 25

�3x� �7


● Solve rational equations in one variable.● Solve rational inequalities in one variable.

Key Terms● rational equation● extraneous solution

7.4 SolutionsSolving Rational Equations and Inequalities


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c.

d.

e. Does your solution check in part (d)? Explain.

f. When working with rational equations, you must limit your solution spacebased on the domain of the rational expressions. Explain.

In some rational equations, a rational expression is equal to a constant or to anotherrational expression. You can think of these equations as proportions. To solve them,you can cross multiply.

2. Solve each of the following rational equations. (Make sure to list the restrictionsto your solution set.)

a.12

x � 5� �2

5x� 25 �

5x

5x� 12 �

25x

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b.

c.

d.

e.x2 � 5x

4�

8x2

x � 5x � 2

�89

x � 7x � 4

� �3

7x � 3

�8

x � 2


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f.

3. Solve the equation.

a.

b. What was the first step in your solution path in part (a)? Why?

4. Using the same method, solve each of the following rational equations. Makesure to list the restrictions to your solution set.

a.�7x�

47�

8x�

97

5x6�

1112�

3x4�

23

3x3 � 5x3

�2x3 � 10x � 4

2

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b.

c.

d.3y��45y� �11

3a � 1

�2

5a � 5�

�3a2 � 1

3x � 1

�2

5x � 5�

�3x � 1


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5. Solve the rational equation.

a.

b. What happens when you check your answers?

c. While solving an equation, you sometimes multiply by a variable and increasethe degree of the equation, thereby increasing the number of solutions. This iswhat happened in part (a). The extra solution is called an extraneous solution.In part (a), what is the actual solution, and what is the extraneous solution?

x � 3x2 � 1

��2x

x � 1� 1

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6. Solve each of the following rational equations. Make sure to identify anyextraneous solutions, and make sure to list the restrictions to your solution set.

a.

b.�2

x � 3�

3x � 2

�5

x2 � x � 6

3x � 1

�1

x � 1�

2x2 � 1


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c.

d.1

x � 5�

5x2 � 2x � 35

1x � 2

� 1 �8

x2 � 2x � 8

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Problem 2Rational inequalities have the same restrictions to the solution set as rational equations. The roots of the denominators represent values for which the inequality is undefined. Let’s solve the following rational inequality:

First, rewrite the inequality so one side is zero.

Next, determine the restrictions for the denominator by setting it equal to zero.

Then, solve the corresponding equation.

These steps give you three critical points: two undefined points and one solution.These three points (�3, �2, and 3) would divide a number line into four sets:

. Notice that even though the inequality has the sign, the solution set cannot contain �3 or �2. Why?�

x � �3, �3 � x � �2, �2 � x � 3, and 3 � x

x � 3

x � 3 � 0

x � 3x2 � 5x � 6

� 0

(x � 2) (x � 3) � 0 so x � �2, �3

x2 � 5x � 6 � 0

x � 3x2 � 5x � 6

� 0

xx2 � 5x � 6

�3

x2 � 5x � 6


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To determine the solution set, pick numbers in each of the four sets (e.g., �4, �2.5,0, 4), and evaluate the inequalities at these values to determine which of them arepart of the solution set. Graph the solution set on a number line.

Solution set: .x � �3 or �2 � x � 3

142� 0 False

�36� 0 True

4 � 3

(4)2 � 5(4) � 6� 0

0 � 3(0)2 � 5(0) � 6

� 0

x � 3

x2 � 5x � 6� 0

x � 3x2 � 5x � 6

� 0

�5.5�0.25

� 22 � 0 False �72� 0 True

�2.5 � 3

(�2.5)2 � 5(�2.5) � 6� 0

�4 � 3(�4)2 � 5(�4) � 6

� 0

x � 3

x2 � 5x � 6� 0

x � 3x2 � 5x � 6

� 0

Solve each of the following rational inequalities.

1.x

x2 � 9� 0

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2.2

x � 3�

5x � 2


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3.�1x�

x � 3x � 4

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4.x

x2 � 5x � 4�

5x � 4



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Lesson 7.5 ● Graphing Rational Functions and Discontinuities 383

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Problem 1In the first activity of this chapter, you graphed some rational functions. Now you willexamine the graphs of these types of functions in more detail.

1. The simplest rational function is the inverse variation, where k is the constant of variation.

a. Graph this function with k equal to 2.

f(x) �kx


● Graph rational functions.● Determine both nonremovable and

removable discontinuities.● Determine asymptotes.

Key Terms● nonremovable discontinuities● removable discontinuities● greatest integer function● step functions

7.5 Holes and BreaksGraphing Rational Functionsand Discontinuities


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b. What is this function’s domain? Range?

c. Does the graph have any asymptotes? If so, what are they?

d. Describe the graph’s end behavior.

2. Graph the function .g(x) �2

x � 2

a. What is this function’s domain? Range?

b. What are its asymptote(s)?

c. Describe the relationship of this graph to the graph in Question 1.


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3. Graph the function .g(x) �2x� 2



c. Describe the graph’s end behavior.

d. Describe the relationship of this graph to the graph in Question 1.

e. Rewrite the function as single rational term.


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4. Graph the function .g(x) �x � 1x2 � 1


b. Does the graph have any asymptotes? If so, what are they?

c. What is g(1) equal to? g(�1) equal to?


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5. Graph the function .h(x) �1

x � 1



c. What is h(1) equal to? h(�1) equal to?



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e. Are and the same function? Do they have the

same graph? Explain.

f. What could you do to to make it the same function as

?

Questions 4 and 5 illustrate two different types of discontinuities of rational functions.

In both Questions 4 and 5, the functions have a discontinuity at the asymptotes.This type of discontinuity is called a nonremovable discontinuity.

In Question 4, however, g(x) has an additional discontinuity at x = 1. There is noasymptote at x = 1; instead, there is a “hole,” or missing point. This type ofdiscontinuity is called a removable discontinuity because it can be “removed”by redefining the function with the missing value.

For example, if we redefined

then

The distinction between nonremovable and removable discontinuities is fundamentalto understanding and graphing rational functions. Removable discontinuities are holesin the graph found at values for which a factor that is common to both the numeratorand denominator is equal to zero. Nonremovable discontinuities are asymptotes foundat values where a factor of the denominator is equal to zero.

6. What characterizes removable discontinuities on the graph of a rational function?Nonremovable discontinuities?

g(x) � h(x)

g(x) as g(x) � �

x � 1x2 � 1

for x � �1

12

for x � 1

h(x) �1

x � 1

g(x) �x � 1x2 � 1

g(x) �x � 1x2 � 1

h(x) �1

x � 1

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7. For each of the following rational functions, list the removable and nonremovablediscontinuities.

a.

b.

c.

d.

8. Graph each of the following rational functions. Then list the removable andnonremovable discontinuities, the domain, the range, and the asymptotes. If thefunction has removable discontinuities, rewrite the function to remove them.Describe each graph’s end behavior.

a. g(x) �3

(x � 4) (x � 4)

k(x) �x2 � x � 6

x � 3

h(x) �x � 3x � 1

g(x) �(x � 5) (x � 2)

x � 2

f(x) �1

(x � 2) (x � 2)


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b. h(x) �x � 1

(x � 1) (x � 3)

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c. j(x) �x2 � 3x

x2 � 5x � 6


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9. For each of the following rational functions, list the removable and nonremovablediscontinuities, the domain, the range, and the asymptotes. If the function hasany removable discontinuities, rewrite the function to remove them. Describeeach graph’s end behavior.

a. p(x) �x

x2 � 1

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b. r(x) �x � 3x2 � 9

Problem 2There are a number of other functions with discontinuities, including one called thegreatest integer function. The greatest integer function is defined as f(x) � [x]where f(x) is the greatest integer less than or equal to x. This means that for everynumber x, f(x) rounds the number down to the nearest integer.


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1. In the table, calculate the following values of the greatest integer function.

Expression

2. Create a graph using the values in the table in Question 1.

3. On the same grid, graph f(x) � [x] for �4 x � 4.�

x

�2.3

�2

�1.5

�1

0.37

2

2.3

2.75

3

[ ]x

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4. Why is this a function?

5. How many discontinuities does this function have, and where are they?

6. Are these discontinuities removable or nonremovable? Explain.

7. This function is one of a family of functions called step functions. Explain why.


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Lesson 7.6 ● Applications of Rational Equations and Functions 399

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● Solve work problems.● Solve mixture problems.● Solve cost problems.

Key Terms● work problems● mixture problems● cost problems

7.6 Work, Mixture, and MoreApplications of Rational Equationsand Functions

Problem 1 Work ProblemsThere is a class of problems that can be solved using rational equations andfunctions commonly called work problems.

1. Two teams of bricklayers are working on a new development of townhouses.Each quad of townhouses has the exact same block foundation. One team ofbricklayers can complete this foundation in 30 hours, and the second team cancomplete it in 40 hours. If both teams work on the same foundation at once, howlong will it take them to finish one set of quad townhouses? Complete parts (a)through (f) to answer this question.

a. How much of the foundation will the first team of bricklayers complete in10 hours? 20 hours? 1 hour? x hours?

b. How long will it take the first team to finish the job? of the job? of the job?130

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c. How much of the foundation will the second team of bricklayers complete in10 hours? 20 hours? 1 hour? x hours?

d. How long will it take the second team to finish the job? of the job?

of the job?

e. If the two teams work together for x hours, write an expression that representshow much of the job they would complete.

f. Using the expression you wrote in part (e), write and solve an equation thatrepresents one completed job. Then answer the following question: If bothteams work on the same foundation at once, how long will it take them to fin-ish one set of quad townhouses?

140

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2. If Elaine is working alone, she can complete a job in 10 minutes. If she is workingwith José, the two of them can complete this job in 6 minutes. How long wouldit take José to complete this job if he is working alone? Complete parts (a)through (d) to answer this question.

a. Define a variable for the time it takes José to do the job alone.

b. Using the variable you defined in part (a), what fraction of the job can Josécomplete in one minute?

c. What fraction of the job can Elaine complete in 6 minutes?

d. Write and solve an equation that represents one completed job when Elaineand José are working together. Use this equation to calculate the number ofminutes it would take José to complete the job working alone.


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b. What would the salt concentration of the solution be if you added 80 mL ofwater? 180 mL of water?

c. Define a variable for the amount of water you need to add to the originalsolution. Use this variable to write and solve an equation to find how muchwater needs to be added to make the solution a 2% solution. Then answer thefollowing question: How much water would need to be added to this solutionfor it to contain only 2% salt?

2. For the mixture described in Problem 2, define a function C(x) for theconcentration where x is the amount of water added.

Problem 2 Mixture ProblemsAnother class of problems using rational equations and functions is concentration,or mixture problems.

1. A saline or salt solution of 120 mL contains 10% salt. How much water wouldneed to be added to this solution for it to contain only 2% salt? Completeparts (a) through (c) to answer this question.

a. In the original saline solution, how many milliliters are there of salt? Of water?Explain how you know.

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a. Using C(x), graph the problem situation. Then identify the domain and range ofthe graph.

b. Now graph C(x) as a mathematical function. Determine the domain, range,asymptotes, discontinuities, and end behavior of this function.


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3. A 20% sulfuric acid 20 mL solution will be mixed with a 5% solution to produceother concentration solutions.

a. Will the new solution ever have a concentration of 20%? 5%? Explain.

b. If 20 mL of 20% solution is mixed with 10 mL of 5% solution, what will be theconcentration of the resulting solution? Explain.

c. Define a variable for the amount of 5% solution added, and then define afunction for the concentration of the resulting solution.

d. Using S(x), graph the problem situation. Then identify the domain and range ofthe graph.

S(x)

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e. Now graph S(x) as a mathematical function. Determine the domain, range,asymptotes, discontinuities, and end behavior of this function.

Problem 3 Cost of Ownership ProblemsAnother class of problems has to do with the total cost of owning something overtime. These problems are called cost problems.

1. A new High Definition television costs $1600 and uses about $15 of electricity a year.

a. Assuming that this television is reliable and its only costs of ownership are theoriginal cost and the cost of electricity, what would be the average annual costof owning this television after 5 years? 10 years?


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b. After how many years would the average annual cost be $115?

c. Determine the function of the cost of ownership of this television and definethe variables.

d. Using C(t), graph the problem situation. Then identify the domain and range ofthe graph.

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e. Now graph C(t) as a mathematical function. Determine the domain, range,asymptotes, discontinuities, and end behavior of this function.

2. A new luxury automobile costs $75,000 to purchase, and it is estimated to costabout $8500 a year to own including fuel, service, repairs, and insurance.

a. Assuming that these estimates are reliable and will not change, what would bethe average annual cost of ownership after 5 years? 10 years?


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c. Determine the function of the cost of ownership of this automobile and definethe variables.

d. Using C(t), graph the problem situation. Then identify the domain and range ofthe graph.

b. After how many years would the average annual cost be $10,000?

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e. Now graph C(t) as a mathematical function. Determine the domain, range,asymptotes, discontinuities, and end behavior of this function.

CHAPTER 7 Rational Equations and Functions -...

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