AND STATISTICS Inverse and Joint Variation · 534 Chapter 9 Rational Equations and Functions...

534 Chapter 9 Rational Equations and Functions

Inverse and Joint VariationUSING INVERSE VARIATION

In Lesson 2.4 you learned that two variables x and y show direct variation if y = kxfor some nonzero constant k. Another type of variation is called inverse variation.Two variables x and y show if they are related as follows:

y = �kx�, k ≠ 0

The nonzero constant k is called the and y is said to varyinversely with x.

Classifying Direct and Inverse Variation

Tell whether x and y show direct variation, inverse variation, or neither.

GIVEN EQUATION REWRITTEN EQUATION TYPE OF VARIATION

a. �5y

� = x y = 5x Direct

b. y = x + 2 Neither

c. xy = 4 y = �4x� Inverse

Writing an Inverse Variation Equation

The variables x and y vary inversely, and y = 8 when x = 3.

a. Write an equation that relates x and y.

b. Find y when x = º4.

SOLUTION

a. Use the given values of x and y to find the constant of variation.

y = �xk

� Write general equation for inverse variation.

8 = �3k

� Substitute 8 for y and 3 for x.

24 = k Solve for k.

� The inverse variation equation is y = �2x4�.

b. When x = º4, the value of y is:

y = �º24

4�

= º6

E X A M P L E 2

E X A M P L E 1

constant of variation,

inverse variation

GOAL 1

Write and useinverse variation models, asapplied in Example 4.

Write and use jointvariation models, as appliedin Example 6.

� To solve real-lifeproblems, such as finding thespeed of a whirlpool’s currentin Example 3.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

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9.1E X P L O R I N G D ATAA N D S TAT I S T I C S

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Writing an Inverse Variation Model

The speed of the current in a whirlpool varies inversely with the distance from thewhirlpool’s center. The Lofoten Maelstrom is a whirlpool located off the coast ofNorway. At a distance of 3 kilometers (3000 meters) from the center, the speed of thecurrent is about 0.1 meter per second. Describe the change in the speed of the currentas you move closer to the whirlpool’s center.

SOLUTION

First write an inverse variation model relating distance from center d and speed s.

s = �dk

� Model for inverse variation

0.1 = �30k00� Substitute 0.1 for s and 3000 for d.

300 = k Solve for k.

The model is s = �30d

0�. The table shows some speeds for different values of d.

� From the table you can see that the speed of the current increases as you movecloser to the whirlpool’s center.

. . . . . . . . . .

The equation for inverse variation can be rewritten as xy = k. This tells you that a setof data pairs (x, y) shows inverse variation if the products xy are constant orapproximately constant.

Checking Data for Inverse Variation

BIOLOGY CONNECTION The table compares the wing flapping rate r (in beats persecond) to the wing length l (in centimeters) for several birds. Do these data showinverse variation? If so, find a model for the relationship between r and l.

SOLUTION

Each product rl is approximately equal to 117. For instance, (3.6)(32.5) = 117 and (5.0)(23.5) = 117.5. So, the data do show inverse variation. A model for the

relationship between wing flapping rate and wing length is r = �11l7

�.

E X A M P L E 4

E X A M P L E 3

9.1 Inverse and Joint Variation 535

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Oceanography

Distance from center (meters), d 2000 1500 500 250 50

Speed (meters per second), s 0.15 0.2 0.6 1.2 6

� Source: Smithsonian Miscellaneous Collections

COMMONSCOTER The

common scoter migratesfrom the Quebec/Labradorborder in Canada to coastalcities such as Portland,Maine, and Galveston,Texas. To reach its winterdestination, the scoter willtravel up to 2150 miles.

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FOCUS ONAPPLICATIONS

Bird r (beats per second) l (cm)

Carrion crow 3.6 32.5

Common scoter 5.0 23.5

Great crested grebe 6.3 18.7

Curlew 4.0 29.2

Lesser black-backed gull 2.8 42.2

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EARTH AND SUNEarth’s orbit around

the sun is elliptical, so itsdistance from the sun varies.The shortest distance p is1.47 ª 1011 meters and thelongest distance a is 1.52 ª 1011 meters.

APPLICATION LINKwww.mcdougallittell.com

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Earth‘s orbital path

p a

Not drawn to scale

sunEarth

Look Back

For help with directvariation, see p. 94.

STUDENT HELP

USING JOINT VARIATION

occurs when a quantity varies directly as the product of two or moreother quantities. For instance, if z = kxy where k ≠ 0, then z varies jointly with x andy. Other types of variation are also possible, as illustrated in the following example.

Comparing Different Types of Variation

Write an equation for the given relationship.

RELATIONSHIP EQUATION

a. y varies directly with x. y = kx

b. y varies inversely with x. y = �xk

�

c. z varies jointly with x and y. z = kxy

d. y varies inversely with the square of x. y = �xk2�

e. z varies directly with y and inversely with x. z = �kxy�

Writing a Variation Model

SCIENCE CONNECTION The law of universal gravitation states that thegravitational force F (in newtons) between two objects varies jointly with theirmasses m1 and m2 (in kilograms) and inversely with the square of the distance d (in meters) between the two objects. The constant of variation is denoted by G and is called the universal gravitational constant.

a. Write an equation for the law of universal gravitation.

b. Estimate the universal gravitational constant. Use the Earth and sun facts given at the right.

SOLUTION

a. F = �Gm

d21m2�

b. Substitute the given values and solve for G.

F =

3.53 ª 1022 =

3.53 ª 1022 ≈ G(5.29 ª 1032)6.67 ª 10º11 ≈ G

� The universal gravitational constant is about 6.67 ª 10º11 �N

k•gm2

2�.

G(5.98 ª 1024)(1.99 ª 1030)��(1.50 ª 1011)2

Gm1m2�d2

E X A M P L E 6

E X A M P L E 5

Joint variation

GOAL 2

Mass of Earth:m1 = 5.98 ª 1024 kg

Mass of sun:m2 = 1.99 ª 1030 kg

Mean distance between Earthand sun:d = 1.50 ª 1011 m

Force between Earth and sun:F = 3.53 ª 1022 N

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1. Complete this statement: If w varies directly as the product of x, y, and z, then wvaries ��? with x, y, and z.

2. How can you tell whether a set of data pairs (x, y) shows inverse variation?

3. Suppose z varies jointly with x and y. What can you say about �xzy�?

Tell whether x and y show direct variation, inverse variation, or neither.

4. xy = �14� 5. �

xy� = 5 6. y = x º 3 7. x = �y

7�

8. �yx� = 12 9. �

12�xy = 9 10. y = �

1x� 11. 2x + y = 4

Tell whether x varies jointly with y and z.

12. x = 15yz 13. �xz� = 0.5y 14. xy = 4z 15. x = �

y2z�

16. x = �3yz� 17. 2yz = 7x 18. �

xy� = 17z 19. 5x = 4yz

20. TOOLS The force F needed to loosen a bolt with a wrench varies inverselywith the length l of the handle. Write an equation relating F and l, given that 250 pounds of force must be exerted to loosen a bolt when using a wrench with a handle 6 inches long. How much force must be exerted when using a wrenchwith a handle 24 inches long?

DETERMINING VARIATION Tell whether x and y show direct variation, inversevariation, or neither.

21. xy = 10 22. xy = �110� 23. y = x º 1 24. �9

y� = x

25. x = �5y� 26. 3x = y 27. x = 5y 28. x + y = 2.5

INVERSE VARIATION MODELS The variables x and y vary inversely. Use the

given values to write an equation relating x and y. Then find y when x = 2.

29. x = 5, y = º2 30. x = 4, y = 8 31. x = 7, y = 1

32. x = �12�, y = 10 33. x = º�

23�, y = 6 34. x = �

34�, y = �8

3�

INTERPRETING DATA Determine whether x and y show direct variation,

inverse variation, or neither.

35. 36. 37. 38.

PRACTICE AND APPLICATIONS

GUIDED PRACTICE


STUDENT HELP

HOMEWORK HELPExample 1: Exs. 21–28Example 2: Exs. 29–34Example 3: Exs. 51–54Example 4: Exs. 35–38,

48, 49Example 5: Exs. 45–47Example 6: Exs. 55–58

Extra Practice

to help you masterskills is on p. 952.

STUDENT HELP

Vocabulary Check ✓

Concept Check ✓

Skill Check ✓

x y

1.5 20

2.5 12

4 7.5

5 6

x y

31 217

20 140

17 119

12 84

x y

3 36

7 105

5 50

16 48

x y

4 16

5 12.8

1.6 40

20 3.2

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JOINT VARIATION MODELS The variable z varies jointly with x and y. Use the

given values to write an equation relating x, y, and z. Then find z when x = º4

and y = 7.

39. x = 3, y = 8, z = 6 40. x = º12, y = 4, z = 2

41. x = 1, y = �13�, z = 5 42. x = º6, y = 3, z = �5

2�

43. x = �56�, y = �1

30�, z = 8 44. x = �

38�, y = �

11

67�, z = �2

3�

WRITING EQUATIONS Write an equation for the given relationship.

45. x varies inversely with y and directly with z.

46. y varies jointly with z and the square root of x.

47. w varies inversely with x and jointly with y and z.

HOME REPAIR In Exercises 48–50, use the following information.

On some tubes of caulking, the diameter of the circularnozzle opening can be adjusted to produce lines ofvarying thickness. The table shows the length l ofcaulking obtained from a tube when the nozzle openinghas diameter d and cross-sectional area A.

48. Determine whether l varies inversely with d. If so,write an equation relating l and d.

49. Determine whether l varies inversely with A. If so,write an equation relating l and A.

50. Find the length of caulking you get from a tube whose nozzle opening has a diameter of �

34� inch.

ASTRONOMY In Exercises 51–53, use the following information.

A star’s diameter D (as a multiple of the sun’s diameter) varies directly with thesquare root of the star’s luminosity L (as a multiple of the sun’s luminosity) andinversely with the square of the star’s temperature T (in kelvins).

51. Write an equation relating D, L, T, and a constant k.

52. The luminosity of Polaris is 10,000 times the luminosity of the sun. The surfacetemperature of Polaris is about 5800 kelvins. Using k = 33,640,000, find howthe diameter of Polaris compares with the diameter of the sun.

53. The sun’s diameter is 1,390,000 kilometers. What is the diameter of Polaris?

54. INTENSITY OF SOUND The intensity I of a sound (in watts per squaremeter) varies inversely with the square of the distance d (in meters) from thesound’s source. At a distance of 1 meter from the stage, the intensity of the sound at a rock concert is about 10 watts per square meter. Write an equationrelating I and d. If you are sitting 15 meters back from the stage, what is theintensity of the sound you hear?

55. The work W (in joules) done when lifting an objectvaries jointly with the mass m (in kilograms) of the object and the height h (inmeters) that the object is lifted. The work done when a 120 kilogram object islifted 1.8 meters is 2116.8 joules. Write an equation that relates W, m, and h.How much work is done when lifting a 100 kilogram object 1.5 meters?

SCIENCE CONNECTION


d (in.) A (in.2) l (in.)

�18� �2

π56� 1440

�14� �6

π4� 360

�38� �2

95π6� 160

�12� �1

π6� 90

STEPHENHAWKING, a

theoretical physicist, hasspent years studying blackholes. A black hole isbelieved to be formed whena star’s core collapses. Thegravitational pull becomesso strong that even the star’s light, as discussed inExs. 51–53, cannot escape.

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FOCUS ONPEOPLE

HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor help with Exs. 45–47.

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HEAT LOSS In Exercises 56 and 57, use the following information.

The heat loss h (in watts) through a single-pane glass window varies jointly with thewindow’s area A (in square meters) and the difference between the inside and outsidetemperatures d (in kelvins).

56. Write an equation relating h, A, d, and a constant k.

57. A single-pane window with an area of 1 square meter and a temperaturedifference of 1 kelvin has a heat loss of 5.7 watts. What is the heat loss through a single-pane window with an area of 2.5 square meters and a temperaturedifference of 20 kelvins?

58. The area of a trapezoid varies jointly with the heightand the sum of the lengths of the bases. When the sum of the lengths of the basesis 18 inches and the height is 4 inches, the area is 36 square inches. Find aformula for the area of a trapezoid.

59. MULTI-STEP PROBLEM The load P (in pounds) that can be safely supportedby a horizontal beam varies jointly with the width W (in feet) of the beam andthe square of its depth D (in feet), and inversely with its length L (in feet).

a. How does P change when the widthand length of the beam are doubled?

b. How does P change when the widthand depth of the beam are doubled?

c. How does P change when all threedimensions are doubled?

d. Writing Describe several ways abeam can be modified if the safe load it is required to support is increasedby a factor of 4.

60. LOGICAL REASONING Suppose x varies inversely with y and y varies inverselywith z. How does x vary with z? Justify your answer algebraically.

SQUARE ROOT FUNCTIONS Graph the function. Then state the domain and

range. (Review 7.5 for 9.2)

61. y = �x�+� 2� 62. y = �x� º 4 63. y = �x�+� 1� º 3

SOLVING RADICAL EQUATIONS Solve the equation. Check for extraneous

solutions. (Review 7.6)

64. �x� = 22 65. �42�x� + 2 = 6 66. x1/3 º 7 = 0

67. �3x�+� 1�2� = 5 68. (x º 2)3/2 = º8 69. �3�x�+� 1� = �x�+� 1�5�

70. COLLEGE ADMISSION The number of admission applications received by acollege was 1152 in 1990 and increased 5% per year until 1998. (Review 8.1 for 9.2)

a. Write a model giving the number A of applications t years after 1990.

b. Graph the model. Use the graph to estimate the year in which there were 1400 applications.

MIXED REVIEW

GEOMETRY CONNECTION

L W

D

P

TestPreparation

★★Challenge

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Graphing SimpleRational Functions

GRAPHING A SIMPLE RATIONAL FUNCTION

A is a function of the form

ƒ(x) = �pq

((xx))�

where p(x) and q(x) are polynomials and q(x) ≠ 0. In this lesson you will learn tograph rational functions for which p(x) and q(x) are linear. For instance, consider thefollowing rational function:

y = �1x�

The graph of this function is called a and is shown below. Notice thefollowing properties.

• The x-axis is a horizontal asymptote.

• The y-axis is a vertical asymptote.

• The domain and range are all nonzero real numbers.

• The graph has two symmetrical parts called For each point (x, y) on onebranch, there is a corresponding point (ºx, ºy) on the other branch.

branches.

hyperbola

rational function

GOAL 1

Graph simplerational functions.

Use the graph of arational function to solvereal-life problems, such asfinding the average cost percalendar in Example 3.

� To solve real-lifeproblems, such as finding thefrequency of an approachingambulance siren in Exs. 47 and 48.


GOAL 2

GOAL 1


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x y

º4 º�41

�

º3 º�31

�

º2 º�21

�

º1 º1

º�12� º2

x y

4 �41

�

3 �31

�

2 �21

�

1 1

�12� 2

1

1

y

x

y � 1x

Investigating Graphs of Rational FunctionsGraph each function.

a. y = �2x� b. y = �

3x� c. y = �ºx

1� d. y = �ºx

2�

Use the graphs to describe how the sign of a affects the graph of y = �ax�.

Use the graphs to describe how |a| affects the graph of y = �ax�.3

2

1

DevelopingConcepts

ACTIVITY

Page 1 of 6

9.2 Graphing Simple Rational Functions 541

All rational functions of the form y = �x ºa

h� + k have graphs that are hyperbolas

with asymptotes at x = h and y = k. To draw the graph, plot a couple of points oneach side of the vertical asymptote. Then draw the two branches of the hyperbola thatapproach the asymptotes and pass through the plotted points.

Graphing a Rational Function

Graph y = �xº+

23� º 1. State the domain and range.

SOLUTION

Draw the asymptotes x = º3 and y = º1.

Plot two points to the left of the verticalasymptote, such as (º4, 1) and (º5, 0), and two points to the right, such as

(º1, º2) and �0, º�53��.

Use the asymptotes and plotted points todraw the branches of the hyperbola.

The domain is all real numbers except º3, and the range is all real numbers except º1.

. . . . . . . . . .

All rational functions of the form y = �acxx

++

db

� also have graphs that are hyperbolas.

The vertical asymptote occurs at the x-value that makes the denominator zero.

The horizontal asymptote is the line y = �ac�.

Graphing a Rational Function

Graph y = �2xx+º

14�. State the domain and range.

SOLUTION

Draw the asymptotes. Solve 2x º 4 = 0 for xto find the vertical asymptote x = 2. The

horizontal asymptote is the line y = �ac� = �2

1�.

Plot two points to the left of the vertical

asymptote, such as �0, º�14�� and (1, º1), and

two points to the right, such as (3, 2) and �4, �54��.

Use the asymptotes and plotted points todraw the branches of the hyperbola.

The domain is all real numbers except 2, and

the range is all real numbers except �12�.

E X A M P L E 2

E X A M P L E 1

Look Back

For help withasymptotes, see p. 465.

STUDENT HELP

1

1

x

y

(�5, 0)(�4, 1)

(�1, �2) 53�0, � �

3

1

y

x

(3, 2)

(1, �1)

14�0, � �

54�4, �


www.mcdougallittell.comfor extra examples.

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USING RATIONAL FUNCTIONS IN REAL LIFE

Writing a Rational Model

For a fundraising project, your math club is publishing afractal art calendar. The cost of the digital images andthe permission to use them is $850. In addition to these“one-time” charges, the unit cost of printing eachcalendar is $3.25.

a. Write a model that gives the average cost percalendar as a function of the number of calendarsprinted.

b. Graph the model and use the graph to estimate thenumber of calendars you need to print before theaverage cost drops to $5 per calendar.

c. Describe what happens to the average cost as thenumber of calendars printed increases.

SOLUTION

a. The average cost is the total cost of making the calendars divided by the numberof calendars printed.

=

Average cost = (dollars per calendar)

One-time charges = 850 (dollars)

Unit cost = 3.25 (dollars per calendar)

Number printed = (calendars)

=

b. The graph of the model is shown at the right. The A-axis is the vertical asymptote and theline A = 3.25 is the horizontal asymptote. Thedomain is x > 0 and the range is A > 3.25.When A = 5 the value of x is about 500. So,you need to print about 500 calendars beforethe average cost drops to $5 per calendar.

c. As the number of calendars printed increases,the average cost per calendar gets closer andcloser to $3.25. For instance, when x = 5000the average cost is $3.42, and when x = 10,000the average cost is $3.34.

A

x

A

Averagecost

E X A M P L E 3

GOAL 2

January


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Business

LABELS

VERBALMODEL

ALGEBRAICMODEL

PPROBLEMSOLVING

STRATEGY

+ •

Number printed

Number printedUnit costOne-time charges

850 + 3.25x

x

Aver

age

cost

(dol

lars

)

Number printed200

2

A

x00 400 600 800

4

6

8

(486, 5)

A � 850 � 3.25xx

S M T W T F S

13 14 15 16 17 18 19

20 21 22 23 24 25 26

27 28 29 30 31

6 7 8 9 10 11 12

1 2 3 4 5

Page 3 of 6


1. Complete this statement: The graph of a function of the form y = �x ºa

h� + k is called a(n) ��? .

2. ERROR ANALYSIS Explain why the graph

shown is not the graph of y = �x +6

3� + 7.

3. If the graph of a rational function is a hyperbola with the x-axis and the y-axis asasymptotes, what is the domain of the function? What is the range?

Identify the horizontal and vertical asymptotes of the graph of the function.

4. y = �x º2

3� + 4 5. y = �2xx++

43

� 6. y = �xx

+º

33

�

7. y = �2xx+º

54� 8. y = �x +

38� º 10 9. y = �x

ºº

46� º 5

10. CALENDAR FUNDRAISER Look back at Example 3 on page 542. Supposeyou decide to generate your own fractals on a computer to save money. The costfor the software (a “one-time” cost) is $125. Write a model that gives the averagecost per calendar as a function of the number of calendars printed. Graph themodel and use the graph to estimate the number of calendars you need to printbefore the average cost drops to $5 per calendar.

IDENTIFYING ASYMPTOTES Identify the horizontal and vertical asymptotes of

the graph of the function. Then state the domain and range.

11. y = �3x� + 2 12. y = �x º

43� + 2 13. y = �x

º+

23� º 2

14. y = �xx+º

23� 15. y = �23

xx

++

21� 16. y = �º

º43xx

º+ 2

5�

17. y = �xº+

2423� º 17 18. y = �31

46

xx

º+

24� 19. y = �x º

46� + 19

MATCHING GRAPHS Match the function with its graph.

20. y = �x º3

2� + 3 21. y = �xºº

32� + 3 22. y = �x

x++

32

�

A. B. C.

1

1

y

x

1

y

x3

1

2

y

x


GUIDED PRACTICE


Concept Check ✓

Skill Check ✓

x = 3

4

y

x

2

y = 7

Extra Practice


STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 11–31Example 2: Exs. 11–22,

32–40Example 3: Exs. 42–48

Page 4 of 6


GRAPHING FUNCTIONS Graph the function. State the domain and range.

23. y = �4x� 24. y = �x º

33� + 1 25. y = �x

º+

45� º 8

26. y = �x º1

7� + 3 27. y = �x +6

2� º 6 28. y = �5x� + 4

29. y = �4x +1

12� º 2 30. y = �23x� 31. y = �3x

4º 6� + 5

GRAPHING FUNCTIONS Graph the function. State the domain and range.

32. y = �xx++

23� 33. y = �4x

x+ 3� 34. y = �3

xxºº

78�

35. y = �93xx

+º

12� 36. y = �º4

3xxº+

1120

� 37. y = �5x4+x

2�

38. y = �2x3ºx

4� 39. y = �ºx7ºx

15� 40. y = �º21x4x

ºº1

4�

41. CRITICAL THINKING Write a rational function that has the vertical asymptote x = º4 and the horizontal asymptote y = 3.

RACQUETBALL In Exercises 42 and 43, use the following information.

You’ve paid $120 for a membership to a racquetball club. Court time is $5 per hour.

42. Write a model that represents your average cost per hour of court time as afunction of the number of hours played. Graph the model. What is an equation ofthe horizontal asymptote and what does the asymptote represent?

43. Suppose that you can play racquetball at the YMCA for $9 per hour withoutbeing a member. How many hours would you have to play at the racquetball clubbefore your average cost per hour of court time is less than $9?

44. LIGHTNING Air temperature affects how long it takes sound to travel a givendistance. The time it takes for sound to travel one kilometer can be modeled by

t = �0.6T10

+00

331�

where t is the time (in seconds) and T is the temperature (in degrees Celsius). Youare 1 kilometer from a lightning strike and it takes you exactly 3 seconds to hear thesound of thunder. Use a graph to find the approximate air temperature. (Hint: Usetick marks that are 0.1 unit apart on the t-axis.)

ECONOMICS In Exercises 45 and 46, use the following information.

Economist Arthur Laffer argues that beyond a certain percent pm, increased taxes willproduce less government revenue. His theory is illustrated in the graph below.

45. Using Laffer’s theory, an economist modelsthe revenue generated by one kind of tax by

R = �80

ppºº

1810000

�

where R is the government revenue (in tens of millions of dollars) and p is the percent tax rate (55 ≤ p ≤ 100). Graph the model.

46. Use your graph from Exercise 45 to find thetax rate that yields $600 million of revenue.

LIGHTNINGThe equation given

in Ex. 44 uses the fact that at 0°C, sound travels at 331 meters per second.Since light travels at about300,000 kilometers persecond, a flash of lightning is seen before it is heard.

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Gov

ernm

ent

reve

nue

Tax rate (percent)

100

Maximum revenueat tax rate pm

pm

The Laffer Curve

0

Page 5 of 6


DOPPLER EFFECT In Exercises 47 and 48, use the following information.

When the source of a sound is moving relative to a stationary listener, the frequencyƒl (in hertz) heard by the listener is different from the frequency ƒs (in hertz) of thesound at its source. An equation for the frequency heard by the listener is

ƒl = �774400ºƒs

r�

where r is the speed (in miles per hour) of the sound source relative to the listener.

47. The sound of an ambulance siren has a frequency of about 2000 hertz. You arestanding on the sidewalk as an ambulance approaches with its siren on. Write thefrequency that you hear as a function of the ambulance’s speed.

48. Graph the function from Exercise 47 for 0 ≤ r ≤ 60. What happens to thefrequency you hear as the value of r increases?

49. Writing In what line(s) is the graph of y = �1x� symmetric? What does this

symmetry tell you about the inverse of the function ƒ(x) = �1x�?

50. MULTIPLE CHOICE What are the asymptotes of the graph of y = �x ºº7

1341� + 27?

¡A x = 141, y = 27 ¡B x = º141, y = 27 ¡C x = º73, y = 27

¡D x = º73, y = 141 ¡E None of these

51. MULTIPLE CHOICE Which of the following is a function whose domain andrange are all nonzero real numbers?

¡A ƒ(x) = �2xx+ 1� ¡B ƒ(x) = �

23

xx

ºº

12� ¡C ƒ(x) = �

1x� + 1

¡D ƒ(x) = �x ºx

2� ¡E None of these

52. EQUIVALENT FORMS Show algebraically that the function ƒ(x) = �x º3

5� + 10

and the function g(x) = �10

xxºº

547

� are equivalent.

GRAPHING POLYNOMIALS Graph the polynomial function. (Review 6.2 for 9.3)

53. ƒ(x) = 3x554. ƒ(x) = 4 º 2x3

55. ƒ(x) = x6 º 1

56. ƒ(x) = 4x4 + 1 57. ƒ(x) = 6x758. ƒ(x) = x3 º 5

FACTORING Factor the polynomial. (Review 6.4 for 9.3)

59. 8x3 º 125 60. 3x3 + 81 61. x3 + 3x2 + 3x + 9

62. 5x3 + 10x2 + x + 2 63. 81x4 º 1 64. 4x4 º 4x2 º 120

SIMPLIFYING EXPRESSIONS Simplify the expression. (Review 8.3)

65. 66. 7eº5e867. exe4x + 1

68. 69. e4e2xeº3x70. e3eº56ex

�

ex�

MIXED REVIEW

TestPreparation

★★Challenge

Page 6 of 6

9.3 Graphing General Rational Functions 547

RE

AL LIFE

RE

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Graphing GeneralRational Functions

GRAPHING RATIONAL FUNCTIONS

In Lesson 9.2 you learned how to graph rational functions of the form

ƒ(x) =

for which p(x) and q(x) are linear polynomials and q(x) ≠ 0. In this lesson you willlearn how to graph rational functions for which p(x) and q(x) may be higher-degreepolynomials.

Graphing a Rational Function (m < n)

Graph y = . State the domain and range.

SOLUTION

The numerator has no zeros, so there is no x-intercept. The denominator has no real zeros, so there is no vertical asymptote. The degree ofthe numerator (0) is less than the degree of thedenominator (2), so the line y = 0 (the x-axis) is a horizontal asymptote. The bell-shaped graphpasses through the points (º3, 0.4), (º1, 2), (0, 4), (1, 2), and (3, 0.4). The domain is all realnumbers, and the range is 0 < y ≤ 4.

4�x2 + 1

E X A M P L E 1

p(x)�q(x)

GOAL 1

Graph generalrational functions.

Use the graph of a rational function to solvereal-life problems, such asdetermining the efficiency ofpackaging in Example 4.

� To solve real-lifeproblems, such as finding theenergy expenditure of a para-keet in Ex. 39.


GOAL 2

GOAL 1


9.3RE

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RE

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Let p(x) and q(x) be polynomials with no common factors other than 1. The graph of the rational function

ƒ(x) = �pq

((xx

))� =

has the following characteristics.

1. The x-intercepts of the graph of ƒ are the real zeros of p(x).

2. The graph of ƒ has a vertical asymptote at each real zero of q(x).

3. The graph of ƒ has at most one horizontal asymptote.

• If m < n, the line y = 0 is a horizontal asymptote.

• If m = n, the line y = �ab

m

n� is a horizontal asymptote.

• If m > n, the graph has no horizontal asymptote. The graph’s end behavior

is the same as the graph of y = �ab

m

n�x m º n.

amxm + am º 1xm º 1 + . . . + a1x + a0��

bnxn + bn º 1xn º 1 + . . . + b1x + b0

GRAPHS OF RATIONAL FUNCTIONS

1

1

y

x

CONCEPT

SUMMARY

Page 1 of 7


Graphing a Rational Function (m = n)

Graph y = �x2

3ºx2

4�.

SOLUTION

The numerator has 0 as its only zero, so the graph has one x-intercept at (0, 0). Thedenominator can be factored as (x + 2)(x º 2), so the denominator has zeros º2 and2. This implies that the lines x = º2 and x = 2 are vertical asymptotes of the graph.The degree of the numerator (2) is equal to the degree of the denominator (2), so the

horizontal asymptote is y = = 3. To draw the graph, plot points between and

beyond the vertical asymptotes.

To the left of x = º2

Between x = º2and x = 2

To the right of x = 2

Graphing a Rational Function (m > n)

Graph y = �x2 º

x +2x

4º 3

�.

SOLUTION

The numerator can be factored as (x º 3)(x + 1), so the x-intercepts of the graph are3 and º1. The only zero of the denominator is º4, so the only vertical asymptote is x = º4. The degree of the numerator (2) is greater than the degree of thedenominator (1), so there is no horizontal asymptote and the end behavior of thegraph of ƒ is the same as the end behavior of the graph of y = x2 º 1 = x. To draw thegraph, plot points to the left and right of the vertical asymptote.

To the left of x = º4

To the right of x = º4

E X A M P L E 3

am�bn

E X A M P L E 2

Look Back

For help with findingzeros, see p. 259.

STUDENT HELP

x y

º12 º20.6

º9 º19.2

º6 º22.5

º2 2.5

0 º0.75

2 º0.5

4 0.63

6 2.1

x y

º4 4

º3 5.4

º1 º1

0 0

1 º1

3 5.4

4 4

4

4

y

x

5

5

x

y

Page 2 of 7


USING RATIONAL FUNCTIONS IN REAL LIFE

Manufacturers often want to package their products in a way that uses the leastamount of packaging material. Finding the most efficient packaging sometimesinvolves finding a local minimum of a rational function.

Finding a Local Minimum

A standard beverage can has a volume of 355 cubic centimeters.

a. Find the dimensions of the can that has this volume and uses the least amount ofmaterial possible.

b. Compare your result with the dimensions of an actual beverage can, which has aradius of 3.1 centimeters and a height of 11.8 centimeters.

SOLUTION

a. The volume must be 355 cubic centimeters, so you can write the height h of eachpossible can in terms of its radius r.

V = πr2h Formula for volume of cylinder

355 = πr2h Substitute 355 for V.

= h Solve for h.

Using the least amount of material is equivalent to having a minimum surfacearea S. You can find the minimum surface area by writing its formula in terms ofa single variable and graphing the result.

S = 2πr2 + 2πrh Formula for surface area of cylinder

= 2πr2 + 2πr� � Substitute for h.

= 2πr2 + �71r0

� Simplify.

Graph the function for the surface area S using a graphing calculator. Then use the Minimumfeature to find the minimum value of S. Whenyou do this, you get a minimum value of about 278, which occurs when r ≈ 3.84 centimeters and

h ≈ �π(

335.854)2� ≈ 7.66 centimeters.

b. An actual beverage can is tallerand narrower than the can withminimal surface area—probably to make it easier to hold the can in one hand.

355�πr2

355�πr2

E X A M P L E 4

GOAL 2

RE

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RE

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Manufacturing

Look Back

For help with rewriting anequation with more thanone variable, see p. 26.

STUDENT HELP

MinimumX=3.8372 Y=277.54

3.1 cm

7.7 cm11.8 cm

3.8 cm

Actualbeverage can

Can with minimalsurface area

Page 3 of 7


1. Let ƒ(x) = �qp((xx))

� where p(x) and q(x) are polynomials with no common factors

other than 1. Complete this statement: The line y = 0 is a horizontal asymptoteof the graph of ƒ when the degree of q(x) is ��? the degree of p(x).

2. Let ƒ(x) = �qp((xx))

� where p(x) and q(x) are polynomials with no common factors

other than 1. Describe how to find the x-intercepts and the vertical asymptotes ofthe graph of ƒ.

3. Let ƒ(x) = �qp((xx))

� where p(x) and q(x) are both cubic polynomials with no

common factors other than 1. The leading coefficient of p(x) is 8 and the leadingcoefficient of q(x) is 2. Describe the end behavior of the graph of ƒ.

Graph the function.

4. y = �x2 +

63

� 5. y = �xx2

+º

14

� 6. y = �xx

2

2

+º

27

�

7. y = �x2

x+

3

7� 8. y = �

x22ºx2

1� 9. y = �

x2 ºx

16�

10. SOUP CANS The can for a popular brand of soup has a volume of about342 cubic centimeters. Find the dimensions of the can with this volume that usesthe least metal possible. Compare these dimensions with the dimensions of theactual can, which has a radius of 3.3 centimeters and a height of 10 centimeters.

ANALYZING GRAPHS Identify the x-intercepts and vertical asymptotes of the

graph of the function.

11. y = �x2 º

x9

� 12. y = �2xx2

º+

13

� 13. y = �2x2

x2º

º9x

16º 5

�

14. y = �2xx+3

3� 15. y = 16. y = �3x2 º

x21+3x

8+ 10

�

17. y = �32xx2+º

89

� 18. y = �2xx2

2

++

1x

� 19. y = �x3

2ºx

27�


20. y = �xx2

2º+

72

� 21. y = �x2

ºº8

4� 22. y = �

x2xº

3

4�

A. B. C.

6

y

x64

6

y

x

1

2y

x

x2 + 4x º 5��x º 6


GUIDED PRACTICE

STUDENT HELP

Extra Practice



Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELPExamples 1–3: Exs. 11–37Example 4: Exs. 38–45

Page 4 of 7



23. y = 24. y = 25. y =

A. B. C.

GRAPHING FUNCTIONS Graph the function.

26. y = 27. y = 28. y =

29. y = 30. y = 31. y =

32. y = 33. y = 34. y =

35. y = 36. y = 37. y =

38. GARDEN FENCING Suppose you want to make a rectangular garden withan area of 200 square feet. You want to use the side of your house for one side ofthe garden and use fencing for the other three sides. Find the dimensions of thegarden that minimize the length of fencing needed.

GRAPHING MODELS In Exercises 39–45, you may find it helpful to use a

graphing calculator to graph the models.

39. ENERGY EXPENDITURE The total energy expenditure E (in joules pergram mass per kilometer) of a typical budgerigar parakeet can be modeled by

E =

where v is the speed of the bird (in kilometers per hour). Graph the model. What speed minimizes a budgerigar’s energy expenditure?� Source: Introduction to Mathematics for Life Scientists

40. OCEANOGRAPHY The mean temperature T (in degrees Celsius) of theAtlantic Ocean between latitudes 40°N and 40°S can be modeled by

T =

where d is the depth (in meters). Graph the model. Use your graph to estimatethe depth at which the mean temperature is 4°C. � Source: Practical Handbook of Marine Science

41. HOSPITAL COSTS For 1985 to 1995, the average daily cost per patient C (in dollars) at community hospitals in the United States can be modeled by

C =

where x is the number of years since 1985. Graph the model. Would you use this model to predict patient costs in 2005? Explain. � Source: Hospital Statistics

º22,407x + 462,048��5x2 º 122x + 1000

17,800d + 20,000��3d 2 + 740d + 1000

0.31v2 º 21.7v + 471.75��v

x3 + 5x2 º 1��

x2 º 4xx2 º 9x + 20��2x

º4x2�x2 º 16

4 º x��5x2 º 4x º 1

x2 º 11x º 12��

x3 + 273x3 + 1�4x3 º 32

º2x2�3x + 6

2x2 + 3x + 1��x2 º 5x + 4

4x + 1�x2 º 1

x2 º 4�x2 + 3

º24�x2 + 8

2x2 º 3�x + 2

2

2

y

x

4

1

x

y

3

1

x

y

x2 + 4x�2x º 1

ºx3�x2 + 9

3�x3 º 27

HOSPITALADMINISTRATOR

A hospital administratoroversees the quality of careand finances at a hospital. In 1996 there wereapproximately 329,000hospital administrators in the United States.

CAREER LINKwww.mcdougallittell.com

INTE

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RE

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RE

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FOCUS ONCAREERS


www.mcdougallittell.comfor help with Exs. 26–37.

INTE

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STUDENT HELP

Page 5 of 7


42. AUTOMOTIVE INDUSTRY For 1980 to 1995, the total revenue R (in billions of dollars) from parking and automotive service and repair in theUnited States can be modeled by

R =

where x is the number of years since 1980. Graph the model. In what year wasthe total revenue approximately $75 billion?

DATA UPDATE of U.S. Bureau of the Census data at www.mcdougallittell.com

In Exercises 43–45, use the following information.

The acceleration due to gravity g§ (in meters per second squared) of a falling objectat the moment it is dropped is given by the function

g§ =

where h is the object’s altitude (in meters) above sea level.

43. Graph the function.

44. What is the acceleration due to gravity for an object dropped at an altitude of5000 kilometers?

45. Describe what happens to g§ as h increases.

46. CRITICAL THINKING Give an example of a rational function whose graph hastwo vertical asymptotes: x = 2 and x = 7.

47. MULTIPLE CHOICE What is the horizontal asymptote of the graph of thefollowing function?

y =

¡A y = º10 ¡B y = 0 ¡C y = 2

¡D y = 10 ¡E No horizontal asymptote

48. MULTIPLE CHOICE Which of the following functions is graphed?

¡A y = ¡B y =

¡C y = ¡D y =

49. Consider the following two functions:

ƒ(x) = and g(x) =

Notice that the numerator and denominator of g have a common factor of x º 3.

a. Make a table of values for each function from x = 2.95 to x = 3.05 inincrements of 0.01.

b. Use your table of values to graph each function for 2.95 ≤ x ≤ 3.05.

c. As x approaches 3, what happens to the graph of ƒ(x)? to the graph of g(x)?

d. What do you think is true about the graph of a function g(x) = �qp((xx))

� wherep(x) and q(x) have a common factor x º k?

(x + 2)(x º 3)��(x º 3)(x º 5)

(x + 1)(x + 2)��(x º 3)(x º 5)

º5x2�x2 º 9

5x2�x2 + 9

5x2�x2 º 9

º5x2�x2 + 9

10x2 º 1�

x3 + 8

3.99 ª 1014

��h2 + (1.28 ª 107)h + 4.07 ª 1013

SCIENCE CONNECTION

INTE

RNET

427x2 º 6416x + 30,432��º0.7x3 + 25x2 º 268x + 1000

★★Challenge

TestPreparation

2

4y

x

EXTRA CHALLENGE

www.mcdougallittell.com

Page 6 of 7


SIMPLIFYING ALGEBRAIC EXPRESSIONS Simplify the expression. Tell which

properties of exponents you used. (Review 6.1 for 9.4)

50. 51. 52.

53. 54. � �255. � �3

JOINT VARIATION MODELS The variable z varies jointly with x and y. Use the

given values to write an equation relating x, y, and z. Then find z when

x = º3 and y = 2. (Review 9.1)

56. x = 3, y = º6, z = 2 57. x = º5, y = 2, z = �43

�

58. x = º8, y = 4, z = �83� 59. x = 1, y = �

12�, z = 4

VERIFYING INVERSES Verify that ƒ and g are inverse functions. (Review 7.4)

60. ƒ(x) = �12�x º 3, g(x) = 2x + 6 61. ƒ(x) = º3x + 2, g(x) = º�

13�x + �3

2�

62. ƒ(x) = 5x3 + 2, g(x) = � �1/363. ƒ(x) = 16x4, x ≥ 0; g(x) =

The variables x and y vary inversely. Use the given values to write an equation

relating x and y. Then find y when x = º3. (Lesson 9.1)

1. x = 6, y = º2 2. x = 11, y = 6 3. x = �15�, y = 30

The variable x varies jointly with y and z. Use the given values to write an

equation relating x, y, and z. Then find y when x = 4 and z = 1. (Lesson 9.1)

4. x = 5, y = º5, z = 6 5. x = 12, y = 6, z = �12� 6. x = º10, y = 2, z = 4

Graph the function. (Lessons 9.2 and 9.3)

7. y = �1x0� 8. y = º 7 9. y =

10. y = 11. y = 12. y =

13. HOTEL REVENUE For 1980 to 1995, the total revenue R (in billions ofdollars) from hotels and motels in the United States can be modeled by

R =

where x is the number of years since 1980. Graph the model. Use your graph tofind the year in which the total revenue from hotels and motels was approximately$68 billion. (Lesson 9.2)

2.76x + 26.88��º0.01x + 1

x2 º 4x º 5��x + 2

3x2�x2 º 25

6x�x2 º 36

3x + 5�2x º 11

2�x + 9

QUIZ 1 Self-Test for Lessons 9.1–9.3

�4 x��2

x º 2�5

5x3�25xy2

x2y2�x3y

12x5yº2�3xº2y5

3x3y3�6xº1y

x6y5

�xyxº3y�xy4

MIXED REVIEW

Page 7 of 7


Multiplying and DividingRational Expressions

WORKING WITH RATIONAL EXPRESSIONS

A rational expression is in provided its numerator and denominatorhave no common factors (other than ±1). To simplify a rational expression, apply thefollowing property.

Simplifying a rational expression usually requires two steps. First, factor thenumerator and denominator. Then, divide out any factors that are common to both the numerator and denominator. Here is an example:

�x2 +

x25x

� = �x(x

x+• x

5)� = �x +

x5

�

Notice that you can divide out common factors in the second expression above, butyou cannot divide out like terms in the third expression.

Simplifying a Rational Expression

Simplify: �x2 º

x24ºx º

412

�

SOLUTION

= �((xx

++

22

))((xx

ºº

62

))� Factor numerator and denominator.

= �((xx

++

22

))((xx

ºº

62

))� Divide out common factor.

= �xxºº

62� Simplified form

. . . . . . . . . .

The rule for multiplying rational expressions is the same as the rule for multiplyingnumerical fractions: multiply numerators, multiply denominators, and write the newfraction in simplified form.

�ab� • �d

c� = �b

adc� Simplify }ab

cd} if possible.

x2 º 4x º 12��

x2 º 4

E X A M P L E 1

simplified form

GOAL 1

Multiply and dividerational expressions.

Use rationalexpressions to model real-life quantities, such asthe heat generated by arunner in Exs. 50 and 51.

� To solve real-lifeproblems, such as finding the average number of acres per farm in Exs. 52 and 53.


GOAL 2

GOAL 1


9.4RE

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RE

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Let a, b, and c be nonzero real numbers or variable expressions. Then thefollowing property applies:

�bac

c� = �ba

� Divide out common factor c.

SIMPLIFYING RATIONAL EXPRESSIONS

Page 1 of 7

9.4 Multiplying and Dividing Rational Expressions 555

Multiplying Rational Expressions Involving Monomials

Multiply: • �61x0

3

yy2

�

SOLUTION

• �61x0

3yy

2� = Multiply numerators and denominators.

= Factor and divide out common factors.

= �32xy


Multiplying Rational Expressions Involving Polynomials

Multiply: �x2

4+x º

2x4ºx2

3� • �x

2 +4xx

º 6�

SOLUTION

�x2

4+x º

2x4ºx2

3� • �x

2 +4xx

º 6�

= �(x4ºx(

11)(ºx +

x)3)� • �

(x + 34)(xx º 2)� Factor numerators and denominators.

= Multiply numerators and denominators.

= Rewrite (1 º x) as (º1)(x º 1).

= Divide out common factors.

= ºx + 2 Simplified form

Multiplying by a Polynomial

Multiply: �8xx3

+º

31

� • (4x2 + 2x + 1)

SOLUTION

�8xx3

+º

31

� • (4x2 + 2x + 1)

= • �4x2 +12x + 1� Write polynomial as rational expression.

= Factor and multiply numerators and denominators.

= Divide out common factors.

= �2xx+º


(x + 3)(4x2 + 2x + 1)��(2x º 1)(4x2 + 2x + 1)

(x + 3)(4x2 + 2x + 1)��(2x º 1)(4x2 + 2x + 1)

x + 3�8x3 º 1

E X A M P L E 4

4x(º1)(x º 1)(x + 3)(x º 2)��(x º 1)(x + 3)(4x)

4x(º1)(x º 1)(x + 3)(x º 2)��(x º 1)(x + 3)(4x)

4x(1 º x)(x + 3)(x º 2)��(x º 1)(x + 3)(4x)

E X A M P L E 3

3 • 10 • x • x4 • y3��2 • 10 • x • y • y3

30x5y3�20xy4

5x2y�2xy3

5x2y�2xy3

E X A M P L E 2

Look Back

For help with factoring adifference of two cubes,see p. 345.

STUDENT HELP

Page 2 of 7


To divide one rational expression by another, multiply the first expression by thereciprocal of the second expression.

�ab� ÷ �d

c� = �

ab� • �

dc� = �

abdc� Simplify }ab

dc} if possible.

Dividing Rational Expressions

Divide: �3x5ºx

12� ÷ �x2

xº

2 º6x

2+x

8�

SOLUTION

�3x5ºx

12� ÷ = �3x5ºx

12� • Multiply by reciprocal.

= �3(x5ºx

4)� • �(x º

x(x2)

º(x

2º)

4)� Factor.

= �35(xx(ºx º

4)2(x)()x(x

ºº

42))� Divide out common factors.

= �53� Simplified form

Dividing by a Polynomial

Divide: ÷ (2x2 + 3x)

SOLUTION

÷ (2x2 + 3x) = • �2x2 +

13x

�

= �(3x º 1

6

)

x

(2

2x + 3)� • �x(2x

1+ 3)�

=

= �3x6x

º3

1�

Multiplying and Dividing

Simplify: �x +x

5� • (3x º 5) ÷ �9xx

2

+º

525

�

SOLUTION

�x +x

5� • (3x º 5) ÷ �9xx

2

+º

525

� = �x +x

5� • �3x1º 5� • �

9xx2+º

525

�

=

= �3xx+ 5�

x(3x º 5)(x + 5)��(x + 5)(3x º 5)(3x + 5)

E X A M P L E 7

(3x º 1)(2x + 3)��(6x2)(x)(2x + 3)

6x2 + 7x º 3��

6x26x2 + 7x º 3��

6x2

6x2 + 7x º 3��

6x2

E X A M P L E 6

x2 º 6x + 8��

x2 º 2xx2 º 2x

��x2 º 6x + 8

E X A M P L E 5



INTE

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STUDENT HELP

Page 3 of 7


USING RATIONAL EXPRESSIONS IN REAL LIFE

Writing and Simplifying a Rational Model

SKYDIVING A falling skydiver accelerates until reaching a constant falling speed,called the terminal velocity. Because of air resistance, the ratio of a skydiver’svolume to his or her cross-sectional surface area affects the terminal velocity: the larger the ratio, the greater the terminal velocity.

a. The diagram shows a simplifiedgeometric model of a skydiver withmaximum cross-sectional surfacearea. Use the diagram to write amodel for the ratio of volume tocross-sectional surface area for askydiver.

b. Use the result of part (a) to comparethe terminal velocities of twoskydivers: one who is 60 inches talland one who is 72 inches tall.

SOLUTION

a. The volume and cross-sectional surface area of each part of the skydiver aregiven in the table below. (Assume that the front side of the skydiver’s body isparallel with the ground when falling.)

Using these volumes and cross-sectional surface areas, you can write the ratio as:

�SuVrf

oalcuemaerea� =

=

= 2x

b. The overall height of the geometric model is 14x. For the skydiver whose heightis 60 inches, 14x = 60, so x ≈ 4.3. For the skydiver whose height is 72 inches,14x = 72, so x ≈ 5.1. The ratio of volume to cross-sectional surface area foreach skydiver is:

60 inch skydiver: = 2x ≈ 2(4.3) = 8.6

72 inch skydiver: = 2x ≈ 2(5.1) = 10.2

� The taller skydiver has the greater terminal velocity.

Volume��Surface area

Volume��Surface area

104x3�52x2

4(6x3) + 8x3 + 72x3

��4(6x2) + 4x2 + 24x2

E X A M P L E 8

GOAL 2

GREGORYROBERTSON

made a daring rescue whileskydiving in 1987. To reach anovice skydiver in trouble,he increased his velocity bydiving headfirst towards theother diver. Just 10 secondsbefore he would have hit theground, he was able todeploy both of their chutes.

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2x

x

6x

3x

6x

xx

4x

2x

2x

x 6x

headarm

trunk

leg

Body part Volume Cross-sectionalsurface area

Arm or leg V = 6x3 S = 6x(x) = 6x2

Head V = 8x3 S = 2x(2x) = 4x2

Trunk V = 72x3 S = 6x(4x) = 24x2

Page 4 of 7

1. Explain how you know when a rationalexpression is in simplified form.

2. ERROR ANALYSIS Explain what iswrong with the simplification of therational expression shown.

If possible, simplify the rational expression.

3. 4. 5.

6. 7. �2x2

xº+

91� 8.

Perform the indicated operation. Simplify the result.

9. �156yx9

3� • 10. �

75xx

4yy

3

� • �221xy

7

5� 11. �x2 +

2xx2º 6

� • �x2 +

2x7+x +

812

�

12. �144x4y� ÷ 13. ÷ �

98xxy

2

7� 14. �x52xº

2 +x

1º0x

6� ÷ �15x

x

3

2+º

495x2

�

15. SKYDIVING Look back at Example 8 on page 557. Some skydivers wear“wings” to increase their surface area. Suppose a skydiver who is 65 inches tall iswearing wings that add 18x2 of surface area and an insignificant amount ofvolume. Calculate the skydiver’s volume to surface area ratio with and withoutthe wings.

SIMPLIFYING If possible, simplify the rational expression.

16. 17. 18.

19. 20. 21.

22. �xx3ºº

28

� 23. 24.

25. 26. 27.

MULTIPLYING Multiply the rational expressions. Simplify the result.

28. • �8yx� 29. �

80y3

x4� •

30. • �3x

x2+º

215

� 31. �2xxºº

38� •

32. �x2 º

4xx3º 6

� • �x2 +

x +5x

1+ 6

� 33. �x2

2ºx2

6ºx º

27

� • (x2 º 10x + 21)

34. • (x + 1) 35. • (x2 + 2x + 1)x º 3��ºx3 + 3x2

x3 + 5x2 º x º 5��

x2 º 25

6x2 º 96�

x2 º 92x2 º 10�

x + 1

xy�5x2

4xy3�x2y

x3 + 3x2 º 2x º 6��

x3 + 27x3 º 2x2 + x º 2��

3x2 º 3x º 815x2 º 8x º 18��º20x2 + 14x + 12

x2 + 6x + 9��

x2 º 9x3 º 27

��x3 + 3x2 + 9x

3x2 º 3x º 6��

x2 º 4x2 º 2x º 3��x2 º 7x + 12

x2 + 2x º 4��x2 + x º 6

x2 º 3x + 2��x2 + 5x º 6

x2 º x º 6��x2 + 8x + 16

3x3��12x2 + 9x


16xy�3x5y5

54y3�3x3y

x5y8�80x3y

2x3 º 32x��x2 + 8x + 16

6x2 º 4x º 3��

3x2 + x

x2 + 10x º 4��

x2 + 10xx2 + 4x º 5��

x2 º 14x2

��4x3 + 12x

GUIDED PRACTICE



Concept Check ✓

Skill Check ✓

�5x2

5x+2 +

8x5+x

3� = �(5x5

+x(x

3)+(x

1+)

1)�

= �5x5+x

3�

= �31� = 3

Extra Practice


STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 16–27Examples 2–4: Exs. 28–35,

44–49Examples 5–7: Exs. 36–49Example 8: Exs. 50–55

Page 5 of 7


DIVIDING Divide the rational expressions. Simplify the result.

36. �32

y

x9

3y� ÷ �

8yx6

4� 37. �

2

x

x2z

y2

z� ÷ �3

6xyz

3

�

38. ÷ �x2+x

2� 39. �x2 º

x21º4x

6+x

48� ÷ (3x º 24)

40. ÷ �3x2ºx

3� 41. �x2 +

x8+x

2+ 16� ÷ �x

2 +x2

6ºx

4+ 8

�

42. ÷ �x 6+x

7� 43. (x2 + 6x º 27) ÷ �3x

x

2

++

527x

�

COMBINED OPERATIONS Perform the indicated operations. Simplify the result.

44. (x º 5) ÷ • (x º 6) 45. �x2 º

8xx2

º 12� ÷ �

8xx

3

3+º

32xx

2

2� ÷ �4xx+º

21

�

46. ÷ (3x2 + 6x) • 47. �2

2x2

x2

º+

1x1x

ºº

1521

� • (6x + 9) ÷ �32xxºº

251�

48. (x3 + 8) • ÷ 49. �x2 +

4x122ºx +

920

� • �6x3

x3

+º

190

xx

2

2� • (2x + 3)

HEAT GENERATION In Exercises 50 and 51, use the following information.

Almost all of the energy generated by a long-distance runner is released in the formof heat. The rate of heat generation hg and the rate of heat released hr for a runner ofheight H can be modeled by

hg = k1H3V2 and hr = k2H2

where k1 and k2 are constants and V is the runner’s speed.

50. Write the ratio of heat generated to heat released.

51. When the ratio of heat generated to heat released equals 1, how is height relatedto velocity? Does this mean that a taller or a shorter runner has an advantage?

FARMLAND In Exercises 52 and 53, use the following information.

From 1987 to 1996, the total acres of farmland L (in millions) and the total number offarms F (in hundreds of thousands) in the United States can be modeled by

L = �403.0

.34

t82

+t +

9991� and F = �0

0.1.00510t2

0t+2 +

2.210

�

where t represents the number of years since 1987. � Source: U.S. Bureau of the Census

52. Write a model for the average number of acres A per farm as a function of the year.

53. What was the average number of acres per farm in 1993?

WEIGHT IN GOLD In Exercises 54 and 55, use the following information.

From 1990 to 1996, the price P of gold (in dollars per ounce) and the weight W ofgold mined (in millions of ounces) in the United States can be modeled by

P =

W = º0.0112t5 + 0.193t4 º 1.17t3 + 2.82t2 º 1.76t + 10.4

where t represents the number of years since 1990. � Source: U.S. Bureau of the Census

54. Write a model for the total value V of gold mined as a function of the year.

55. What was the total value of gold mined in the United States in 1994?

53.4t2 º 243t + 385��0.00146t3 + 0.122t2 º 0.586t + 1

x2 º 4�x º 6

x º 2��x2 º 2x + 4

x2 º 4�x + 11

x2 + 11x�x º 2

x2 º 11x + 30��x2 + 7x + 12

x2 + 6x º 7��

3x2

2x2 º 12x��x2 º 7x + 6

3x2 + x º 2��x2 + 3x + 2

FARMERIn 1996 there were

1.3 million farmers and farmmanagers in the UnitedStates. In addition to know-ing about crops and animals,farmers must keep up withchanging technology andpossess strong businessskills.


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Page 6 of 7


56. Use the diagram at the right. Find the ratio of the volume of the rectangular prism to the volume of the inscribed cylinder. Write your answer in simplified form.

57. MULTI-STEP PROBLEM The surface area S and the volume V of a tin can are given by S = 2πr2 + 2πrhand V = πr2h where r is the radius of the can and h is the height of the can. One measure of the efficiency of a tin can is the ratio of its surface area to its volume.

a. Find a general formula (in simplified form) for the ratio �VS

�.

b. Find the efficiency of a can when h = 2r.

c. Calculate the efficiency of each can.

• A soup can with r = 2�58� inches and h = 3�

78� inches.

• A 2 pound coffee can with r = 5�18� inches and h = 6�

12� inches.

• A 3 pound coffee can with r = 6�136� inches and h = 7 inches.

d. Writing Rank the three cans in part (c) by efficiency (most efficient to leastefficient). Explain your rankings.

58. Find two rational functions ƒ(x) and g(x) such that ƒ(x) • g(x) = x2 and

�ƒg

((xx))� = �

(

(

x

x

º

+

1

2

)

)

2

2�.

59. Find two rational functions ƒ(x) and g(x) such that ƒ(x) • g(x) = x º 1

and �ƒg

((xx))� = .

GCFS AND LCMS Find the greatest common factor and least common

multiple of each pair of numbers. (Skills Review, p. 908)

60. 96, 160 61. 120, 165 62. 48, 108

63. 72, 84 64. 238, 51 65. 480, 600

MULTIPLYING POLYNOMIALS Find the product. (Review 6.3 for 9.5)

66. x(x2 + 7x º 1) 67. (x + 7)(x º 1) 68. (x + 10)(x º 3)

69. (x + 3)(x2 + 3x + 2) 70. (2x º 2)(x3 º 4x2) 71. x(x2 º 4)(5 º 6x3)

BICYCLE DEPRECIATION In Exercises 72 and 73, use the following information.

You bought a new mountain bike for $800. The value of the bike decreases by about14% each year. (Review 8.2)

72. Write an exponential decay model for the value of the bike. Use the model toestimate the value after 4 years.

73. Graph the model. Use the graph to estimate when the bike will be worth $300.

MIXED REVIEW

(x + 1)2(x º 1)��

x4

GEOMETRY CONNECTION

7x � 1

4x4x

★★Challenge

TestPreparation

EXTRA CHALLENGE


Page 7 of 7


Addition, Subtraction, andComplex Fractions

WORKING WITH RATIONAL EXPRESSIONS

As with numerical fractions, the procedure used to add (or subtract) two rationalexpressions depends upon whether the expressions have like or unlike denominators.

To add (or subtract) two rational expressions with like denominators, simply add (or subtract) their numerators and place the result over the common denominator.

Adding and Subtracting with Like Denominators

Perform the indicated operation.

a. �34x� + �3

5x� b. �x

2+x

3� º �x +4

3�

SOLUTION

a. �34x� + �3

5x� = �4 3

+x

5� = �3

9x� = �

3x� Add numerators and simplify expression.

b. �x2+x

3� º �x +4

3� = �2xx+º

34

� Subtract numerators.

. . . . . . . . . .

To add (or subtract) rational expressions with unlike denominators, first find the least common denominator (LCD) of the rational expressions. Then, rewrite eachexpression as an equivalent rational expression using the LCD and proceed as withrational expressions with like denominators.

Adding with Unlike Denominators

Add: �65x2� + �

4x2 ºx

12x�

SOLUTION

First find the least common denominator of �65x2� and �

4x2 ºx

12x�.

It helps to factor each denominator: 6x2 = 6 • x • x and 4x2 º 12x = 4 • x • (x º 3).

The LCD is 12x2(x º 3). Use this to rewrite each expression.

+ �4x2 º

x12x

� = �65x2� + �4x(x

xº 3)� = +

= �12

1x0

2x(xº

º30

3)� + �

12x23(xx2

º 3)�

= �3x1

2

2+x2(

1x0x

ºº3)

30�

x(3x)��4x(x º 3)(3x)

5[2(x º 3)]��6x2[2(x º 3)]

5�6x2

E X A M P L E 2

E X A M P L E 1

GOAL 1

Add and subtractrational expressions, asapplied in Example 4.

Simplify complexfractions, as applied inExample 6.

� To solve real-lifeproblems, such as modelingthe total number of malecollege graduates in Ex. 47.


GOAL 2

GOAL 1


9.5RE

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Skills Review

For help with LCDs,see p. 908.

STUDENT HELP

Page 1 of 6

9.5 Addition, Subtraction, and Complex Fractions 563

Subtracting With Unlike Denominators

Subtract: �x2 +

x +4x

1+ 4

� º �x2 º

24

�

SOLUTION

�x2 +

x +4x

1+ 4

� º �x2

2º 4� = �

(xx

++

21)2� º �(x º 2)

2(x + 2)�

= º

=

= �(x

x+

2 º2)

32(xxºº

62)

�

Adding Rational Models

The distribution of heights for American men and women aged 20–29 can bemodeled by

y1 = American men’s heights

y2 = American women’s heights

where x is the height (in inches) and y is the percent (in decimal form) of adults aged20–29 whose height is x ± 0.5 inches. � Source: Statistical Abstract of the United States

a. Graph each model. What is the most common height for men aged 20–29? Whatis the most common height for women aged 20–29?

b. Write a model that shows the distribution of the heights of all adults aged 20–29.Graph the model and find the most common height.

SOLUTION

a. From the graphing calculator screen shown at the top right, you can see that the most common heightfor men is 70 inches (14.3%). The second mostcommon heights are 69 inches and 71 inches (14.2% each). For women, the curve has the sameshape, but is shifted to the left so that the mostcommon height is 64 inches. The second mostcommon heights are 63 inches and 65 inches.

b. To find a model for the distribution of all adults aged 20–29, add the two models and divide by 2.

y = �12�� + �

From the graph shown at the bottom right, you can see that the most common height is 67 inches.

0.143��1 + 0.008(x º 64)4

0.143��1 + 0.008(x º 70)4

0.143��1 + 0.008(x º 64)4

0.143��1 + 0.008(x º 70)4

E X A M P L E 4

x2 º x º 2 º (2x + 4)��

(x + 2)2(x º 2)

2(x + 2)��(x º 2)(x + 2)(x + 2)

(x + 1)(x º 2)��(x + 2)2(x º 2)

E X A M P L E 3

0

0.1

0.2

54 64 70 80

women men

0

0.1

0.2

54 67 80

men and women

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Statistics

Look Back

For help with multiplyingpolynomials, see p. 338.

STUDENT HELP

Page 2 of 6


SIMPLIFYING COMPLEX FRACTIONS

A is a fraction that contains a fraction in its numerator ordenominator. To simplify a complex fraction, write its numerator and its denominatoras single fractions. Then divide by multiplying by the reciprocal of the denominator.

Simplifying a Complex Fraction

Simplify:

SOLUTION

= Add fractions in denominator.

= �x +2

2� • �x3(xx

++

24)

� Multiply by reciprocal.

= �(x +2x

2(x)(

+3x

2+)

4)� Divide out common factor.

= �3x2+x

4� Write in simplified form.

. . . . . . . . . .

Another way to simplify a complex fraction is to multiply the numerator anddenominator by the least common denominator of every fraction in the numeratorand denominator.

Simplifying a Complex Fraction

PHOTOGRAPHY The focal length ƒ of a thin camera lens is given by

ƒ =

where p is the distance between an object being photographed and the lens and q isthe distance between the lens and the film. Simplify the complex fraction.

SOLUTION

ƒ = Write equation.

= �pp

qq� • Multiply numerator and denominator by pq.

= �qp+q

p� Simplify.

1��1p� + �q

1�

1��1p� + �q

1�

1��1p� + �q

1�

E X A M P L E 6

�x +2

2��x3(xx

++

24)�

�x +2

2��x +

12� + �x

2�

�x +2

2��x +

12� + �x

2�

E X A M P L E 5

complex fraction

GOAL 2

PHOTOGRAPHYThe focal length of a

camera lens is the distancebetween the lens and thepoint where light raysconverge after passingthrough the lens.

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qp

imagelensobject

f



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1. Give two examples of a complex fraction.

2. How is adding (or subtracting) rational expressions similar to adding (orsubtracting) numerical fractions?

3. Describe two ways to simplify a complex fraction.

4. Why isn’t (x + 1)3 the LCD of �x +1

1� and �(x +

11)2�? What is the LCD?

Perform the indicated operation and simplify.

5. �x2+x

5� + �x +7

5� 6. �57x� + �3

8x� 7. �x º

x4� º �x +

63�

Simplify the complex fraction.

8. 9. 10.

11. FINANCE For a loan paid back over t years, the monthly payment is given

by M = where P is the principal and i is the annual interest rate.

Show that this formula is equivalent to M = �(1

P

+

i(1

i)

+12t

i)

º

12t

1�.

OPERATIONS WITH LIKE DENOMINATORS Perform the indicated operation

and simplify.

12. �67x� + �

161x� 13. �

1203x2� º �

10xx2� 14. �x

4+x

1� º �x +3

1�

15. + �x5+x

8� 16. �x6ºx2

2� º �x1º2x

2� 17. �x2 º

x5x

� º �x2 º

55x

�

FINDING LCDS Find the least common denominator.

18. �4(x1+4

1)�, �47x� 19. �

214x2�, �

3x2 ºx

15x�

20. , �3x�, �2x

9+x

1� 21. �x(x1º 6)�, �

x2 º 31x2

º 18�

22. �x3(xx

º+

71)�, �

x2 º 63x º 7� 23. �

x2 º 31x º 28�, �

x2 + 6xx + 8�

LOGICAL REASONING Tell whether the statement is always true, sometimestrue, or never true. Explain your reasoning.

24. The LCD of two rational expressions is the product of the denominators.

25. The LCD of two rational expressions will have a degree greater than or equal tothat of the denominator with the higher degree.

5x + 2�4x2 º 1

5x2�x + 8


Pi��

1 º ��1 +1

i��12t

�2x1+5

2��

�6x� º �2

1�

�x +

52

� º 5��

8 + �4x�

�5x

� + 4�8 + �

1x�

GUIDED PRACTICE

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 12–17Examples 2, 3: Exs. 18–23,

26–37Example 4: Exs. 47–51Example 5: Exs. 38–46Example 6: Exs. 52, 53

Extra Practice


STUDENT HELP

Page 4 of 6


OPERATIONS WITH UNLIKE DENOMINATORS Perform the indicated

operation(s) and simplify.

26. + �52x� 27. º�7

4x� º �3

5x�

28. �6(x7º 2)� º �x 6

+x

3� 29. �

6xx2 º

+91

� + �x º4

3�

30. + �x º2

7� 31. �x2

5+x

2ºx

1º 8

� º �x +6

4�

32. º �x1+0

8� 33. �2x º

º150x

� + �3x1+ 2�

34. + �x3ºx

8� 35. �x2 +

2x8+x +

116

� º �x2 º

316

�

36. �x4+x

1� + �2x5º 3� º �

4x� 37. �

3x120ºx

3� + �x º

41� + �6

5x�

SIMPLIFYING COMPLEX FRACTIONS Simplify the complex fraction.

38. 39. 40.

41. 42. 43.

44. 45. 46.

47. COLLEGE GRADUATES From the 1984–85 school year through the1993–94 school year, the number of female college graduates F and the totalnumber of college graduates G in the United States can be modeled by

F = and G =

where t is the number of school years since the 1984–85 school year. Write amodel for the number of male college graduates. � Source: U.S. Department of Education

DRUG ABSORPTION In Exercises 48–51, use the following information.

The amount A (in milligrams) of an oral drug, such as aspirin, in a person’sbloodstream can be modeled by

A =

where t is the time (in hours) after one dose is taken. � Source: Drug Disposition in Humans

48. Graph the equation using a graphing calculator.

49. A second dose of the drug is taken 1 hour after the first dose. Write an equationto model the amount of the second dose in the bloodstream.

50. Write and graph a model for the total amount of the drug in the bloodstream afterthe second dose is taken.

51. About how long after the second dose has been taken is the greatest amount ofthe drug in the bloodstream?

391t2 + 0.112��0.218t4 + 0.991t2 + 1

7560t2 + 978,000��

0.00418t2 + 1º19,600t + 493,000��º0.0580t + 1

�2x2 + 6

3x + 18� + �

x3 ºx

27�

��3x

5ºx

9� º �x º3

3�

�x3 +

164

�

��x2 º

516

� º �3x2 +

212x�

�x2 º

49

� + �x º2

3�

��x +

13� + �x º

13�

�4x1+ 3� º �3(4x

5+ 3)�

��4x

x+ 3�

�1

xº

4x

�

��xº2 º �

x3 +2

x2�

�1x� º �

xº1x+ 1�

��3x�

�2x2

1º 2�

��x +

21� + �

x2 º 2xx º 3�

�x2+0

1��

�14� º �x +

71�

�2x

� º 5�6 + �

3x�

x2 + x º 3��x2 º 12x + 32

4x2�3x + 5

10��x2 º 5x º 14

6�4x2

Look Back

For help with the negativeexponents in Exs. 41 and 42, see p. 323.

STUDENT HELP

PHARMACIST In addition to mixing

and dispensing prescriptiondrugs, pharmacists advisepatients and physicians onthe use of medications. Thisincludes warning of possibleside effects and recommend-ing drug dosages, asdiscussed in Exs. 48–51.


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Page 5 of 6


ELECTRONICS In Exercises 52 and 53, use the following information.

If three resistors in a parallel circuit have resistances R1, R2, and R3 (all in ohms),then the total resistance Rt (in ohms) is given by this formula:

Rt =

52. Simplify the complex fraction.

53. You have three resistors in a parallel circuit with resistances 6 ohms, 12 ohms,and 24 ohms. What is the total resistance of the circuit?

54. MULTI-STEP PROBLEM From 1988 through 1997, the total dollar valueV (in millions of dollars) of the United States sound-recording industry can

be modeled by

V = �5718+3 +

0.012153t4t

�

where t represents the number of years since 1988.� Source: Recording Industry Association of America

a. Calculate the percent change in dollar value from 1988 to 1989.

b. Develop a general formula for the percent change in dollar value from year tto year t + 1.

c. Enter the formula into a graphing calculator or spreadsheet. Observe thechanges from year to year for 1988 through 1997. Describe what you observefrom the data.

CRITICAL THINKING In Exercises 55 and 56, use the following expressions.

2 + , 2 + , 2 +

55. The expressions form a pattern. Continue the pattern two more times. Thensimplify all five expressions.

56. The expressions are getting closer and closer to some value. What is it?

SOLVING LINEAR EQUATIONS Solve the equation. (Review 1.3 for 9.6)

57. �12�x º 7 = 5 58. 6 º �1

10�x = º1 59. �

34�x + �

12� = x º �6

5�

60. �38�x + 4 = º8 61. º�1

12�x º 3 = �

52� 62. 2 = º�

43�x + 10

63. º5x º �34�x = �

521� 64. 2x + �

78�x = º23 65. x = 12 + �

56�x

SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.2, 5.3 for 9.6)

66. x2 º 5x º 24 = 0 67. 5x2 º 8 = 4(x2 + 3) 68. 6x2 + 13x º 5 = 0

69. 3(x º 5)2 = 27 70. 2(x + 7)2 º 1 = 49 71. 2x(x + 6) = 7 º x

MIXED REVIEW

2�3 + �4

3�

1��

2 +

1��1 +1

�2 + �

23�

1��1 +

1�1 + �

12�

1��R11� + �R

12� + �R

13

�

TestPreparation

★★Challenge

EXTRA CHALLENGE


Page 6 of 6


Solving Rational EquationsSOLVING A RATIONAL EQUATION

To solve a rational equation, multiply each term on both sides of the equation by theLCD of the terms. Simplify and solve the resulting polynomial equation.

An Equation with One Solution

Solve: �4x� + �

52� = º�

1x1�

SOLUTION

The least common denominator is 2x.

�4x� + �

52� = º�

1x1� Write original equation.

2x��4x� + �

52�� = 2x�º�

1x1�� Multiply each side by 2x.

8 + 5x = º22 Simplify.

5x = º30 Subtract 8 from each side.

x = º6 Divide each side by 5.

� The solution is º6. Check this in the original equation.

An Equation with an Extraneous Solution

Solve: �x5ºx

2� = 7 + �x1º0

2�

SOLUTION

The least common denominator is x º 2.

�x5ºx

2� = 7 + �x1º0

2�

(x º 2) • �x5ºx

2� = (x º 2) • 7 + (x º 2) • �x1º0

2�

5x = 7(x º 2) + 10

5x = 7x º 4

x = 2

� The solution appears to be 2. After checking it in the original equation, however,you can conclude that 2 is an extraneous solution because it leads to division byzero. So, the original equation has no solution.

E X A M P L E 2

E X A M P L E 1

GOAL 1

Solve rationalequations.

Use rationalequations to solve real-lifeproblems, such as findinghow to dilute an acid solutionin Example 5.

� To solve real-lifeproblems, such as finding the year in which a certainamount of rodeo prize money was earned inExample 6.


GOAL 2

GOAL 1


9.6RE

AL LIFE

RE

AL LIFE



INTE

RNET

STUDENT HELP

Page 1 of 7

9.6 Solving Rational Equations 569

An Equation with Two Solutions

Solve: �4xx++

11

� = �x2

1º2

1� + 3

SOLUTION

Write each denominator in factored form. The LCD is (x + 1)(x º 1).

�4xx++

11

� = �(x + 11)2(x º 1)� + 3

(x + 1)(x º 1) • �4xx++

11

� = (x + 1)(x º 1) • �(x + 11)2(x º 1)� + (x + 1)(x º 1) • 3

(x º 1)(4x + 1) = 12 + 3(x + 1)(x º 1)

4x2 º 3x º 1 = 12 + 3x2 º 3

x2 º 3x º 10 = 0

(x + 2)(x º 5) = 0

x + 2 = 0 or x º 5 = 0

x = º2 or x = 5

� The solutions are º2 and 5. Check these in the original equation.

. . . . . . . . . .

You can use to solve a simple rational equation for which eachside of the equation is a single rational expression.

Solving an Equation by Cross Multiplying

Solve: �x2

2º x� = �x º

11�

SOLUTION

= �x º1

1� Write original equation.

2(x º 1) = 1(x2 º x) Cross multiply.

2x º 2 = x2 º x Simplify.

0 = x2 º 3x + 2 Write in standard form.

0 = (x º 2)(x º 1) Factor.

x = 2 or x = 1 Zero product property

� The solutions appear to be 2 and 1. After checkingthem in the original equation, however, you see thatthe only solution is 2. The apparent solution x = 1 isextraneous. A graphic check shows that the graphs of

the left and right sides of the equation, y = �x2 º

2x

�

and y = �x º1

1�, intersect only at x = 2. At x = 1,

the graphs have a common vertical asymptote.

2�x2 º x

E X A M P L E 4

cross multiplying

E X A M P L E 3

(2, 1)

Page 2 of 7

=

Concentration of new solution = 12 (moles per liter)

Moles of acid in original solution = 16(0.2) (moles)

Volume of original solution = 0.2 (liters)

Volume of water added = x (liters)

12 = �01.62(0+.2

x)

� Write equation.

12(0.2 + x) = 16(0.2) Multiply each side by 0.2 + x.

2.4 + 12x = 3.2 Simplify.

12x = 0.8 Subtract 2.4 from each side.

x ≈ 0.067 Divide each side by 12.

Concentration ofnew solution


USING RATIONAL EQUATIONS IN REAL LIFE

Writing and Using a Rational Model

CHEMISTRY You have 0.2 liter of an acid solution whose acid concentration is 16 moles per liter. You want to dilute the solution with water so that its acidconcentration is only 12 moles per liter. How much water should you add to thesolution?

SOLUTION

� You should add about 0.067 liter, or 67 milliliters, of water.

Using a Rational Model

RODEOS From 1980 through 1997, the total prize money P (in millions ofdollars) at Professional Rodeo Cowboys Association events can be modeled by

P = �ºt

328+0t

3+1t

5+ 1

�

where t represents the number of years since 1980. During which year was the totalprize money about $20 million? � Source: Professional Rodeo Cowboys Association

SOLUTION

Use a graphing calculator to graph the equation

y = �ºx

328+0x

3+1x

5+ 1

�. Then graph the line y = 20.

Use the Intersect feature to find the value of x thatgives a y-value of 20. As shown at the right, thisvalue is x ≈ 12. So, the total prize money was about$20 million 12 years after 1980, in 1992.

E X A M P L E 6

E X A M P L E 5

GOAL 2

CHEMICALENGRAVING

Acid mixtures are used toengrave, or etch, electroniccircuits on silicon wafers. A wafer like the one shownabove can be cut into asmany as 1000 computerchips.

RE

AL LIFE

RE

AL LIFE


LABELS

VERBALMODEL

ALGEBRAICMODEL

STUDENT HELP

Study Tip

Example 6 can also besolved by setting theexpression for P equal to20 and solving theresulting equationalgebraically.

IntersectionX=12.06 Y=20

+Volume of

water addedVolume of

original solution

Moles of acid in original solution

Page 3 of 7


1. Give an example of a rational equation that can be solved using crossmultiplication.

2. A student solved the equation �x º2

3� = �x ºx

3� and got the solutions 2 and 3.

Which, if either, of these is extraneous? Explain how you know.

3. Describe two methods that can be used to solve a rational equation. Whichmethod can always be used? Why?

4. Solve the equation �1x� = �

x22�. Check the apparent solutions graphically. Explain

how a graph can help you identify actual and extraneous solutions.

Solve the equation using any method. Check each solution.

5. �7x� + �

34� = �

5x� 6. �

x º6

2� = �xx

ºº

21� 7. 3x + �3

x� = 5

8. �x ºx

3� = 2 º �x º2

3� 9. �x º5

3� = 10. �x5ºx

1� + 5 = �x1º5

1�

11. �x2+x

3� = �x3ºx

3� 12. �x2ºx

4� = �x º8

4� + 3 13. �2x2+x

4� = �x3+x

2�

14. BASKETBALL STATISTICS So far in the basketball season you have made12 free throws out of the 20 free throws you have attempted, for a free-throwshooting percentage of 60%. How many consecutive free-throw shots would youhave to make to raise your free-throw shooting percentage to 80%?

CHECKING SOLUTIONS Determine whether the given x-value is a solution of

the equation.

15. �2xx+º

33

� = �x3+x

4�; x = º 1 16. �2xx+ 1� = �4 º

5x�; x = º1

17. �4xxºº

43

� + 1 = �x ºx

3�; x = 2 18. �x3ºx

6� = 5 + �x1º8

6�; x = 6

19. �x ºx

3� = �x º6

3�; x = 6 20. �x(x2+ 2)� + �

3x� = �x º

42�; x = 2

LEAST COMMON DENOMINATOR Solve the equation by using the LCD. Check

each solution.

21. �32� + �

1x� = 2 22. �

3x� + x = 4 23. �2

3x� º �

92� = 6x

24. �x +8

2� + �82� = 5 25. �x

3+x

1� + �26x� = �

7x� 26. �3

2x� + �

23� = �x +

86�

27. �x6+x

4� + 4 = �2xxº+

12

� 28. �xx

ºº

34� + 4 = �

3xx� 29. �

72

xx

++

15� + 1 = �

10x3xº 3�

30. + �4x� = �x º

52� 31. �

4(xxºº

11)

� = �2xx+º

12

� 32. �2(

xx++

47)

� º 2 = �22xx++

280

�10

�x2 º 2x


2x�x2 º 9

GUIDED PRACTICE


Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELPExamples 1–3: Exs. 15–32,

42–50Example 4: Exs. 15, 16, 19,

33–50Examples 5, 6: Exs. 54–59

Extra Practice


STUDENT HELP

Page 4 of 7


CROSS MULTIPLYING Solve the equation by cross multiplying. Check each

solution.

33. �43x� = �x +

52� 34. �x

º+

31� = �x º

41� 35. �

x2 ºx

8� = �x

2�

36. �2xx+ 7� = �xx

ºº

51� 37. �x

ºº

21� = �xx

º+

81� 38. �

x2 º

2(x

10

º

x +

2)

16� = �x +

22�

39. �8

x

(2

x

º

º

4

1)� = �x º

42� 40. = �x º

23

� 41. �xºº

13� = �

xx2 º

º247

�

CHOOSING A METHOD Solve the equation using any method. Check each solution.

42. �xx

º+

22� = �

3x� 43. �x +

32� = �x º

61� 44. �x

3+x

1� = �x2

1º2

1� + 2

45. = �xx+º

12� 46. �

x ºx

4� = 47. �4

2ºx

x� = �x ºx2

4�

48. �x2ºx

3� = + 2 49. �2xxº 6� = �x º

24� 50. �x +

21� + �x º

x1� = �

x2 º2

1�

LOGICAL REASONING In Exercises 51–53, a is a nonzero real number. Tell

whether the algebraic statement is always true, sometimes true, or never true.

Explain your reasoning.

51. For the equation �x º1

a� = �x ºx

a�, x = a is an extraneous solution.

52. The equation �x º3

a� = �x ºx

a� has exactly one solution.

53. The equation �x º1

a� = �x +2

a� + �x2

2ºa

a2� has no solution.

54. FOOTBALL STATISTICS At the end of the 1998 season, the NationalFootball League’s all-time leading passer during regular season play was DanMarino with 4763 completed passes out of 7989 attempts. In his debut 1998season, Peyton Manning made 326 completed passes out of 575 attempts. How many consecutive completed passes would Peyton Manning have to maketo equal Dan Marino’s pass completion percentage?

55. PHONE CARDS A telephone company offers you an opportunity to sellprepaid, 30 minute long-distance phone cards. You will have to pay the companya one-time setup fee of $200. Each phone card will cost you $5.70. How manycards would you have to sell before your average total cost per card falls to $8?

56. RIVER CURRENT It takes a paddle boat 53 minutes to travel 5 miles up ariver and 5 miles back, going at a steady speed of 12 miles per hour (with respectto the water). Find the speed of the current.

57. The number f of flies eaten by a praying mantis in 8 hours can be modeled by

ƒ = �d +26

0..60d017�

where d is the density of flies available (in flies percubic centimeter). Approximate the density of flies (in flies per cubic meter) when a praying mantis eats15 flies in 8 hours. (Hint: There are 1,000,000 cm3

in 1 m3.) � Source: Biology by Numbers

BIOLOGY CONNECTION

3x�x2 º 9

6�x2 º 3x

3x + 6�x2 º 4

x2 º 3�x + 2

DATA UPDATE of National Football League data at www.mcdougallittell.com INTE

RNET

The praying mantis blends inwith its environment.

Page 5 of 7


FUEL EFFICIENCY In Exercises 58 and 59, use the following information.

The cost of fueling your car for one year can be calculated using this equation:

Fuel cost for one year =

58. Last year you drove 9000 miles, paid $1.10 per gallon of gasoline, and spent atotal of $412.50 on gasoline. What is the fuel efficiency rate of your car?

59. How much would you have saved if your car’s fuel efficiency rate were 25 milesper gallon?

QUANTITATIVE COMPARISON In Exercises 60 and 61, choose the statement

below that is true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

60.

61.

62. You have 0.5 liter of an acid solution whose acidconcentration is 16 moles per liter. To decrease the acid concentration to 12 moles per liter, you plan to add a certain amount of a second acid solutionwhose acid concentration is only 10 moles per liter. How many liters of thesecond acid solution should you add?

SLOPES OF LINES Find the slope of a line parallel to the given line and the

slope of a line perpendicular to the given line. (Review 2.4 for 10.1)

63. y = x + 3 64. y = 3x º 4 65. y = º�23�x + 15

66. y + 3 = 3x + 2 67. 2y º x = 7 68. 4x º 3y = 17

PROPERTIES OF SQUARE ROOTS Simplify the expression. (Review 5.3 for 10.1)

69. �4�8� 70. �1�8� 71. �1�0�8� 72. �4�3�2�

73. �6� • �4�5� 74. ��17�6

2�� 75. �7�5� • �3� 76. ��48�9��

77. GEOLOGY You can find the pH of a soil by using the formula

pH = ºlog [H+]

where [H+] is the soil’s hydrogen ion concentration (in moles per liter). Find thepH of a layer of soil that has a hydrogen ion concentration of 1.6 ª 10º7 molesper liter. (Review 8.4)

MIXED REVIEW

SCIENCE CONNECTION

Miles driven ª Price per gallon of fuel��Fuel efficiency rate

★★Challenge

TestPreparation

Column A Column B

The solution of �x3 +

x1

� = 2x2 The solution of �xº+

23� = �x º

42�

The solution of �1x� + 3 = �2

9x� The solution of �

12� + �

3x� = �

41

34�

EXTRA CHALLENGE


Page 6 of 7


Perform the indicated operation and simplify. (Lessons 9.4 and 9.5)

1. �3

2

x

x

3

y

y2� • �

109xx

4y2

� 2. �x2 º 3

5xx

º 40� ÷ (x + 5)

3. + �x2+x

4� 4. �25x

82xº

2

36� º �10x

1+ 12�

Simplify the complex fraction. (Lesson 9.5)

5. 6. 7. 8.

9. AVERAGE COST You bought a potholder weaving frame for $10. A bag ofpotholder material costs $4 and contains enough material to make a dozenpotholders. How many dozens of potholders must you make before your average total cost per dozen falls to $4.50? (Lesson 9.6)

�x º

15

� º �x2 º

x25

�

��

�25x�

�x2 º

21

� º �x +1

1�

��12x2

1º 3�

36 º �x12�

��61x2� º 6

�8x� + 11��61x� º 1

18x��x2 º 5x º 36

QUIZ 2 Self-Test for Lessons 9.4–9.6

Deep Water Diving

THENTHEN IN 1530 the invention of the diving bell provided the first effective means of breathingunderwater. Like many other diving devices, a diving bell uses air that is compressedby the pressure of the water. Because oxygen under high pressure (at great depths) canhave toxic effects on the body, the percent of oxygen in the air must be adjusted. Therecommended percent p of oxygen (by volume) in the air that a diver breathes is

p = �d6+60

33�

where d is the depth (in feet) at which the diver is working.

1. Graph the equation.

2. At what depth is the recommended percent of oxygen 5%?

3. What value does the recommended percent of oxygen approachas a diver’s depth increases?

TODAY diving technology makes it easier for scientists like Dr. Sylvia Earle to study ocean life. Using one-person submarines,Earle has undertaken a five-year study of marine sanctuaries.

APPLICATION LINKwww.mcdougallittell.com

INTE

RNET

NOWNOW

The diving bell isinvented. It is open tothe water at the bottomand traps air at the top.

Sylvia Earle walksuntethered on the oceanfloor at a record depthof 1250 feet.

Augustus Siebeinvents theclosed hard-hatdiving suit.

William Beebedescends 1426 feetin a bathysphere.

19301837 1979

1530

Sylvia Earle, marine biologist.

Page 7 of 7

AND STATISTICS Inverse and Joint Variation · 534 Chapter 9 Rational Equations and Functions...

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