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6: More Fractional Equations and Problem Solving

6: More Fractional Equations and Problem Solving

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7.6 • More Fractional Equations and Problem Solving



Classroom Example

9

1

Solve x ϩ

ϭ .

2x

4



EXAMPLE 3



Solve n ϩ



299



10

1

ϭ .

n

3



Solution





10

1

ϭ ,

n

3



n϶ 0



10

1

3n a n ϩ b ϭ 3n a b

n

3

3n2 ϩ 3 ϭ 10n

3n Ϫ 10n ϩ 3 ϭ 0

13n Ϫ 121n Ϫ 32 ϭ 0

3n Ϫ 1 ϭ 0

or

nϪ3ϭ0

3n ϭ 1

or

nϭ3

1

n ϭ     or  

nϭ3

3

2



Remember when we used the factoring

techniques to help solve equations of this

type in Chapter 6?



1

The solution set is e , 3 f .

3



Problem Solving

2

3

and are called multiplicative inverses, or reciprocals, of each other because

3

2

1

their product is 1. In general, the reciprocal of any nonzero real number n is the number .

n

Let’s use this idea to solve a problem.

Recall that



Classroom Example

The sum of a number and its

17

reciprocal is . Find the number.

4



EXAMPLE 4



The sum of a number and its reciprocal is



26

. Find the number.

5



Solution

We let n represent the number. Then



1

represents its reciprocal.

n



Number



ϩ



Its reciprocal



ϭ



26

5



n



ϩ



1

n



ϭ



26

,

5



n϶ 0



1

26

5n a n ϩ b ϭ 5n a b

n

5



Multiply both sides by 5n, the LCD



5n2 ϩ 5 ϭ 26n

5n2 Ϫ 26n ϩ 5 ϭ 0



15n Ϫ 121n Ϫ 52 ϭ 0

5n Ϫ 1 ϭ 0



or



5n ϭ 1  or

1

n ϭ     or  

5



nϪ5ϭ0

nϭ5

nϭ5



1

1

1

If the number is , its reciprocal is ϭ 5. If the number is 5, its reciprocal is .

5

1

5

5



300



Chapter 7 • Algebraic Fractions



Now let’s consider another uniform motion problem, which is a slight variation of those

we studied in the previous section. Again, keep in mind that we always use the distance–rate–

time relationships in these problems.

Classroom Example

To travel 280 miles, it takes Gary

one hour less than it takes Wayne

to travel 250 miles. Gary travels

20 miles per hour faster than Wayne.

Find the times and rates of both

travelers.



EXAMPLE 5

To travel 60 miles, it takes Sue, riding a moped, 2 hours less than it takes LeAnn, riding a bicycle, to travel 50 miles (see Figure 7.2). Sue travels 10 miles per hour faster than LeAnn. Find

the times and rates of both girls.

Sue



M



O



PE

D



LeAnn



50 miles

60 miles

Figure 7.2



Solution

We let t represent LeAnn’s time. Then t Ϫ 2 represents Sue’s time. We can record the information from Example 5 in the table.



Distance



Time



LeAnn



50



t



Sue



60



tϪ2



Rate a r ϭ



d

b

t



50

t

60

tϪ2



We use the fact that Sue travels 10 miles per hour faster than LeAnn as a guideline to set up

an equation.

Sue’s Rate



ϭ LeAnn’s Rate ϩ 10



60

tϪ2



ϭ



50

ϩ 10,

t



t ϶ 2 and t ϶ 0



Solving this equation yields

60

50

ϭ

ϩ 10

tϪ2

t

t1t Ϫ 22 a



50

60

b ϭ t1t Ϫ 22 a

ϩ 10b

tϪ2

t



60t ϭ 501t Ϫ 22 ϩ 10t1t Ϫ 22

60t ϭ 50t Ϫ 100 ϩ 10t2 Ϫ 20t

0 ϭ 10t2 Ϫ 30t Ϫ 100

0 ϭ t2 Ϫ 3t Ϫ 10

0 ϭ 1t Ϫ 521t ϩ 22

t Ϫ 5 ϭ 0   or   t ϩ 2 ϭ 0

t ϭ 5   or  

t ϭ Ϫ2



Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.



7.6 • More Fractional Equations and Problem Solving



301



We must disregard the negative solution, so LeAnn’s time is 5 hours, and Sue’s time is 5 Ϫ 2 ϭ

50

60

3 hours. LeAnn’s rate is

ϭ 10 miles per hour, and Sue’s rate is

ϭ 20 miles per hour.

5

3

(Be sure that all of these results check back into the original problem!)



There is another class of problems that we commonly refer to as work problems, or

sometimes as rate-time problems. For example, if a certain machine produces 120 items

120

in 10 minutes, then we say that it is producing at a rate of

ϭ 12 items per minute.

10

Likewise, if a person can do a certain job in 5 hours, then that person is working at a rate of

1

of the job per hour. In general, if Q is the quantity of something done in t units of time, then

5

Q

the rate r is given by r ϭ . The rate is stated in terms of so much quantity per unit of time.

t

The uniform-motion problems we discussed earlier are a special kind of rate-time problem in

which the quantity is distance. The use of tables to organize information, as we illustrated

with the uniform-motion problems, is a convenient aid for some rate-time problems. Let’s

consider some examples.



Classroom Example

Printing press A can produce

30 posters per minute, and press B

can produce 20 posters per minute.

Printing press A is set up and starts

a job, and then 20 minutes later

printing press B is started, and both

presses continue printing until

2850 posters are produced. How long

would printing press B be used?



EXAMPLE 6

Printing press A can produce 35 fliers per minute, and press B can produce 50 fliers per

minute. Printing press A is set up and starts a job, and then 15 minutes later printing press B

is started, and both presses continue printing until 2225 fliers are produced. How long would

printing press B be used?



Solution

We let m represent the number of minutes that printing press B is used. Then m ϩ 15

represents the number of minutes that press A is used. The information in the problem can be

organized in a table.



Press A

Press B



Rate



Time



Quantity ‫ ؍‬Rate ؋ Time



35

50



m ϩ 15

m



35(m ϩ 15)

50m



Since the total quantity (total number of fliers) is 2225 fliers, we can set up and solve the following equation:

351m ϩ 152 ϩ 50m ϭ 2225

35m ϩ 525 ϩ 50m ϭ 2225

85m ϭ 1700

m ϭ 20

Therefore, printing press B must be used for 20 minutes.

Classroom Example

Sandy can shovel the walk in

50 minutes, and Ashley can shovel

the same walk in 75 minutes. How

long would it take the two of them

working together to shovel the walk?



EXAMPLE 7

Bill can mow a lawn in 45 minutes, and Jennifer can mow the same lawn in 30 minutes. How

long would it take the two of them working together to mow the lawn? (See Figure 7.3.)



302



Chapter 7 • Algebraic Fractions



Figure 7.3



Remark: Before you look at the solution of this problem, estimate the answer. Remember

that Jennifer can mow the lawn by herself in 30 minutes.



Solution

1

1

of the lawn per minute, and Jennifer’s rate is

of the lawn per minute.

45

30

1

If we let m represent the number of minutes that they work together, then represents the rate

m

when working together. Therefore, since the sum of the individual rates must equal the rate

working together, we can set up and solve the following equation:

Bill’s rate is



1

1

1

ϩ

ϭ ,

m϶ 0

m

30

45

1

1

1

90m a

ϩ b ϭ 90m a b

m

30

45

3m ϩ 2m ϭ 90

5m ϭ 90

m ϭ 18



Multiply both sides by 90m, the LCD



It should take them 18 minutes to mow the lawn when working together. (How close was your

estimate?)



Classroom Example

It takes Jake three times as long to

mow the lawn as it does Zack. How

long would it take each boy by

himself if they can mow the lawn

together in 36 minutes?



EXAMPLE 8

It takes Amy twice as long to deliver papers as it does Nancy. How long would it take each

girl by herself if they can deliver the papers together in 40 minutes?



Solution

We let m represent the number of minutes that it takes Nancy by herself. Then 2m represents

1

1

Amy’s time by herself. Therefore, Nancy’s rate is , and Amy’s rate is

. Since the combined

m

2m

1

rate is , we can set up and solve the following equation:

40

Nancy’s

Amy’s Combined

rate ϩ rate ϭ rate



1

1

1

    ϩ   

  ϭ     ,

m

2m

40

40m a



m϶ 0



1

1

1

ϩ

b ϭ 40m a b

m

2m

40



7.6 • More Fractional Equations and Problem Solving



303



40 ϩ 20 ϭ m

60 ϭ m

Therefore, Nancy can deliver the papers by herself in 60 minutes, and Amy can deliver them

by herself in 2(60) ϭ 120 minutes.

One final example of this section outlines another approach that some people find meaningful for work problems. This approach represents the fractional parts of a job. For example,

if a person can do a certain job in 7 hours, then at the end of 3 hours, that person has finished

3

5

of the job. (Again, we assume a constant rate of work.) At the end of 5 hours, of the

7

7

h

job has been done—in general, at the end of h hours, of the job has been completed. Let’s

7

use this idea to solve a work problem.

Classroom Example

It takes Chris 7 hours to install a

wood railing. After working for

2 hours he is joined by Carlos, and

together they finish the railing in

3 hours. How long would it take

Carlos to install the railing by

himself?



EXAMPLE 9

It takes Pat 12 hours to install a wood floor. After he had been working for 3 hours, he was

joined by his brother Mike, and together they finished the floor in 5 hours. How long would

it take Mike to install the floor by himself?



Solution

Let h represent the number of hours that it would take Mike to install the floor by himself.

The fractional part of the job that Pat does equals his working rate times his time. Because

1

it takes Pat 12 hours to do the entire floor, his working rate is . He works for 8 hours (3 hours

12

1

8

before Mike and then 5 hours with Mike). Therefore, Pat’s part of the job is

(8) ϭ .

12

12

The fractional part of the job that Mike does equals his working rate times his time. Because h

1

represents Mike’s time to install the floor, his working rate is . He works for 5 hours.

h

5

1

Therefore, Mike’s part of the job is (5) ϭ . Adding the two fractional parts together results

h

h

in 1 entire job being done. Let’s also show this information in chart form and set up our guideline. Then we can set up and solve the equation.

Time to do

entire job



Pat



12



Mike



h



Fractional part of

the job that Pat does



Working

rate



Time

working



1

12

1

h

Fractional part of

the job that Mike does



8

5

ϩ ϭ1

12

h

8

5

12h a

ϩ b ϭ 12h112

12

h

8

5

12h a b ϩ 12h a b ϭ 12h

12

h

8h ϩ 60 ϭ 12h



8

5



Fractional part

of the job done



8

12

5

h



304



Chapter 7 • Algebraic Fractions



60 ϭ 4h

15 ϭ h

It would take Mike 15 hours to install the floor by himself.

We emphasize a point made earlier. Don’t become discouraged if solving word problems

is still giving you trouble. The development of problem-solving skills is a long-term objective. If you continue to work hard and give it your best shot, you will gradually become more

and more confident in your approach to solving problems. Don’t be afraid to try some different approaches on your own. Our problem-solving suggestions simply provide a framework

for you to build on.



Concept Quiz 7.6

For Problems 1–10, answer true or false.

1. Assuming uniform motion, the rate at which a car travels is equal to the time traveled

divided by the distance traveled.

2. If a worker can lay 640 square feet of tile in 8 hours, we can say his rate of work is

80 square feet per hour.

5

3. If a person can complete 2 jobs in 5 hours, then the person is working at the rate of

2

of the job per hour.

4. In a time-rate problem involving two workers, the sum of their individual rates must

equal the rate working together.

2

5. If a person works at the rate of

of the job per hour, then at the end of 3 hours the

15

6

job would be

completed.

15

1

6. The solution set for x ϩ ϭ 4 is {2, 4}.

x

1

ϩ

xϩ2

x

2

8. The solution set for

Ϫ

xϪ1

x

7. The solution set for



1

5 Ϫ 2x

3

is e f .

ϭ 2

Ϫ3

2

x ϪxϪ6

3

Ϫx

is л.

ϭ 2

ϩ1

x Ϫ1



9. If Kim can do a certain job in 5 hours, then at the end of h hours she will have

5

completed of the job.

h

x

4

1

10. The solution set for

ϩ 2

ϭ is 5Ϫ86.

3x Ϫ 6

3

x Ϫ4



Problem Set 7.6

For Problems 1–32, solve each equation. (Objective 1)

1.



4

7

2

1

ϩ ϭ ϩ

x

x

6

3x



2.



2

25

9

Ϫ ϭϪ

x

3x

9



5.



5

3

Ϫ

ϭ1

2n Ϫ 10

nϪ5



6.



7

2

Ϫ

ϭ2

3x ϩ 6

xϩ2



3.



3

4

11

ϩ

ϭ

2x ϩ 2

xϩ1

12



7.



3

5

7

Ϫ ϭ ϩ1

2t

t

5t



4.



1

7

5

ϩ

ϭ

2x Ϫ 6

xϪ3

2



9.



x

4

ϩ

ϭ1

xϪ2

xϩ2



8.



2

3

5

ϩ ϭ1Ϫ

3t

4t

2t



7.6 • More Fractional Equations and Problem Solving



10.



2x

3

Ϫ

ϭ2

xϩ1

xϪ1



11.



x

2x

Ϫ

ϭ Ϫ1

xϪ4

xϩ4



12.



2x

x

ϩ

ϭ3

xϩ2

xϪ2



13.



3n

n

Ϫ

ϭ2

nϩ3

nϪ3



14.



4n

2n

Ϫ

ϭ2

nϪ5

nϩ5



15.



3

5

2

ϩ

ϭ

tϩ2

tϪ2

t Ϫ4



16.



t

16

1

ϩ 2

ϭ

2t Ϫ 8

2

t Ϫ 16



32. 3 ϩ



34. The sum of a number and three times its reciprocal is 4.

Find the number.

35. A number is



37. Suppose that Celia rides her bicycle 60 miles in

2 hours less time than it takes Tom to ride his bicycle

85 miles. If Celia rides 3 miles per hour faster than Tom,

find their respective rates.



3x Ϫ 1

4

5

ϩ

ϭ

2

xϩ3

xϪ3

x Ϫ9

5

3

ϭ

yϩ2

y2 ϩ 2y



20. 2 ϩ



4

4

ϭ 2

yϪ1

y Ϫy



21. n ϩ



1

17

ϭ

n

4



23.



15

15

ϩ

ϭ1

4n

41n ϩ 42



24.



10

10

ϩ

ϭ1

7x

71x ϩ 32



25. x Ϫ

26.



21

larger than its reciprocal. Find the number.

10



36. Suppose that the reciprocal of a number subtracted from

5

the number yields . Find the number.

6



2



19. 8 ϩ



6

6

ϭ 2

tϪ3

t Ϫ 3t



For Problems 33–50, set up an equation and solve the

problem. (Objective 2)

9

33. The sum of a number and twice its reciprocal is .

2

Find the number.



4

2x Ϫ 3

6

17.

Ϫ 2

ϭ

xϪ1

xϩ1

x Ϫ1

18.



22. n ϩ



38. To travel 300 miles, it takes a freight train 2 hours

longer than it takes an express train to travel 280 miles.

The rate of the express train is 20 miles per hour faster

than the rate of the freight train. Find the rates of both

trains.



3

ϭ4

n



39. One day, Jeff rides his bicycle out into the country

40 miles (see Figure 7.4). On the way back, he takes a

different route that is 2 miles longer, and it takes him an

hour longer to return. If his rate on the way out to the

country is 4 miles per hour faster than his rate back, find

both rates.



5x

Ϫ10

ϭ

xϪ2

xϪ2



xϩ1

3

12

Ϫ ϭ 2

x

xϪ3

x Ϫ 3x



40



t

5

1

27.

ϩ 2

ϭ

4t Ϫ 4

4

t Ϫ1

28.



305



42



x

4

1

ϩ 2

ϭ

3x Ϫ 6

3

x Ϫ4



N

E

W



3

4

2n ϩ 11

29.

ϩ

ϭ 2

nϪ5

nϩ7

n ϩ 2n Ϫ 35

30.



2

3

2n Ϫ 1

ϩ

ϭ 2

nϩ3

nϪ4

n Ϫ n Ϫ 12



31.



a

3

14

ϩ

ϭ 2

aϩ2

aϩ4

a ϩ 6a ϩ 8



S



Figure 7.4



Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.



306



Chapter 7 • Algebraic Fractions



40. Rita jogs for 8 miles and then walks an additional

12 miles. She jogs at a rate twice her walking rate, and

she covers the entire distance of 20 miles in 4 hours.

Find the rate she jogs and the rate she walks.

41. A water tank can be filled by an inlet pipe in 5 minutes.

A drain pipe will empty the tank in 6 minutes. If

by mistake the drain is left open as the tank is

being filled, how long will it take before the tank

overflows?

42. Betty can do a job in 10 minutes. Doug can do the same

job in 15 minutes. If they work together, how long will

it take them to complete the job?

43. It takes Barry twice as long to deliver papers as it does

Mike. How long would it take each if they can deliver

the papers together in 40 minutes?

44. Working together, Cindy and Sharon can address

envelopes in 12 minutes. Cindy could do the addressing

by herself in 20 minutes. How long would it take

Sharon to address the envelopes by herself?

45. Mark can overhaul an engine in 20 hours, and Phil can

do the same job by himself in 30 hours. If they both

work together for a time, and then Mark finishes the job

by himself in 5 hours, how long did they work

together?

46. Working together, Pam and Laura can complete a

1

job in 1 hours. When working alone, it takes Laura

2



4 hours longer than Pam to do the job. How long does it

take each of them working alone?

47. A copy center has two copiers. Copier A can produce

copies at a rate of 40 pages per minute, and copier B

does 30 pages per minute. How long will copier B need

to run if copier A has been copying by itself for 6 minutes, and then both copier A and B are used until 520

copies are made?

48. It takes two pipes 3 hours to fill a water tank. Pipe B

can fill the tank alone in 8 hours more than it takes pipe

A to fill the tank alone. How long would it take each

pipe to fill the tank by itself?

49. In a survivor competition, the Pachena tribe can shuck

300 oysters in 10 minutes less time than it takes the

Tchaika tribe. If the Pachena tribe shucks oysters at a

rate of 5 oysters per minute faster than the Tchaika

tribe, find the rate of each tribe.

50. Machine A can wrap 600 pieces of candy in 5 minutes

less time than it takes machine B to wrap 600 pieces

of candy. If the rate of machine A is 20 candies

per minute faster than machine B, find the rate of each

machine.



Additional word problems can be found in Appendix B.

All of the problems in the Appendix marked as (7.5)

or (7.6) are appropriate for your practice.



Thoughts Into Words

51. Write a paragraph or two summarizing the new ideas

about problem solving that you have acquired thus far

in this course.



Further Investigations

For Problems 52–54, solve each equation.

52.



3x Ϫ 1

4

7

ϩ

ϭ

2

xϩ3

xϪ3

x Ϫ9



53.



xϪ2

3

Ϫ5

ϩ

ϭ

xϩ1

xϪ1

x2 Ϫ 1



Answers to the Concept Quiz

1. False

2. True

3. False

9. False

10. True



54.



4. True



5. True



7x Ϫ 12

5

2

Ϫ

ϭ

2

x

ϩ

4

x

Ϫ

4

x Ϫ 16



6. False



7. True



8. True



Chapter 7 Summary

OBJECTIVE



SUMMARY



Simplify rational expressions

using factoring techniques.



The fundamental principle of fractions

ak

a

a ϭ b provides the basis for simplifybk

b

ing rational expressions. For many problems, the numerator and denominator will

have to be factored before you can apply

the fundamental principle of fractions.



(Section 7.1/Objective 1)



Multiply rational expressions.

(Section 7.2/Objective 1)



To multiply algebraic fractions, multiply

the numerators, multiply the denominators,

and express the product in simplified form.



EXAMPLE



Simplify



3x2 Ϫ 15x

.

x Ϫ 3x Ϫ 10

2



Solution



3x1x Ϫ 52

3x2 Ϫ 15x

ϭ

1x ϩ 221x Ϫ 52

x2 Ϫ 3x Ϫ 10

3x

ϭ

xϩ2



Multiply



y

y ϩ 4y ϩ 3

2



Solution



y

y ϩ 4y ϩ 3

2



Divide rational expressions.

(Section 7.2/Objective 2)



To divide algebraic fractions, invert the

divisor and multiply.



и



3y ϩ 3

.

8



3y ϩ 3

8



ϭ



y

(y ϩ 1)(y ϩ 3)



ϭ



3y

81y ϩ 32



Divide



и



и



3(y ϩ 1)

8



a2 Ϫ 36

6a ϩ 6

Ϭ 2

.

2

a ϩ 8a ϩ 12

a ϩ 3a ϩ 2



Solution



a2 Ϫ 36

6a ϩ 6

Ϭ 2

2

a ϩ 8a ϩ 12

a ϩ 3a ϩ 2

ϭ

ϭ

ϭ

Combine rational expressions

with common denominators.

(Section 7.3/Objective 1)



Addition and subtraction of algebraic

fractions are based on the following

definitions:

a

c

aϩc

ϩ ϭ

Addition

b

b

b

a

c

aϪc

Ϫ ϭ

Subtraction

b

b

b

The final answer should always be in

simplified form.



a2 Ϫ 36

a2 ϩ 8a ϩ 12



и



1a ϩ 621a Ϫ 62



1a ϩ 621a ϩ 22



a2 ϩ 3a ϩ 2

6a ϩ 6



и



1a ϩ 221a ϩ 12

61a ϩ 12



aϪ6

6



Subtract



8n ϩ 1

5n Ϫ 8

Ϫ

.

6

6



Solution



8n ϩ 1 5n Ϫ 8 8n ϩ 1 Ϫ (5n Ϫ 8)

Ϫ

ϭ

6

6

6

8n ϩ 1 Ϫ 5n ϩ 8

ϭ

6

3n ϩ 9

ϭ

6

31n ϩ 32

nϩ3

ϭ

ϭ

6

2

(continued)



307



308



Chapter 7 • Algebraic Fractions



OBJECTIVE



SUMMARY



EXAMPLE



Add and subtract rational

expressions with different

denominators.



Use the following procedure when adding

and subtracting fractions.



Subtract



(Section 7.3/Objectives 2 and

3; Section 7.4/Objective 1)



Simplify complex fractions.

(Section 7.4/Objective 2)



x

1

Ϫ

.

xϪ4

x Ϫ 16

2



1. Find the least common denominator.

2. Change each fraction to an equivalent

fraction that has the LCD as its denominator.

3. Add or subtract the numerators, and

place this result over the LCD.

4. Look for possibilities to simplify the

final fraction.



Solution



Fractional forms that contain fractions in

the numerator and/or the denominator are

complex fractions. To simplify a complex

fraction means to express it as a single

fraction. One method for simplifying is to

multiply the entire complex fraction by a

form of 1. Another method is to simplify

the numerator, simplify the denominator,

and then proceed as with a division of

fractions problem.



3

2

ϩ

x

y

Simplify

.

1

1

ϩ

x2

y2



x

1

Ϫ

xϪ4

x Ϫ 16

x

1

ϭ

Ϫ

1x ϩ 421x Ϫ 42

xϪ4

11x ϩ 42

x

ϭ

Ϫ

1x ϩ 421x Ϫ 42

1x Ϫ 421x ϩ 42

x Ϫ 11x ϩ 42

ϭ

1x ϩ 421x Ϫ 42

xϪxϪ4

Ϫ4

ϭ

ϭ

1x ϩ 421x Ϫ 42

1x ϩ 421x Ϫ 42

2



Solution



3

2

3

2

ϩ

ϩ

2 2

x

y

x y

x

y

ϭ 2 2±



1

1

1

1

x y

ϩ

ϩ

x2

y2

x2

y2

3

2

x2y2 a b ϩ x2y2 a b

x

y

ϭ

1

1

x2y2 a 2 b ϩ x2y2 a 2 b

x

y

2

2

3xy ϩ 2x y

ϭ

y2 ϩ x2



Solve rational equations that

have constants in the

denominator.

(Section 7.5/Objective 1)



To solve rational equations that have

constants in the denominator, multiply

both sides of the equation by the LCD of

all the denominators in the equation.



Solve



xϩ3

xϪ4

21

ϩ

ϭ .

2

5

10



Solution



xϩ3

xϪ4

21

ϩ

ϭ

2

5

10

xϩ3

xϪ4

21

10 a

ϩ

b ϭ 10 a b

2

5

10

51x ϩ 32 ϩ 21x Ϫ 42 ϭ 21

5x ϩ 15 ϩ 2x Ϫ 8 ϭ 21

7x ϩ 7 ϭ 21

7x ϭ 14

xϭ2

The solution set is {2}.

(continued)



Chapter 7 • Summary



OBJECTIVE



SUMMARY



Solve rational equations

that have variables in the

denominator.



If an equation contains a variable in one or

more of the denominators, we must avoid

any value of the variable that makes the

denominator zero.



(Section 7.5/Objective 1;

Section 7.6/Objective 1)



EXAMPLE



Solve



3

5

4

ϩ

ϭ .

n

2n

3



Solution



First we need to realize that n can not

equal zero.

3

5

4

6n a ϩ b ϭ 6n a b

n

2n

3

3

5

4

6n a b ϩ 6n a b ϭ 6n a b

n

2n

3

18 ϩ 15 ϭ 8n

33 ϭ 8n

33

ϭn

8

The solution set is e



Solve rational equations

that are in the form of a

proportion.

(Section 7.5/Objective 1)



c

a

ϭ if

b

d

and only if ad ϭ bc where b ϶ 0

and d ϶ 0, can be applied to solve

proportions. This property of proportions

is often referred to as cross products

are equal.

The property of proportions,



309



Solve



33

f.

8



6

3

ϭ

.

xϩ1

xϪ2



Solution



3

6

ϭ

xϩ1

xϪ2

61x Ϫ 22 ϭ 31x ϩ 12

6x Ϫ 12 ϭ 3x ϩ 3

3x ϭ 15

xϭ5

The solution set is {5}.



Solve proportion word

problems.

(Section 7.5/Objective 2)



Some of the word problems in this chapter

translate into equations that are proportions.



The numerator of a fraction is 4 less than

the denominator. The fraction in its

9

simplest form is . Find the fraction.

10

Solution



Let x represent the denominator. Then

x Ϫ 4 represents the numerator.

xϪ4

9

ϭ

x

10

101x Ϫ 42 ϭ 9x

10x Ϫ 40 ϭ 9x

x ϭ 40

The fraction is



36

.

40

(continued)



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