SECTION 8.5-8.6 Solving Rational Equations and their Applications

Solving equations containing fractions:Key: GET RID OF THE FRACTIONS!Solve:

1486

)7(2)2(4)2(3

6

712

3

212

4

212

6

7

3

2

4

2

)1(6

7

)1(3

2

)1(4

1)1(3

equation. original in thesolution your Check :4 Step

1

99

1459

14839

2)7()2(4)13(3

for x. solve to termslike combine andproperty vedistributi Use:3 Step

126

7

3

212

4

1312

LCM. by theequation theof SIDES BOTH ON EVERYTHINGMultiply :2 Step

12x is6x and 3x, 4x, of CML The

).numerators about thet worry (don' RSDENOMINATO theall of LCM theFind :1 Step6

7

3

2

4

13

x

x

x

x

x

xxx

xx

xx

xxx

x

3 42

When solving proportions (1 rational expression set equal to another),then just cross-multiply.

YES

Check

x

x

xx

xx

multiplyCrossx

Yes

Check

x

x

x

?)3(6)3(83

4

6

83

4

33

8:

3

124

1248

)3(48

:

4

3x

8

Example

?)3(4)2(63

2

6

4:

6

122

)3(423

2

x

4

:Solve

Applications of proportions:Rates: A ratio of two numbers (usually in different units), where one number depends on the other in some manner.When setting up a proportion (one ratio = another ratio), make sure the units of the numerators match and the ratios of the denominators match.

The monthly loan payment for car is $29.50 for each $1000 borrowed. At this rate, find the monthly payment for a $9000 loan.

The phrase “for each” is like “per” in a rate. Since it is $29.50“for each” $1000 borrowed, the rate is written:

$265.50 ispayment Monthly

50.265$

)9000($9000$1000$

50.29$)9000($

$9000 by sidesboth gmultiplyinby itselfby Pget just

g,multiplyin-cross of instead case, In this

9000$1000$

50.29$

9000$1000$

50.29$

P

P

P

P

dAmtBorrowe

Payment

Example:An investment of $1200 earns $96 each year At the same rate,How much additional money must be invested to earn $128each year?

What are we asked to find? How much ADDITONAL money must be invested in order to earn $128 each year.Let x = additional money to be invested.Given info: Rate of investment = Investment/Amount Earned =

Also, amount earned from new investment is $128.

$400. additionalan invest shouldinvestor The:CONCLUSION

invest. amount to additional theyeah,Oh anyway?for stand x didWhat

400

3840096

11520015360096x

15360096115200

1281200120096

:multiply-Cross128$

1200$

96$

1200$128$

1200$

EarnedAmt New

Investment New

96$

1200$

EarnedAmt

Investment

x

x

x

)(x)(

x

x

96$

1200$

Work Problems

Savannah can paint a room in seven hours. Jordan can paint the same room in nine hours. How long does it take for both Savannah and Jordan to paint the room if they are working together?

Rate of Work * Time Worked = Part of Job CompletedRate of Work is the portion of a job that can get done in 1 unit of time (usually hours).If Savannah can paint a room in seven hours, then her rate of work is 1/7 = 1 job/7 hours = 1/7 of a job in 1 hour.Jordan’s rate of work = 1 room/9 hours = 1/9

Savannah’s Part + Jordan’s Part = Whole Job = 1

What are we being asked to find? How long it takes for both Savannah and Jordan to paint the room working together. That is, TIME. Let t = time to work together.

hours 16153

16

63

6316

6379

)63(19

637

63

63

197

19

1

7

1

t

t

tt

tt

LCD

tt

tt

Megan and Julia can finish a piece of work in 15 days. Megan can do the job herself in twenty days. If Julia wanted to do the job alone, how long would it take her?

In this case, let t = time for Julia to do the job alone.Julia’s rate of work =

Given information. The whole job can get done in 15 days. Megan’s rate of work is ____

Megan’s Part + Julia’s Part job = 1

t

1

dayst

t

tt

tt

tt

tLCDt

t

t

60

160

4603

)4(115

)4(4

3)4(

4

115

4

3

115

20

15

1)15(1

)15(20

1

Megan’s Rate of

Work

x Time Worked Together

+ Julia’s Rate of

Work

x Time Worked Together

= 1