9.6 solving rational equations

9.6 Solving Rational Equations9.6 Solving Rational Equationsp.569p.569

What is the easiest way to solve a rational What is the easiest way to solve a rational equation?equation?

What is an extraneous solution?What is an extraneous solution?

How do you know when to cross multiply or How do you know when to cross multiply or multiply out the denominator?multiply out the denominator?

Hint for solving:Hint for solving:

• A quick way to solve a rational A quick way to solve a rational equation is to multiply everything equation is to multiply everything through by the LCD. through by the LCD.

• This will get rid of all the fractions!This will get rid of all the fractions!

• Beware of extraneous solutions!Beware of extraneous solutions!

Example: Solve.

xx

12

2

13

LCD: 2x

Multiply each fraction through by the LCD

x

xx

x

x 12*2

2

23*2

246 x

18 x

18xCheck your solution!Check your solution!

18

12

2

1

18

3

1293

Solve.1

54

1

5

xx

x LCD: ?LCD: (x+1)

)1(

)1(5)1(4

)1(

)1(5

x

xx

x

xx

5445 xx

145 xx

1x

Check your solution!Check your solution!

11

54

11

)1(5

0

54

0

5

?

No Solution!No Solution!

Solve. 14

6

2

232

xx

x

Factor 1st!

1)2)(2(

6

2

23

xxx

x

LCD: (x+2)(x-2)

)2)(2()2)(2(

)2)(2(6

)2(

)2)(2)(23(

xx

xx

xx

x

xxx

42264263 22 xxxxxx

2443 22 xxx

0642 2 xx0322 xx

0)1)(3( xx

01or 03 xx

1or 3 xx

Check your solutions!Check your solutions!

Short Cut!

When there is only one fraction on each When there is only one fraction on each side of the =, just cross multiply as if side of the =, just cross multiply as if you are solving a proportion.you are solving a proportion.

Example: Solve.

4

1

4

32

xxx

xx 42 12342 xxx

0122 xx0)3)(4( xx

03or 04 xx3or 4 xx

Check your solutions!Check your solutions!

)4(3 x

Last Example: Solve. 1

2

22

62

x

x

xx

)1(6 x)2)(1(2 xxx

6)2(2 xx3)2( xx

0322 xx0)1)(3( xx

01or 03 xx1or 3 xx

1

2

)1(2

6

x

x

xx

Check your solutions!Check your solutions!

• What is the easiest way to solve a rational What is the easiest way to solve a rational equation?equation?

Multiply the equation by a value that will Multiply the equation by a value that will make the denominator equal to 1.make the denominator equal to 1.

• What is an extraneous solution?What is an extraneous solution?

A solution that does not make a true A solution that does not make a true statement in the original equation.statement in the original equation.

• How do you know when to cross multiply How do you know when to cross multiply or multiply out the denominator?or multiply out the denominator?

Cross multiply if there is only one fraction Cross multiply if there is only one fraction on each side of the equal sign, otherwise on each side of the equal sign, otherwise multiply out the denominator.multiply out the denominator.

Assignment 9.6

p. 571

15-49 odd

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