10.6 Solving Rational Equations

• Goals: To solve problems involving rational expressions

Rational Equation

An equation containing one or more rational expressions

Steps to solve Rational Equations

1. Find the LCD2. Multiply every term on both sides of the

equation by the LCD over 1(objective is to cancel out the denominators)

3. Solve for the variable a. If it is a linear equation get variables on one

side and constants on the other b. If it is a quadratric set your equation = 0 and

factor.

Extraneous Solutions

• When both sides of the equation are mult by a variable, the equation is transformed into a new equation and may have an extra solution.

• Check each solution in the original rational equation

• Make sure that your answer does not make the denominator 0

Solving Rational EquationsSolving Rational Equations

Multiply both sides of the equation by the LCM of the denominators.

xx

411

Least Common Multiple: Each factor raised to the greatest exponent.

xx 4

LCM is 4x x

xx 4

xx 4x24 x2

Solve for x:

1285

43 x

LCM =

x21518

x233

22 32

32 3 24 241

241

22 3

Solve for x:

xx2

11

LCM =

x

(x + 1)(x)

(x + 1)(x)•

•(x + 1)(x)

22 x

x 2

1x 1x

0 2x

Solve for x:

12121

xx

LCM =

21

2x

2x•

•2x

x24

x81

Solve for x:

21

xx

12x

x• • x

x2

1x

0122 xx 01 2 x

Solve for x:

24

2

2

xxx

2x42 x

2x

-2 is an extraneous solution.

Solve for x:

24

2

2

xxx

2x42 x

2x

-2 is an extraneous solution.

Cross productsShort cut:

7 12 4

xx

4 7x

extraneous solution?

1 2x 4 28 2x x 1x 1x

3 28 2x 3 30x

10x

Cross products: 2 11 2x x

1 2 4x x 5 x

Cross products: 3 13 2 5xx

5 15 3 2x x 2 17x

172

x

Cross products: 2 13 1

x xx x

2 2x x 2 4 3x x 2x 2x

2 4 3x x

3 2 3x 53

x

4x 4x

3 5x

Assignment:Page 453

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