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

4

11

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

xx 4

LCM is 4x x

xx 4

xx 4

x24 x2

Solve for x:

128

5

4

3 x LCM =

x21518

x2

33

22 32

32 3 24 24

124

1

22 3

Solve for x:

xx

2

1

1

LCM =

x

(x + 1)(x)

(x + 1)(x)•

•(x + 1)(x)

22 x

x 2

1x 1x

0 2x

Solve for x:

121

2

1

xx

LCM =

21

2x

2x•

•2x

x24

x8

1

Solve for x:

21

x

x

12x

x• • x

x2

1x

0122 xx

01 2 x

Solve for x:

2

4

2

2

xx

x

2x42 x

2x

-2 is an extraneous solution.

Solve for x:

2

4

2

2

xx

x

2x42 x

2x

-2 is an extraneous solution.

Cross productsShort cut:

7 1

2 4

x

x

4 7x

extraneous solution?

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

3 28 2x 3 30x

10x

Cross products: 2 1

1 2x x

1 2 4x x 5 x

Cross products: 3 1

3 2 5

x

x

5 15 3 2x x 2 17x

17

2x

Cross products: 2 1

3 1

x x

x x

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

2 4 3x x

3 2 3x 5

3x

4x 4x

3 5x

Assignment:Page 453

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