Solving Solving Rational Rational

EquationsEquationsOn to Section 2.8aOn to Section 2.8a

Solving Rational Equations

Rational Equation – an equation involving rational expressionsor fractions…can always be written in the form:

0f x

g x

where f (x) and g (x) are polynomial functions with no commonfactors. The zeros of f (x) are the solutions of the equation…

Solving Rational Equations

To solve an equation involving fractions:

• Find the LCD of all the terms in the equation.

• Clear the fractions by multiplying both sides of the equation by the LCD.

• Solve the resulting equation.

Extraneous Solutions – solutions that are not solutions of theoriginal equation; result from multiplying or dividing an equationby an expression containing variables…

Guided Practice

Solve3

4xx

The LCD is x !!!

2 3 4x x Multiply by Multiply by x x 2 4 3 0x x Subtract 4Subtract 4x x

1 3 0x x FactorFactor

1,3x Zero Factor PropertyZero Factor Property

How do we support these answers How do we support these answers numerically numerically ??????

Guided Practice

Solve1

04

xx

The LCD is x – 4!!!

44 0

4

xx x

x

2 3x 2 4 1 0x x

0.268,3.732How could we support these answers How could we support these answers graphically graphically ??????

To solve this quadratic?To solve this quadratic?

Guided Practice

Solve 2

2 1 2

1 3 4 3

x

x x x x

The LCD!!!

2

2 1 21 3 1 3

1 3 4 3

xx x x x

x x x x

1 3x x

22 5 3 0x x 2 3 1 2x x x

First, factor the last denominator…First, factor the last denominator…

Guided Practice

Solve 2

2 1 2

1 3 4 3

x

x x x x

22 5 3 0x x

2 1 3 0x x Factor!!!Factor!!!

1

2x

Confirm NumericallyConfirm Numerically

Support GraphicallySupport Graphically3x

Extraneous SolutionExtraneous Solution

Guided Practice

Solve2

3 3 60

2 2

x

x x x x

LCD:LCD: 2x x

Clear the fractions:Clear the fractions: 3 2 3 6 0x x x 2 6 3 6 0x x x

2 2 0x x 2 0x x

0, 2x

Guided Practice

Solve2

3 3 60

2 2

x

x x x x

Algebraic solutions:Algebraic solutions: 0, 2x

Both solutions are extraneous!!!Both solutions are extraneous!!!

Substituting either solution into the original equationSubstituting either solution into the original equationresults in division by zero…results in division by zero…

The original equation has no solution!!!The original equation has no solution!!!

Practice ProblemsPractice Problems

Solve

Clear fractions:

ExtraneousExtraneous

2

3 2 6

3 3

x

x x x x

23 2 6x x 2 4 3 0x x

1 3 0x x

1, 3x

Practice ProblemsPractice Problems

Solve

Clear fractions:

2

2

6 53

2

x x

x

2 26 5 3 2x x x

2 26 5 3 6x x x 22 6 11 0x x

3 311.284, 4.284

2x

Practice ProblemsPractice Problems

Solve

Clear fractions:

2

3 2 5

2 1 2

x

x x x x

3 1 2 2 5x x x

23 3 2 4 5x x x 23 1 0x x

1 130.768, 0.434

6x

Practice ProblemsPractice Problems

Solve

Clear fractions:

2

3 5 15

1 2 2

x

x x x x

3 2 5 1 15x x x

23 10 0x x

5,23

x

3 5 2 0x x

ExtraneousExtraneous

Practice ProblemsPractice Problems

Solve

Clear fractions:

2 37x

x

3 3 7x x 3 7 3 0x x

2.398, 0.441,2.838x Solve graphically!