Solving Rational Equations

2

A rational expression is a fraction with polynomials for the numerator and denominator.

1

3 and ,

3

2 ,

1 2

x

x

xxare rational expressions. For example,

3

A rational equation is an equation between rational expressions.

For example, and are rational equations.3

21

xx xx

x

x

x

2

1

1

33

2

We are never allowed to divide by zero, so we need to exclude values that make the denominator zero.

To do this we take each denominator that contains a variable and say that it should never be zero. We then solve this to get the excluded values also known as extraneous solutions

Example:

3

030

3

21

x

xandx

SolutionsExtraneousxx

4

There are two ways to solve a rational equation.Both ways will give you the same answer and you can choose which one you prefer

The first way involves multiplying each term by the LCD to get rid of the denominators.

The second way involves making all the denominators into the LCD and then “getting rid” of the denominator

I will show you both and you decide which one you prefer.

Both ways require you to find extraneous solutions first!

5

First way.

5. Check the solutions against the extraneous solutions.

4. Solve the resulting polynomial equation.

3. Clear denominators by multiplying each term on both sides of the equation by the LCD.

1. Find the extraneous solutions.

To solve a rational equation:

2. Find the LCD of the denominators.

3

21

xx

3

030

x

xandx

SolutionsExtraneous

)3( xxLCD

1

)3(

3

2

1

)3(1

xx

x

xx

x

3

3

23

x

x

xx

6

Second way.

5. Check the solutions against the extraneous solutions.

4. Now that the denominators are equal we can reason logically that the numerators then have to be equal.

3. Turn each term’s denominator into the LCD.

1. Find the extraneous solutions.

To solve a rational equation:

2. Find the LCD of the denominators.

3

21

xx

3

030

x

xandx

SolutionsExtraneous

)3( xxLCD

)3(

2

)3(

3

3

2

)3(

)3(1

xx

x

xx

x

x

x

xx

x

x

3

3

23

x

x

xx

8

Examples: 1. Solve: 3

1

3

1

x

x

x

Find the LCD.

Turn each denominator into LCD

Numerators have to be equal.

LCD = x – 3.

x = 0

Find the extraneous solutions.

1 = x + 1

03 x

3x

3

1

3

1

x

x

x

Check solution against Extraneous solutions and cross off if it is the same.

Second way

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Examples: 2. Solve:

Find the LCD.

Make each denominator into LCD

If denominators are equal then numerators are too

LCD = x(x – 1).

-x = 1

Find the extraneous solutions.

x - 1 = 2x

001 xorx

01 xorx

Check solution against Extraneous solutions and cross off if it is the same.

1

21

xx

)1(

2

)1(

1

1

2

)1(

)1(1

xx

x

xx

x

x

x

xx

x

x

x = -1

Second way

Solve linear equation

12

Examples: 3. Solve:

Find the LCD.

Turn each term into LCD

Solve for x.

LCD =(x+1)(x – 1).

Find the extraneous solutions.

3x +1 = x - 1

01012 xorx

10)1)(1( xorxx

Check solution against Extraneous solutions and cross off if it is the same.

1

1

1

132

xx

x

10101 xorxorx

111 xorxorx

)1)(1(

1

)1)(1(

13

)1(

)1(

1

1

)1)(1(

13

xx

x

xx

x

x

x

xxx

x

2x = -2

x = -1

Second way

16

Example: Solve: .158

6

3 2

xxx

x

Factor.

Polynomial Equation.

Simplify.

Factor.

The LCM is (x – 3)(x – 5).x2 – 8x + 15 = (x – 3)(x – 5)

x(x – 5) = – 6

x2 – 5x + 6 = 0(x – 2)(x – 3) = 0

x = 2 or x = 3

)5)(3(

6

3

xxx

xOriginal Equation.

5,3 xxSecond way

)5)(3(

6

)5)(3(

)5(

)5)(3(

6

5

5

3

xxxx

xx

xxx

x

x

x

17

Example: Solve: .158

6

3 2

xxx

x

Polynomial Equation.

Check solutions against

extraneous solutions

Factor.

x2 – 5x + 6 = 0

(x – 2)(x – 3) = 0

x = 2 or x = 3

5,3 xxSecond way

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Your turn: Solve: