Beginning Algebra, 11th Edition

Solving an Equation with Rational Expressions

Solve, and check the proposed solution.

Multiply each side by

the LCD,

Distributive property

Simplify.

Combine like terms.

Subtract 2x.

x= 2 Multiply by -1.

• x=- 2

x=- 2 + 2 x

x= 2 + 2 x- 4

1 x- 22 a

x

x- 2

b = 1 x- 22 a

2

x- 2

b + 1 x- 22122

1 x- 22 a x-2.

x

x- 2

b = 1 x- 22 a

2

x- 2

+ 2 b

x

x- 2

=

2

x- 2

+ 2

EXAMPLE 3

458 CHAPTER 7 Rational Expressions and Applications

NOW TRY
EXERCISE 3
Solve, and check the proposed
solution.

4 +

6

x- 3

=

2 x

x- 3

NOW TRY ANSWER

3. 
   

x cannotequal 2, since 2

causes both denominators

to equal 0.

As noted, xcannot equal 2, since replacing xwith 2 in the original equation causes

the denominators to equal 0.

CHECK Original equation

Let.

Subtract in the denominators.

2

0

^2

0

+ 2

x= 2

2

2 - 2

^2

2 - 2

+ 2

x

x- 2

=

2

x- 2

+ 2

Division by 0 is

undefined.

Thus, 2 must be rejected as a solution, and the solution set is 0. NOW TRY

A proposed solution that is not an actual solution of the original equation, such as

2 in Example 3,is called an extraneous solution,or extraneous value.Some

students like to determine which numbers cannot be solutions beforesolving the

equation, as we did in Example 3.

Solving an Equation with Rational Expressions

Step 1 Multiply each side of the equation by the LCDto clear the equa-

tion of fractions. Be sure to distribute to everyterm on bothsides.

Step 2 Solvethe resulting equation.

Step 3 Checkeach proposed solution by substituting it into the original

equation. Reject any that cause a denominator to equal 0.

Solving an Equation with Rational Expressions

Solve, and check the proposed solution.

Step 1

Factor the denominators to

find the LCD, x 1 x+ 121 x- 12.

2

x 1 x- 12

=

1

1 x+ 121 x- 12

2

x^2 - x

=

1

x^2 - 1

EXAMPLE 4

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