Beginning Algebra, 11th Edition

424 CHAPTER 7 Rational Expressions and Applications

In Example 5,notice that y-xis the opposite(or additive inverse) of x-y.

Quotient of Opposites

If the numerator and the denominator of a rational expression are opposites, as in

then the rational expression is equal to - 1.

x-y

y-x^ ,

Based on this result, the following are true.

and

• 5 a+ 2 b
  5 a- 2 b

=- 1

q- 7

7 - q

=- 1

Numerator and

denominator

are opposites.

However, the following expression cannot be simplified further.

Numerator and denominator

are notopposites.

Writing in Lowest Terms (Factors Are Opposites)

Write each rational expression in lowest terms.

(a) Since and are opposites, this expression equals

(b)

Factor the numerator and denominator.

Write in the denominator

as

Fundamental property

or

(c) is notthe opposite of

This rational expression is already in lowest terms. NOW TRY

OBJECTIVE 4 Recognize equivalent forms of rational expressions. The

common fraction can also be written and

Consider the final rational expression from Example 6(b).

The sign representing the factor is in front of the expression, even with the

fraction bar. The factor may instead be placed in the numerator or denominator.

Use parentheses.

and

2 x+ 3

• 2


• 12 x+ 32
  2

- 1

- - 1

-

2 x+ 3

2

5

• 
  
  
  
  6. 
     
  
  
  
  


• 5


• 6
  5
  6

3 - r 3 +r.

3 +r

3 - r

a

• b=-

a

• b

2 x+ 3

2

=

2 x+ 3

• 2

,

=

2 x+ 3

21 - 12

• 112 x- 32.

3 - 2 x

=

12 x+ 3212 x- 32

21 - 1212 x- 32

=

12 x+ 3212 x- 32

213 - 2 x 2

4 x^2 - 9

6 - 4 x

2 - m m- 2 - 1.

2 - m

m- 2

EXAMPLE 6

x- 2

x+ 2

NOW TRY
EXERCISE 6
Write each rational expres-
sion in lowest terms.

(a) (b)

(c)

x+y

x-y

4 m^2 - n^2

2 n- 4 m

p- 4

4 - p

NOW TRY ANSWERS

6. (a)
   (b) or
   (c)already in lowest terms

2 m-+ 2 n , - 2 m 2 +n

• 1

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