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8.5 Adding and subtracting rational expressions

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<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be familiar with the basic rule for adding and subtracting rational expressions, be able to add and subtract fractions with the same and with different denominators.</para>

Overview

  • Basic Rule
  • Fractions with the Same Denominator
  • Fractions with Different Denominators

Basic rule

We are now in a position to study the process of adding and subtracting rational expressions. There is a most basic rule to which we must strictly adhere if we wish to conveniently add or subtract rational expressions.

To add or subtract rational expressions conveniently, they should have the same denominators.

Thus, to add or subtract two or more rational expressions conveniently, we must ensure that they all have the same denominator. The denominator that is most convenient is the LCD.

Fractions with the same denominator

The rule for adding and subtracting rational expressions

To add (or subtract) two or more rational expressions with the same denominators, add (or subtract) the numerators and place the result over the LCD. Reduce if necessary. Symbolically,

a c + b c = a + b c
a c − b c = a − b c

Note that we combine only the numerators.

Sample set a

Add or subtract the following rational expressions.

1 6 + 3 6 The denominators are the same . Add the numerators . 1 6 + 3 6 = 1 + 3 6 = 4 6 Reduce . 1 6 + 3 6 = 2 3

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5 x + 8 x The denominators are the same . Add the numerators . 5 x + 8 x = 5 + 8 x = 13 x

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2 a b y 2 w − 5 b y 2 w The denominators are the same . Subtract the numerators . 2 a b y 2 w − 5 b y 2 w = 2 a b − 5 b y 2 w

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3 x 2 + x + 2 x − 7 + x 2 − 4 x + 1 x − 7 The denominators are the same . Add the numerators . 3 x 2 + x + 2 x − 7 + x 2 − 4 x + 1 x − 7 = 3 x 2 + x + 2 + x 2 − 4 x + 1 x − 7 = 4 x 2 − 3 x + 3 x − 7

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5 y + 3 2 y − 5 − 2 y + 4 2 y − 5 The denominators are the same . Subtract the numerators .  But  b e   c a r e f u l  to subtract the  e n t i r e  numerator . Use parentheses! 5 y + 3 2 y − 5 − 2 y + 4 2 y − 5 = 5 y + 3 − ( 2 y + 4 ) 2 y − 5 = 5 y + 3 − 2 y − 4 2 y − 5 = 3 y − 1 2 y − 5 N o t e : 5 y + 3 2 y − 5 − 2 y + 4 2 y − 5 ︸ Observe this part . The term − 2 y + 4 2 y − 5  could be written as + − ( 2 y + 4 ) 2 y − 5 = − 2 y − 4 2 y − 5

A common mistake is to write

− 2 y + 4 2 y − 5  as  − 2 y + 4 2 y − 5
This is not correct, as the negative sign is not being applied to the entire numerator.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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