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Chapter 7: Problem 80

Explain how to add rational expressions when denominators are opposites. Use an example to support your explanation.

Short Answer

Expert verified
The addition of two rational expressions with opposite denominators \(\frac{1}{a}\) and \(\frac{2}{-a}\) will be \(\frac{-1}{a}\) after simplifying.

Step by step solution

01

Introduction

Consider two rational expressions with opposite denominators. For instance, \(\frac{1}{a}\) and \(\frac{2}{-a}\), where a is not equal to zero.
02

Add the Rational Expressions

Addition of rational expressions with opposite denominators can be done by changing the sign of the second rational expression and treating it as addition. This can be achieved by multiplying the numerator and denominator of the second fraction by -1; yielded as such \(\frac{1}{a} + (\frac{2}{-a} * -\frac{1}{1})\). The expression will now be \(\frac{1}{a} + \frac{-2}{a}\).
03

Simplify the Result

Now treat it as addition of fractions with like denominators, which is very straight-forward: add the numerators and write it over the common denominator. So, boil it down as \(\frac{1-2}{a}\) which simplifies to \(\frac{-1}{a}\). Hence the sum of the initial fractions.

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Most popular questions from this chapter

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{8}-\frac{5}{6}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{6 y^{2}+y}{2 y^{2}-9 y+9}-\frac{2 y+9}{2 y^{2}-9 y+9}-\frac{4 y-3}{2 y^{2}-9 y+9}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-3}+\frac{9}{3-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x+7}-1$$

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}+4 x+3}{x+2}-\frac{5 x+9}{x+2}=x-2, x \neq-2$$
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