Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter R - Elementary Algebra Review - R.6 Rational Expressions and Equations - R.6 Exercise Set - Page 980: 27


The simplified form of the expression $\frac{x}{x-2}+\frac{2}{2-x}$ is 1.

Work Step by Step

$\frac{x}{x-2}+\frac{2}{2-x}$ Obtain the alternative form of the expression by multiplying the rational expression $\frac{x}{x-2}+\frac{2}{2-x}$ by 1 in the form $\frac{-1}{-1}$. $\begin{align} & \frac{x}{x-2}+\frac{2}{2-x}=\frac{x}{x-2}+\frac{2}{2-x}.1 \\ & =\frac{x}{x-2}+\frac{2}{2-x}.\frac{-1}{-1} \end{align}$ Apply the Distributive property: $\begin{align} & \frac{x}{x-2}+\frac{2}{2-x}=\frac{x}{x-2}+\frac{2}{2-x}.\frac{-1}{-1} \\ & =\frac{x}{x-2}+\frac{-2}{-\left( 2-x \right)} \\ & =\frac{x}{x-2}+\frac{-2}{x-2} \end{align}$ Now, the denominators are same. So, add the numerators and keep the common denominator: $\begin{align} & \frac{x}{x-2}+\frac{2}{2-x}=\frac{x}{x-2}+\frac{-2}{x-2} \\ & =\frac{x-2}{x-2} \end{align}$ Remove the factor equal to 1, $\frac{x}{x-2}+\frac{2}{2-x}=1$
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