College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter R - Section R.5 - Rational Expressions - R.5 Exercises - Page 48: 66

Answer

$\dfrac{(x+2)(x-1)}{x(x+1)}$

Work Step by Step

The given expression, $ \dfrac{2+\dfrac{2}{1+x}}{2-\dfrac{2}{1-x}} ,$ simplifies to \begin{array}{l}\require{cancel} \dfrac{\dfrac{2(1+x)+2}{1+x}}{\dfrac{2(1-x)-2}{1-x}} \\\\= \dfrac{\dfrac{2+2x+2}{1+x}}{\dfrac{2-2x-2}{1-x}} \\\\= \dfrac{\dfrac{2x+4}{1+x}}{\dfrac{-2x}{1-x}} \\\\= \dfrac{2x+4}{1+x}\div\dfrac{-2x}{1-x} \\\\= \dfrac{2x+4}{1+x}\cdot\dfrac{1-x}{-2x} \\\\= \dfrac{2(x+2)}{1+x}\cdot\dfrac{1-x}{-2x} \\\\= \dfrac{\cancel{2}(x+2)}{1+x}\cdot\dfrac{1-x}{-\cancel{2}x} \\\\= \dfrac{x+2}{1+x}\cdot\dfrac{1-x}{-x} \\\\= \dfrac{(x+2)(1-x)}{-x(1+x)} \\\\= -\dfrac{(x+2)(1-x)}{x(x+1)} \\\\= \dfrac{(x+2)(x-1)}{x(x+1)} .\end{array}
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