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: 74

Answer

$\text{The simplified form of the given complex fraction is } \dfrac{x^{2} -x-8}{x^{2} +x-2}$.

Work Step by Step

$\begin{array}{ l l } =\dfrac{( x( x-2))\left(\dfrac{x+4}{x} -\dfrac{3}{x-2}\right)}{( x( x-2))\left(\dfrac{x}{x-2} +\dfrac{1}{x}\right)} & \begin{array}{l} \mathrm{Multiply\ the\ numerator\ and\ }\\ \mathrm{denominator\ of\ the\ complex\ }\\ \mathrm{fraction\ by\ its\ LCD}\\ x( x-2) \end{array}\\ & \\ =\dfrac{( x( x-2))\dfrac{x+4}{x} -( x( x-2))\dfrac{3}{x-2}}{( x( x-2))\dfrac{x}{x-2} +( x( x-2))\dfrac{1}{x}} & \mathrm{Apply\ the\ distributive\ property}\\ & \\ =\dfrac{( x-2)( x+4) -3x}{x( x) +( x-2)( 1)} & \mathrm{Cancel\ factors\ of\ one}\\ & \\ =\dfrac{\left( x^{2} +2x-8\right) -3x}{x^{2} +x-2} & \begin{array}{l} \mathrm{Expand} \ ( x-2)( x+4) \ \mathrm{using\ FOIL}\\ \mathrm{Multiply\ the\ all\ other\ terms} \end{array}\\ & \\ =\dfrac{x^{2} +2x-3x-8}{x^{2} +x-2} & \begin{array}{l} \mathrm{Group\ like\ terms\ together\ in\ the\ }\\ \mathrm{numerator} \end{array}\\ & \\ =\dfrac{x^{2} -x-8}{x^{2} +x-2} & \mathrm{Combine\ like\ terms.} \end{array}$
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