College Algebra (11th Edition)

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

Chapter 1 - Section 1.8 - Absolute Value Equations and Inequalities - 1.8 Exercises - Page 154: 46


$x=\left\{ \dfrac{1}{2},\dfrac{3}{2} \right\}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ |4-4x|+2=4 ,$ use the properties of equality to isolate the absolute value expression. Then use the properties of absolute value equality. $\bf{\text{Solution Details:}}$ Using the properties of equality, the equation above is equivalent to \begin{array}{l}\require{cancel} |4-4x|=4-2 \\\\ |4-4x|=2 .\end{array} Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to \begin{array}{l}\require{cancel} 4-4x=2 \\\\\text{OR}\\\\ 4-4x=-2 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} 4-4x=2 \\\\ -4x=2-4 \\\\ -4x=-2 \\\\ x=\dfrac{-2}{-4} \\\\ x=\dfrac{1}{2} \\\\\text{OR}\\\\ 4-4x=-2 \\\\ -4x=-2-4 \\\\ -4x=-6 \\\\ x=\dfrac{-6}{-4} \\\\ x=\dfrac{3}{2} .\end{array} Hence, $ x=\left\{ \dfrac{1}{2},\dfrac{3}{2} \right\} .$
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