Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises: 76

Answer

$x=\left\{ 3,5 \right\}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ 5|x-4|=5 ,$ use the properties of equality to isolate the absolute value expression. Then use the definition of absolute value equality. Use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the properties of equality to isolate the absolute value expression results to \begin{array}{l}\require{cancel} |x-4|=\dfrac{5}{5} \\\\ |x-4|=1 .\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} x-4=1 \\\\\text{OR}\\\\ x-4=-1 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} x-4=1 \\\\ x=1+4 \\\\ x=5 \\\\\text{OR}\\\\ x-4=-1 \\\\ x=-1+4 \\\\ x=3 .\end{array} Hence, $ x=\left\{ 3,5 \right\} .$
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