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: 86


$\left[ -5,9 \right]$

Work Step by Step

Using the properties of inequality, the given statement, $ |x-2|-3\le4 ,$ is equivalent to \begin{array}{l}\require{cancel} |x-2|\le4+3 \\\\ |x-2|\le7 .\end{array} Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a$, then the inequality above is equivalent to \begin{array}{l}\require{cancel} -7\le x-2\le7 .\end{array} Using properties of inequality, then \begin{array}{l}\require{cancel} -7\le x-2\le7 \\\\ -7+2\le x-2+2\le7+2 \\\\ -5\le x\le9 .\end{array} Hence, the solution set is the interval $ \left[ -5,9 \right] .$
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