Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 1-2 - Cumulative Review: 47

Answer

$-5\lt x \lt 9$
1514190563

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $ |x-2|\lt7 ,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. Finally, graph the solution set. In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} -7\lt x-2 \lt7 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -7+2\lt x-2+2 \lt7+2 \\\\ -5\lt x \lt 9 .\end{array} Hence, the solution set is $ -5\lt x \lt 9 .$
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