Intermediate Algebra: Connecting Concepts through Application

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

Chapter 2 - Systems of Linear Equations and Inequalities - 2.5 Absolute Value Equations and Inequalities - 2.5 Exercises - Page 188: 52


$w\lt-7 \text{ OR } w\gt3$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $ |w+2|\gt5 ,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. 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 $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} w+2\gt5 \\\\\text{OR}\\\\ w+2\lt-5 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} w+2\gt5 \\\\ w\gt5-2 \\\\ w\gt3 \\\\\text{OR}\\\\ w+2\lt-5 \\\\ w\lt-5-2 \\\\ w\lt-7 .\end{array} Hence, the solution set is $ w\lt-7 \text{ OR } w\gt3 .$ The graph above confirms the solution set of the inequality.
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