Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 1 - Expressions, Equations, and Inequalities - 1-6 Absolute Value Equations and Inequalities - Practice and Problem-Solving Exercises - Page 46: 29

Answer

$-\dfrac{7}{2}\le w\le\dfrac{1}{2}$

Work Step by Step

Using the properties of inequality, the given, $ 4|2w+3|-7\le9 ,$ is equivalent to \begin{array}{l}\require{cancel} 4|2w+3|-7+7\le9+7 \\ 4|2w+3|\le16 \\\\ \dfrac{4|2w+3|}{4}\le\dfrac{16}{4} \\\\ |2w+3|\le4 .\end{array} 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 implies \begin{array}{l}\require{cancel} -4\le2w+3\le4 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -4-3\le2w+3-3\le4-3 \\ -7\le2w\le1 \\\\ -\dfrac{7}{2}\le\dfrac{2w}{2}\le\dfrac{1}{2} \\\\ -\dfrac{7}{2}\le w\le\dfrac{1}{2} .\end{array} Hence, the solution is $ -\dfrac{7}{2}\le w\le\dfrac{1}{2} .$ The graph of the solution above is shown below.
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