Chapter 1-2 - Cumulative Review - Page 219: 48

$-9\le x \le6$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|2x+3|\le15 ,$ 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} -15\le 2x+3 \le15 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -15\le 2x+3 \le15 \\\\ -15-3\le 2x+3-3 \le15-3 \\\\ -18\le 2x \le12 \\\\ -\dfrac{18}{2}\le \dfrac{2x}{2} \le\dfrac{12}{2} \\\\ -9\le x \le6 .\end{array} Hence, the solution set is $-9\le x \le6 .$

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