# Chapter 9 - Inequalities and Problem Solving - 9.3 Absolute-Value Equations and Inequalities - 9.3 Exercise Set: 84

$-\dfrac{13}{2} \lt a \lt \dfrac{5}{2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $30-4|a+2|\gt12 ,$ isolate first the absolute value expression. Then use the definition of a less than (less than or equal to) absolute value inequality. 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 $\ge.$ $\bf{\text{Solution Details:}}$ Using the properties of inequality to isolate the absolute value expression results to \begin{array}{l}\require{cancel} 30-4|a+2|\gt12 \\\\ -4|a+2|\gt12-30 \\\\ -4|a+2|\gt-18 .\end{array} Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to \begin{array}{l}\require{cancel} -4|a+2|\gt-18 \\\\ |a+2|\lt\dfrac{-18}{-4} \\\\ |a+2|\lt\dfrac{9}{2} .\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 is equivalent to \begin{array}{l}\require{cancel} -\dfrac{9}{2}\lt a+2 \lt\dfrac{9}{2} .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -\dfrac{9}{2}\lt a+2 \lt\dfrac{9}{2} \\\\ -\dfrac{9}{2}-2\lt a+2-2 \lt\dfrac{9}{2}-2 \\\\ -\dfrac{9}{2}-\dfrac{4}{2}\lt a \lt\dfrac{9}{2}-\dfrac{4}{2} \\\\ -\dfrac{13}{2} \lt a \lt \dfrac{5}{2} .\end{array} Hence, the solution set $-\dfrac{13}{2} \lt a \lt \dfrac{5}{2} .$

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