## Intermediate Algebra: Connecting Concepts through Application

$w\lt-7 \text{ OR } w\gt3$
$\bf{\text{Solution Outline:}}$ 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.