## Intermediate Algebra: Connecting Concepts through Application

$-1\le h\le7$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|h-3|\le4 ,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. Graph the solution. 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} -4\le h-3\le4 .\end{array} Using the properties of inequality to isolate the variable results to \begin{array}{l}\require{cancel} -4+3\le h-3+3\le4+3 \\\\ -1\le h\le7 .\end{array} The graph above confirms the solution set of the inequality.