Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$x\lt-2 \text{ or } x\gt8$
$\bf{\text{Solution Outline:}}$ Solve the given inequality, $|x-3|+2\gt7 ,$ by isolating first the absolute value expression. Then use the definition of a greater than absolute value inequality. 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, the given expression is equivalent to \begin{array}{l}\require{cancel} |x-3|+2\gt7 \\\\ |x-3|\gt7-2 \\\\ |x-3|\gt5 .\end{array} 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 given inequality is equivalent to \begin{array}{l}\require{cancel} x-3\gt5 \\\\\text{OR}\\\\ x-3\lt-5 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} x-3\gt5 \\\\ x\gt5+3 \\\\ x\gt8 \\\\\text{OR}\\\\ x-3\lt-5 \\\\ x\lt-5+3 \\\\ x\lt-2 .\end{array} Hence, the solution set is $x\lt-2 \text{ or } x\gt8 .$