## Intermediate Algebra (12th Edition)

$\left( -\infty, -25 \right] \cup \left[ 15,\infty \right)$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|r+5|\ge20 ,$ use the definition of absolute value inequalities. Then use the properties of inequality to isolate the variable. For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$ For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ 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} r+5\ge20 \\\\\text{OR}\\\\ r+5\le-20 .\end{array} Using the properties of inequality to isolate the variable results to \begin{array}{l}\require{cancel} r+5\ge20 \\\\ r\ge20-5 \\\\ r\ge15 \\\\\text{OR}\\\\ r+5\le-20 \\\\ r\le-20-5 \\\\ r\le-25 .\end{array} In interval notation, the solution set is $\left( -\infty, -25 \right] \cup \left[ 15,\infty \right) .$ The colored graph is the graph of the solution set.