Intermediate Algebra (12th Edition)

$\left( -25,15 \right)$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|r+5| \lt 20 ,$ use the definition of absolute value inequalities. Use the properties of inequalities 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 $c\gt0$, $|x|\lt c$ 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} -20 \lt r+5 \lt 20 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -20-5 \lt r+5-5 \lt 20-5 \\\\ -25 \lt r \lt 15 .\end{array} In interval notation, the solution set is $\left( -25,15 \right) .$ The colored graph is the graph of the solution set.