## Intermediate Algebra (12th Edition)

$\left[ 2,8 \right]$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|5-x| \le 3 ,$ 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} -3 \le 5-x \le 3 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -3-5 \le 5-x-5 \le 3-5 \\\\ -8 \le -x \le -2 .\end{array} Dividing by a negative number (and consequently reversing the sign), the inequality above is equivalent to \begin{array}{l}\require{cancel} \dfrac{-8}{-1} \ge \dfrac{-x}{-1} \ge \dfrac{-2}{-1} \\\\ 8 \ge x \ge 2 \\\\ 2 \le x \le 8 .\end{array} In interval notation, the solution set is $\left[ 2,8 \right] .$ The colored graph is the graph of the solution set.