Intermediate Algebra (12th Edition)

$(-\infty,-2]\cup[7,\infty)$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|2x-5|\ge9 ,$ use the definition of the absolute value greater than a constant. $\bf{\text{Solution Details:}}$ 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} 2x-5\ge9 \\\\\text{OR}\\\\ 2x-5\le-9 .\end{array} Solving each inequality results to \begin{array}{l}\require{cancel} 2x-5\ge9 \\\\ 2x\ge9+5 \\\\ 2x\ge14 \\\\ x\ge\dfrac{14}{2} \\\\ x\ge7 \\\\\text{OR}\\\\ 2x-5\le-9 \\\\ 2x\le-9+5 \\\\ 2x\le-4 \\\\ x\le-\dfrac{4}{2} \\\\ x\le-2 .\end{array} Hence, the solution set is the interval $(-\infty,-2]\cup[7,\infty) .$