Answer
$(-\infty,-2]\cup[7,\infty)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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)
.$
