#### Answer

$x=\left\{ -\dfrac{19}{2},\dfrac{9}{2} \right\}$

#### Work Step by Step

Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|2x+5|=14
,$ is equivalent to
\begin{array}{l}\require{cancel}
2x+5=14 \text{ OR } 2x+5=-14
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
2x+5=14
\\\\
2x=14-5
\\\\
2x=9
\\\\
x=\dfrac{9}{2}
\\\\\text{ OR }\\\\
2x+5=-14
\\\\
2x=-14-5
\\\\
2x=-19
\\\\
x=-\dfrac{19}{2}
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{19}{2},\dfrac{9}{2} \right\}
.$