#### Answer

$x=\left\{ -14,\dfrac{4}{3} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|2x+5|=|x-9|
,$ use the definition of absolute value equality. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since $|x|=|y|$ implies $x=y \text{ or } x=-y,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
2x+5=x-9
\\\\\text{OR}\\\\
2x+5=-(x-9)
.\end{array}
Using the properties of equality to isolate the variable in each equation results to
\begin{array}{l}\require{cancel}
2x+5=x-9
\\\\
2x-x=-9-5
\\\\
x=-14
\\\\\text{OR}\\\\
2x+5=-(x-9)
\\\\
2x+5=-x+9
\\\\
2x+x=9-5
\\\\
3x=4
\\\\
x=\dfrac{4}{3}
.\end{array}
Hence, $
x=\left\{ -14,\dfrac{4}{3} \right\}
.$