#### Answer

$x=\left\{ -\dfrac{9}{5},1 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|5x+2|=7
,$ use the definition of an absolute value equality. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
5x+2=7
\\\\\text{OR}\\\\
5x+2=-7
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
5x+2=7
\\\\
5x=7-2
\\\\
5x=5
\\\\
x=\dfrac{5}{5}
\\\\
x=1
\\\\\text{OR}\\\\
5x+2=-7
\\\\
5x=-7-2
\\\\
5x=-9
\\\\
x=-\dfrac{9}{5}
.\end{array}
Hence, $
x=\left\{ -\dfrac{9}{5},1 \right\}
.$