Answer
$x=\dfrac{3}{2}$
Work Step by Step
The given equation, $
|3x+5|=5x+2
,$ is equivalent to
\begin{array}{l}\require{cancel}
3x+5=5x+2
\\\\\text{OR}\\\\
3x+5=-(5x+2)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x+5=5x+2
\\
3x-5x=2-5
\\
-2x=-3
\\
\dfrac{-2x}{-2}=\dfrac{-3}{-2}
\\
x=\dfrac{3}{2}
\\\\\text{OR}\\\\
3x+5=-(5x+2)
\\
3x+5=-5x-2
\\
3x+5x=-2-5
\\
8x=-7
\\
\dfrac{8x}{8}=-\dfrac{7}{8}
\\
x=-\dfrac{7}{8}
.\end{array}
Since the right side of the given equation is not a constant, then checking of solution/s is required.
Substituting $
x=\dfrac{3}{2}
$ in the original equation results to
\begin{array}{l}\require{cancel}
|3x+5|=5x+2
\\
\left| 3\left(\dfrac{3}{2}\right)+5 \right|=5\left(\dfrac{3}{2}\right)+2
\\
\left| \dfrac{9}{2}+5 \right|=\dfrac{15}{2}+2
\\
\left| \dfrac{9}{2}+\dfrac{10}{2} \right|=\dfrac{15}{2}+\dfrac{4}{2}
\\
\left| \dfrac{19}{2} \right|=\dfrac{19}{2}
\\
\dfrac{19}{2}=\dfrac{19}{2}
\text{ (TRUE)}
.\end{array}
Substituting $
x=-\dfrac{7}{8}
$ in the original equation results to
\begin{array}{l}\require{cancel}
|3x+5|=5x+2
\\
\left| 3\left(-\dfrac{7}{8}\right)+5 \right|=5\left(-\dfrac{7}{8}\right)+2
\\
\left| -\dfrac{21}{8}+5 \right|=-\dfrac{35}{8}+2
\\
\left| -\dfrac{21}{8}+\dfrac{40}{8} \right|=-\dfrac{35}{8}+\dfrac{16}{8}
\\
\left| \dfrac{19}{8} \right|=-\dfrac{19}{8}
\\
\dfrac{19}{8}=-\dfrac{19}{8}
\text{ (FALSE)}
.\end{array}
Since the substitution above resulted in a FALSE statement, then $
x=-\dfrac{7}{8}
,$ is not a solution (an extraneous solution).
Hence, the solution is $
x=\dfrac{3}{2}
.$
