Answer
$x=\left\{ \dfrac{1}{6},\dfrac{3}{2} \right\}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|-6x+5|=4
,$ is equivalent to
\begin{array}{l}\require{cancel}
-6x+5=4 \text{ OR } -6x+5=-4
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
-6x+5=4
\\\\
-6x=4-5
\\\\
-6x=-1
\\\\
x=\dfrac{-1}{-6}
\\\\
x=\dfrac{1}{6}
\\\\\text{ OR }\\\\
-6x+5=-4
\\\\
-6x=-4-5
\\\\
-6x=-9
\\\\
x=\dfrac{-9}{-6}
\\\\
x=\dfrac{3}{2}
.\end{array}
Hence, the solutions are $
x=\left\{ \dfrac{1}{6},\dfrac{3}{2} \right\}
.$