Answer
$x=\frac{-7}{3},9$
Work Step by Step
$$\left | 2x-1 \right \vert - \left | x+5 \right \vert=3$$
$$\left | 2x-1 \right \vert=\begin{cases}
2x-1, & \mbox{if }\; 2x-1 \geq0 \;\rightarrow\; x\geq1/2\\
1-2x, & \mbox{if }\; 2x-1\lt0 \;\rightarrow\; x\lt1/2
\end{cases}$$
$$\left | x+5 \right \vert=\begin{cases}
x+5, & \mbox{if }\; x+5\geq0\; \rightarrow\; x\geq-5\\
-x-5, & \mbox{if }\; x+5\lt0 \;\rightarrow\; x\lt-5
\end{cases}$$
$$x\lt5 \qquad -5\leq x\lt1/2 \qquad x\geq1/2$$
$$\mbox{If} \; x\lt-5 \\
1-2x-(-x-5)=3\\
1-2x+x+5=3\\
\color{Red}{x=3} $$
$$\mbox{If} \; -5\leq x\lt1/2 \\
1-2x-(x+5)=3\\
1-2x-x-5=3\\
\color{limegreen}{x=-7/3}$$
$$\mbox{If} \; x\geq1/2 \\
2x-1-(x+5)=3\\
2x-1-x-5=3\\
\color{limegreen}{x=9}$$
$$\mbox{Therefore, the solutions are} \; x=\frac{-7}{3}, 9$$
