Answer
$$\mbox{No solutions}$$
Work Step by Step
$$\left | x-1 \right \vert - \left | x-3 \right \vert\geq5$$
$$\left | x-1 \right \vert=\begin{cases} x-1, & \mbox{if }\; x-1 \geq0 \;\rightarrow\; x\geq1\\ 1-x, & \mbox{if }\; x-1\lt0 \;\rightarrow\; x\lt1 \end{cases}$$
$$\left | x-3 \right \vert=\begin{cases} x-3, & \mbox{if }\; x-3\geq0\; \rightarrow\; x\geq3\\ 3-x, & \mbox{if }\; x-3\lt0 \;\rightarrow\; x\lt3 \end{cases}$$
$$x\lt1 \qquad 1\leq x\lt3 \qquad x\geq3$$
$$\mbox{If} \; x\lt1 \\
1-x-(3-x)\geq5\\
1-x-3+x\geq5\\
\color{Red}{-2\geq5} $$
$$\mbox{If} \; 1\leq x\lt3 \\
x-1-(3-x)\geq5\\
x-1-3+x\geq5\\
\color{red}{x\geq9/2~~and~~x\lt 3}$$
$$\mbox{If} \; x\geq3 \\
x-1-(x-3)\geq5\\
x-1-x+3\geq5\\
\color{red}{2\geq5}$$
$$\mbox{Therefore, there are no solutions.}$$
