$|x-1|-x < 0$
Separate the inequality into $2$ possible cases$\begin{array} { l }\begin{array} { l }x-1-x < 0,& x-1 \geq 0\end{array},\\\begin{array} { l }-\left( x-1 \right)-x < 0,& x-1 < 0\end{array}\end{array}$
Solve the inequality for $x$$\begin{array} { l }\begin{array} { l }x \in ℝ,& x-1 \geq 0\end{array},\\\begin{array} { l }-\left( x-1 \right)-x < 0,& x-1 < 0\end{array}\end{array}$
Solve the inequality for $x$$\begin{array} { l }\begin{array} { l }x \in ℝ,& x \geq 1\end{array},\\\begin{array} { l }-\left( x-1 \right)-x < 0,& x-1 < 0\end{array}\end{array}$
Solve the inequality for $x$$\begin{array} { l }\begin{array} { l }x \in ℝ,& x \geq 1\end{array},\\\begin{array} { l }x > \frac{ 1 }{ 2 },& x-1 < 0\end{array}\end{array}$
Solve the inequality for $x$$\begin{array} { l }\begin{array} { l }x \in ℝ,& x \geq 1\end{array},\\\begin{array} { l }x > \frac{ 1 }{ 2 },& x < 1\end{array}\end{array}$
Find the intersection$\begin{array} { l }x \in \left[ 1, +\infty\right\rangle,\\\begin{array} { l }x > \frac{ 1 }{ 2 },& x < 1\end{array}\end{array}$
Find the intersection$\begin{array} { l }x \in \left[ 1, +\infty\right\rangle,\\x \in \langle\frac{ 1 }{ 2 }, 1\rangle\end{array}$
Find the union$\begin{align*}&x \in \langle\frac{ 1 }{ 2 }, +\infty\rangle \\&\begin{array} { l }x > \frac{ 1 }{ 2 }\end{array}\end{align*}$