Evaluate: |x-1| < x

Expression: $|x-1| < x$

Move the variable to the left-hand side and change its sign

$|x-1|-x < 0$

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

$\begin{align*}&x \in \langle\frac{ 1 }{ 2 }, +\infty\rangle \\&\begin{array} { l }x > \frac{ 1 }{ 2 }\end{array}\end{align*}$