\(\Leftrightarrow\left|2x-1\right|=1+2x\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-1=1+2x\\2x-1=-1-2x\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-2x-1-1=0\\2x-1+1+2x=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}-2=0\left(VL\right)\\4x=0\end{matrix}\right.\)
\(\Leftrightarrow x=0\)