\(\left|5x-1\right|=1+x\)
<=> \(\left[{}\begin{matrix}5x-1=-1-x\\5x-1=1+x\end{matrix}\right.\)
<=> \(\left[{}\begin{matrix}5x+x=-1+1\\5x-x=1+1\end{matrix}\right.\)
<=> \(\left[{}\begin{matrix}6x=0\\4x=2\end{matrix}\right.\)
<=> \(\left[{}\begin{matrix}x=0\\x=\dfrac{1}{2}\end{matrix}\right.\)