$1)$ Điều kiện : $\left| {x - 1} \right| - 1 > 0$
$\begin{array}{l}
{\log _2}\frac{1}{{\left| {x - 1} \right| - 1}} = 1 & \Leftrightarrow \frac{1}{{\left| {x - 1} \right| - 1}} = 2 & \Leftrightarrow \left| {x - 1} \right| - 1 = \frac{1}{2}\\
\Leftrightarrow \left| {x - 1} \right| = \frac{3}{2} & \Leftrightarrow x - 1 = \pm \frac{3}{2} & \Leftrightarrow \left[ \begin{array}{l}
x = \frac{5}{2}\\
x = - \frac{1}{2}
\end{array} \right.
\end{array}$
$2)$ $PT\Leftrightarrow
|x+1|-2=\frac{1}{4}\Leftrightarrow|x+1|=\frac{9}{4}$
$\Leftrightarrow
x+1=\pm\frac{9}{4}\Leftrightarrow\left[\begin{array}{I} x=\frac{5}{4}\\
x=-\frac{13}{4}\end{array}\right.$