check the picture below.
keep in mind that an absolute value expression is in effect a piece-wise.
[tex]\bf |x-4|\le 2\implies
\begin{cases}
+(x-4)\le 2\\
-------\\
-(x-4)\le 2\\
x-4\ge -2
\end{cases}\implies
\begin{array}{llll}
-2\le x-4\le 2\\\\
\textit{adding 4 to all 3 sides}\\\\
\boxed{2\le x \le 6}
\end{array}\\\\
-------------------------------\\\\
|x-5|\ \textgreater \ 3\implies
\begin{cases}
+(x-5)\ \textgreater \ 3\\
-------\\
-(x-5)\ \textgreater \ 3\\
x-5\ \textless \ -3
\end{cases}\implies
\begin{array}{llll}
3\ \textless \ x-5\ \textless \ -3\\\\
\textit{adding 5 to all 3 sides}\\\\
\boxed{8\ \textless \ x\ \textless \ 2}
\end{array}[/tex]
