\(\left|-6\right|-\left(\left|x+1\right|-3\right)=-\left(-3\right)\)
\(6-\left(\left|x+1\right|-3\right)=3\)
\(\left(\left|x+1\right|-3\right)=6-3=3\)
\(\left|x+1\right|=3+3=6\)
\(\orbr{\begin{cases}x+1=6\Rightarrow x=5\\x+1=-6\Rightarrow x=-7\end{cases}}\)