a) \(\left|9+x\right|=2x\)

\(\Rightarrow\left[{}\begin{matrix}9+x=2x\\9+x=-2x\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}2x-x=9\\-2x-x=9\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=9\\-3x=9\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=9\\x=-3\end{matrix}\right.\)

b) \(\left|5x\right|-3x=2\)

\(\Leftrightarrow\left|5x\right|=2+3x\)

\(\Rightarrow\left[{}\begin{matrix}5x=2+3x\\5x=-2-3x\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}5x-3x=2\\5x+3x=-2\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}2x=2\\8x=-2\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=1\\x=-\dfrac{1}{4}\end{matrix}\right.\)

c) \(\left|x+6\right|-9=2x\)

\(\Leftrightarrow\left|x+6\right|=2x+9\)

\(\Rightarrow\left[{}\begin{matrix}x+6=2x+9\\x+6=-2x-9\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x-2x=9-6\\x+2x=-9-6\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}-3x=3\\3x=-15\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=-1\\x=-5\end{matrix}\right.\)

d) \(\left|2x-3\right|+x=21\)

\(\Leftrightarrow\left|2x-3\right|=21-x\)

\(\Rightarrow\left[{}\begin{matrix}2x-3=21-x\\2x-3=-21+x\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}2x+x=21+3\\2x-x=-21+3\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}3x=24\\x=-18\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=8\\x=-18\end{matrix}\right.\)