\begin{equation*}
\ ||z_{1}|-|z_{2}||\le|z_{1}+z_{2}|\le|z_{1}|+|z_{2}|
\end{equation*}
The second inequality follows from the triangle inequality: Draw \(z_{1}\) and \(z_{2}\) as vectors and take their sum to form a triangle.
The first inequality arises from:
\begin{equation*}
\ |x|=|(x-y)+y| \leq|x-y|+|y| \Rightarrow|x|-|y| \leq|x-y|
\end{equation*}
and
\begin{equation*}
\ |y|=|(y-x)+x| \leq|y-x|+|x| \Rightarrow|x|-|y| \geq-|x-y|
\end{equation*}
Combining these two gives us the first inequality.