Let $t \mapsto |t|$ be the D412: Absolute value function.
Let $x, y \in \mathbb{R}$ each be a D993: Real number.
Let $x, y \in \mathbb{R}$ each be a D993: Real number.
Then
\begin{equation}
\bigg| |x| - |x| \bigg|
\leq |x - y| \leq |x| + |y|
\end{equation}