طبق نامساوی مثلثی به بررسی تک‌تک گزینه‌ها می‌پردازیم:

$\begin{align}
  & \left| a+b \right|\le \left| a \right|+\left| b \right| \\
 & 1)\,\,\xrightarrow[{}]{b\to -b}\left| a-b \right|\le \left| a \right|+\left| -b \right|\Rightarrow \left| a-b \right|\le \left| a \right|+\left| b \right| \\
 & 2)\,\,\xrightarrow[{}]{b\to -a+b}\left| a-a+b \right|\le \left| a \right|+\left| -a+b \right|\Rightarrow \left| b \right|\le \left| a \right|+\left| a-b \right|\Rightarrow \left| a-b \right|\ge \left| b \right|-\left| a \right|\,\,\,\,\,(1) \\
 & \xrightarrow[{}]{a\to a-b}\left| a-b+b \right|\le \left| a-b \right|+\left| b \right|\Rightarrow \left| a \right|\le \left| a-b \right|+\left| b \right|\Rightarrow \left| a-b \right|\ge \left| a \right|-\left| b \right|\,\,\,\,(2) \\
 & 3)\,\,\xrightarrow[b\to 2b-a]{a\to a-b}\left| a-b+2b-a \right|\le \left| a-b \right|+\left| 2b-a \right|\Rightarrow \left| b \right|\le \left| a-b \right|+\left| a-2b \right| \\
 & 4)\,\,\xrightarrow[{}]{(2),(1)}\left\{ \begin{matrix}
   \left| a-b \right|\ge \left| a \right|-\left| b \right|  \\
   \left| a-b \right|\ge \left| b \right|-\left| a \right|  \\
\end{matrix}\Rightarrow \left| a-b \right|\ge \left| \left| a \right|-\left| b \right| \right| \right. \\
\end{align}$