$绝对值不等式的公式为：$

$||a|-|b||\le |a\pm b|\le |a|+|b|$

$\begin{array}{cc}-\left|a\right|\le a\le \left|a\right|& \text{①}\\ & \\ -\left|\text{b}\right|\le \text{b}\le \left|\text{b}\right|& \text{②}\\ & \\ -\left|\text{b}\right|\le \text{-b}\le \left|\text{b}\right|& \text{③}\end{array}$

$由①+②得：$
$-(\left|a\right|+\left|b\right|)\le a+b\le \left|a\right|+\left|b\right|\Rightarrow \left|a+b\right|\le \left|a\right|+\left|b\right|$

$由①+③得：$
$-(\left|a\right|+\left|b\right|)\le a-b\le \left|a\right|+\left|b\right|\Rightarrow |a-b|\le \left|a\right|+\left|b\right|$

$而$
$\begin{array}{l}\left|a\right|=\left|(a+b)-b\right|=\left|(a-b)+b\right|\\ \left|b\right|=\left|(b+a)-a\right|=\left|(b-a)+a\right|\end{array}$

$由\left|a+b\right|\le \left|a\right|+\left|b\right|得:$

$\begin{array}{lc}\left|a\right|=\left|(a+b)-b\right|\le \left|a+b\right|+\left|-b\right|\Rightarrow \left|a\right|-\left|b\right|\le \left|a+b\right|& \text{⑥}\\ \left|b\right|=\left|\left(b+a\right)-a\right|\le \left|b+a\right|+\left|-a\right|\Rightarrow \left|a\right|-\left|b\right|\ge -\left|a+b\right|& \text{⑦}\end{array}$

$\begin{array}{lc}\left|a\right|=\left|(a-b)+b\right|\le \left|a-b\right|+\left|b\right|=>\left|a\right|-\left|b\right|\le \left|a-b\right|& \text{⑧}\\ \left|b\right|=\left|(b-a)+a\right|\le \left|b-a\right|+\left|a\right|=>\left|a\right|-\left|b\right|\ge -\left|a-b\right|& \text{⑨}\end{array}$

$$\begin{gathered} 由⑥，⑦得：||a|-|b||\le |a+b| \\ 或者： \\ |a|=|a+b-b|\le |b|+|a+b|\to |a|-|b|\le |a+b| \\ a,b交换\to |b|-|a|\le |a+b| \end{gathered}$$

$$\begin{gathered} 由⑧，⑨得：||a|-|b||\le |a-b| \\ 或者： \\ |a|=|a-b+b|\le |b|+|a-b|\to |a|-|b|\le |a-b| \\ a,b交换\to |b|-|a|\le |a-b| \end{gathered}$$

$$\begin{gathered} 等号成立的条件（特别是求最值），即： \\ \begin{array}{l}\left|a-b\right|=\left|a\right|+\left|b\right|\to ab\le 0\\ \left|a\right|-\left|b\right|=\left|a+b\right|\to b(a+b)\le 0\\ \left|a\right|-\left|b\right|=\left|a-b\right|\to b(a-b)\ge 0\end{array} \end{gathered}$$

$注：利用\left|a-b\right|=\left|a\right|+\left|b\right|\to ab\le 0$

$$\begin{gathered} |a|-|b|=|a+b| \\ \to |a|=|b|+|a+b| \\ \to |a+b-b|=|b|+|a+b| \end{gathered}$$

$$\begin{gathered} \left|a\right|-\left|b\right|=\left|a-b\right| \\ \to |a|=|b|+|a-b| \\ \to |a-b+b|=|b|+|a-b| \end{gathered}$$