绝对值不等式的公式为：绝对值不等式的公式为：绝对值不等式的公式为：
∣∣a∣−∣b∣∣≤∣a±b∣≤∣a∣+∣b∣||a|-|b||\leq|a\pm b|\leq|a|+|b|∣∣a∣−∣b∣∣≤∣a±b∣≤∣a∣+∣b∣
−∣a∣≤a≤∣a∣①−∣b∣≤b≤∣b∣②−∣b∣≤-b≤∣b∣③\begin{matrix}-\left|a\right|\leq a\leq\left|a\right| &\text{①}\\ \\ -\left|\text{b}\right|\leq\text{b}\leq\left|\text{b}\right|&\text{②}\\ \\ -\left|\text{b}\right|\leq\text{-b}\leq\left|\text{b}\right|&\text{③}\end{matrix}−∣a∣≤a≤∣a∣−∣b∣≤b≤∣b∣−∣b∣≤-b≤∣b∣①②③
由①+②得：由①+②得：由①+②得：
−(∣a∣+∣b∣)≤a+b≤∣a∣+∣b∣⇒∣a+b∣≤∣a∣+∣b∣-(\left|a\right|+\left|b\right|)\leq a+b\leq\left|a\right|+\left|b\right|\Rightarrow \left|a+b\right|\le\left|a\right|+\left|b\right|−(∣a∣+∣b∣)≤a+b≤∣a∣+∣b∣⇒∣a+b∣≤∣a∣+∣b∣
由①+③得：由①+③得：由①+③得：
−(∣a∣+∣b∣)≤a−b≤∣a∣+∣b∣⇒∣a−b∣≤∣a∣+∣b∣-(\left|a\right|+\left|b\right|)\leq a-b\leq\left|a\right|+\left|b\right|\Rightarrow \big|a-b\big|\le\big|a\big|+\big|b\big|−(∣a∣+∣b∣)≤a−b≤∣a∣+∣b∣⇒a−b≤a+b
而而而
∣a∣=∣(a+b)−b∣=∣(a−b)+b∣∣b∣=∣(b+a)−a∣=∣(b−a)+a∣\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}∣a∣=∣(a+b)−b∣=∣(a−b)+b∣∣b∣=∣(b+a)−a∣=∣(b−a)+a∣
由∣a+b∣≤∣a∣+∣b∣得:由\left|a+b\right|\le\left|a\right|+\left|b\right|得:由∣a+b∣≤∣a∣+∣b∣得:
∣a∣=∣(a+b)−b∣≤∣a+b∣+∣−b∣⇒∣a∣−∣b∣≤∣a+b∣⑥∣b∣=∣(b+a)−a∣≤∣b+a∣+∣−a∣⇒∣a∣−∣b∣≥−∣a+b∣⑦\begin{array}{l}\left|a\right|=\left|(a+b)-b\right|\leq\left|a+b\right|+\left|-b\right|\Rightarrow \left|a\right|-\left|b\right|\leq\left|a+b\right|&\text{⑥}\\ \left|b\right|=\left|\left(b+a\right)-a\right|\leq\left|b+a\right|+\left|-a\right|\Rightarrow \left|a\right|-\left|b\right|\geq-\left|a+b\right|&\text{⑦}\end{array}∣a∣=∣(a+b)−b∣≤∣a+b∣+∣−b∣⇒∣a∣−∣b∣≤∣a+b∣∣b∣=∣(b+a)−a∣≤∣b+a∣+∣−a∣⇒∣a∣−∣b∣≥−∣a+b∣⑥⑦
∣a∣=∣(a−b)+b∣≤∣a−b∣+∣b∣=>∣a∣−∣b∣≤∣a−b∣⑧∣b∣=∣(b−a)+a∣≤∣b−a∣+∣a∣=>∣a∣−∣b∣≥−∣a−b∣⑨\begin{array}{l}\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}∣a∣=∣(a−b)+b∣≤∣a−b∣+∣b∣=>∣a∣−∣b∣≤∣a−b∣∣b∣=∣(b−a)+a∣≤∣b−a∣+∣a∣=>∣a∣−∣b∣≥−∣a−b∣⑧⑨
由⑥，⑦得：∣∣a∣−∣b∣∣≤∣a+b∣或者：∣a∣=∣a+b−b∣≤∣b∣+∣a+b∣→∣a∣−∣b∣≤∣a+b∣a,b交换→∣b∣−∣a∣≤∣a+b∣由⑥，⑦得：||a|-|b||\leq|a+ b|\\或者：\\|a|=|a+b-b|\leq|b|+|a+b|\rightarrow |a|-|b|\leq|a+ b|\\a,b交换\rightarrow |b|-|a|\leq|a+ b|由⑥，⑦得：∣∣a∣−∣b∣∣≤∣a+b∣或者：∣a∣=∣a+b−b∣≤∣b∣+∣a+b∣→∣a∣−∣b∣≤∣a+b∣a,b交换→∣b∣−∣a∣≤∣a+b∣
由⑧，⑨得：∣∣a∣−∣b∣∣≤∣a−b∣或者：∣a∣=∣a−b+b∣≤∣b∣+∣a−b∣→∣a∣−∣b∣≤∣a−b∣a,b交换→∣b∣−∣a∣≤∣a−b∣由⑧，⑨得：||a|-|b||\leq|a- b|\\或者：\\|a|=|a-b+b|\leq|b|+|a-b|\rightarrow |a|-|b|\leq|a- b|\\a,b交换\rightarrow |b|-|a|\leq|a- b|由⑧，⑨得：∣∣a∣−∣b∣∣≤∣a−b∣或者：∣a∣=∣a−b+b∣≤∣b∣+∣a−b∣→∣a∣−∣b∣≤∣a−b∣a,b交换→∣b∣−∣a∣≤∣a−b∣
等号成立的条件（特别是求最值），即：∣a−b∣=∣a∣+∣b∣→ab≤0∣a∣−∣b∣=∣a+b∣→b(a+b)≤0∣a∣−∣b∣=∣a−b∣→b(a−b)≥0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|\rightarrow ab\leq0\\ \left|a\right|-\left|b\right|=\left|a+b\right|\rightarrow b(a+b)\leq0\\ \left|a\right|-\left|b\right|=\left|a-b\right|\rightarrow b(a-b)\geq0\end{array}等号成立的条件（特别是求最值），即：∣a−b∣=∣a∣+∣b∣→ab≤0∣a∣−∣b∣=∣a+b∣→b(a+b)≤0∣a∣−∣b∣=∣a−b∣→b(a−b)≥0
注：利用∣a−b∣=∣a∣+∣b∣→ab≤0注：利用\left|a-b\right|=\left|a\right|+\left|b\right|\rightarrow ab\leq0注：利用∣a−b∣=∣a∣+∣b∣→ab≤0
∣a∣−∣b∣=∣a+b∣→∣a∣=∣b∣+∣a+b∣→∣a+b−b∣=∣b∣+∣a+b∣|a|-|b|=|a+b|\\\rightarrow |a|=|b|+|a+b|\\\rightarrow |a+b-b|=|b|+|a+b|∣a∣−∣b∣=∣a+b∣→∣a∣=∣b∣+∣a+b∣→∣a+b−b∣=∣b∣+∣a+b∣
∣a∣−∣b∣=∣a−b∣→∣a∣=∣b∣+∣a−b∣→∣a−b+b∣=∣b∣+∣a−b∣\left|a\right|-\left|b\right|=\left|a-b\right|\\\rightarrow |a|=|b|+|a-b|\\\rightarrow |a-b+b|=|b|+|a-b|∣a∣−∣b∣=∣a−b∣→∣a∣=∣b∣+∣a−b∣→∣a−b+b∣=∣b∣+∣a−b∣