绝对值不等式绝对值不等式是一个关于绝对值的不等式，在一般的情况下指代的是下面这组不等式。当 a , b {\displaystyle a,b} 为实数时， ∣ ∣ a ∣ − ∣ b ∣ ∣⩽∣ a ± b ∣⩽∣ a ∣ + ∣ b ∣ {\displaystyle \mid \shortmid a\shortmid -\shortmid b\shortmid \mid \leqslant \mid a\pm b\mid \leqslant \mid a\mid +\mid b\mid } 等号成立的充分必要条件： ∣ ∣ a ∣ − ∣ b ∣ ∣=∣ a + b ∣ {\displaystyle \mid \shortmid a\shortmid -\shortmid b\shortmid \mid =\mid a+b\mid } ⇔ a ⋅ b ⩽ 0 {\displaystyle \Leftrightarrow a\cdot b\leqslant 0} ∣ ∣ a ∣ − ∣ b ∣ ∣=∣ a − b ∣ {\displaystyle \mid \shortmid a\shortmid -\shortmid b\shortmid \mid =\mid a-b\mid } ⇔ a ⋅ b ⩾ 0 {\displaystyle \Leftrightarrow a\cdot b\geqslant 0} ∣ a + b ∣=∣ a ∣ + ∣ b ∣ {\displaystyle \mid a+b\mid =\mid a\mid +\mid b\mid } ⇔ a ⋅ b ⩾ 0 {\displaystyle \Leftrightarrow a\cdot b\geqslant 0} ∣ a − b ∣=∣ a ∣ + ∣ b ∣ {\displaystyle \mid a-b\mid =\mid a\mid +\mid b\mid } ⇔ a ⋅ b ⩽ 0 {\displaystyle \Leftrightarrow a\cdot b\leqslant 0}
