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${\displaystyle a,b}$为实数时，

${\displaystyle \mid \shortmid a\shortmid -\shortmid b\shortmid \mid \leqslant \mid a\pm b\mid \leqslant \mid a\mid +\mid b\mid }$

等号成立的充分必要条件：

${\displaystyle \mid \shortmid a\shortmid -\shortmid b\shortmid \mid =\mid a+b\mid }$	${\displaystyle \Leftrightarrow a\cdot b\leqslant 0}$
${\displaystyle \mid \shortmid a\shortmid -\shortmid b\shortmid \mid =\mid a-b\mid }$	${\displaystyle \Leftrightarrow a\cdot b\geqslant 0}$
${\displaystyle \mid a+b\mid =\mid a\mid +\mid b\mid }$	${\displaystyle \Leftrightarrow a\cdot b\geqslant 0}$
${\displaystyle \mid a-b\mid =\mid a\mid +\mid b\mid }$	${\displaystyle \Leftrightarrow a\cdot b\leqslant 0}$