絕對值不等式

絕對值不等式是一個關於絕對值的不等式，在一般的情況下指代的是下面這組不等式。當${\displaystyle a,b}$為實數時，

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

等號成立的充分必要條件：

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