$|a+b|\leq |a|+|b|$

$\left\{\begin{matrix}\\b\in [-a,\infty)\cup (-\infty,0)\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&a<0\end{matrix}\right.$

$\left\{\begin{matrix}\\a\in (-\infty,0)\cup [\frac{-|b|-b}{2},\frac{|b|-b}{2}]\text{, }&\text{unconditionally}\\a\in [-b,\infty)\cup [\frac{-|b|-b}{2},\infty)\text{, }&b\leq 0\\a<-b\text{, }&b\geq 0\\a\in [\frac{-|b|-b}{2},-b)\text{, }&b<0\\a\geq 0\text{, }&b>0\end{matrix}\right.$