Solve |a+b+c|leq|a|+|b|+|c|

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

Solve for c

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

Solve for b

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