Solve a-ableq|a+b|

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

Solve for b

$$\left\{\begin{matrix}\\b=-\frac{2a}{1-a}\text{, }&\text{unconditionally}\\b\geq 0\text{, }&a\geq 0\\b\geq -a\text{, }&a\geq -1\text{ and }a<0\\b\in (-\infty,-a)\cup [0,-\frac{2a}{1-a}]\text{, }&a\geq -1\text{ and }a\leq 0\\b\leq -\frac{2a}{1-a}\text{, }&(a>0\text{ and }a<1)\text{ or }a<-1\\b\in [-1,-\frac{2a}{1-a}]\text{, }&a<-1\text{ or }(a>0\text{ and }a\leq \frac{1}{3})\\b\in [-1,-a)\text{, }&a>-1\text{ and }a<0\\b\geq -\frac{2a}{1-a}\text{, }&a>1\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}\\a=-\frac{b}{2-b}\text{, }&\text{unconditionally}\\a\geq -1\text{, }&b\geq 1\\a\geq -b\text{, }&b\geq 0\text{ and }b<1\\a\in [-1,-\frac{b}{2-b}]\text{, }&(b\geq 0\text{ and }b\leq 1)\text{ or }b<0\\a\geq -\frac{b}{2-b}\text{, }&b>2\\a\in [-2,-\frac{b}{2-b}]\text{, }&b<0\text{ or }(b>1\text{ and }b\leq \frac{4}{3})\\a\in [-2,-b)\text{, }&b>0\text{ and }b<1\\a<-b\text{, }&b\geq 0\text{ and }b\leq 1\\a\leq -\frac{b}{2-b}\text{, }&(b>1\text{ and }b<2)\text{ or }b<0\end{matrix}\right.$$