Simplifying and understanding inequalities in two variables given an equation: $$\frac{x-y}{x+y}\ge0$$ What are the steps to simplify this into an understandable group of inequalities which will yield a solution set as a group of areas? I already know that $x+y$ can't equal 0. I even know the shape of the solution areas, it's basically $$|x|\ge|y|$$ excepting points on the line x=-y. I figured this out through some deductive reasoning and testing some points, but I cannot figure out the operations required to arrive at this result in a legitimate (provable) manner.

We have

$$\frac{x-y}{x+y}\ge0\iff\left(x-y\ge0\land x+y>0\right)\lor \left(x-y\le0\land x+y<0\right)$$ so the first case gives $x\ge|y|$ but $x\
e-y$ and the second case gives $-x\ge |y|$ but $x\
e-y$ so we conclude that $$|x|\ge|y|\quad\text{and}\; x\
e-y$$