This occurs in two situations: when $xy$ has the same sign as $x^2$ and when $xy$ is $0$. Thus, our desired answer is the first quadrant, third quadrant, and the two axes. @below: you can ditch the "or" since the second region contains the first. It also barely counts as describing the region.

In this way is better: $|{{x}^{2}}+xy|\ge |{{x}^{2}}-xy|\Leftrightarrow \left| x \right|\left| x+y \right|\ge \left| x \right|\left| x-y \right|\Leftrightarrow \left| x \right|=0\vee \left| x+y \right|\ge \left| x-y \right|$