Finding Roots of 2 Variable Inequalities

Question

If I happen to have a two variable inequality such as $x+y<xy$ what is the most efficient way of finding out the critical points/roots since I cannot plot 3d functions in my head. For example, in the inequality, $(xy)^2<xy$, we can subtract $xy$ from both sides to get $(xy)^2-xy<0$ or $xy(xy-1)<0$, and get two cases $xy=0$ or $xy-1=0$. But sometimes the inequalities are not so clear cut. Anyone have any clues on how to tackle finding the roots of the inequality $x+y<xy$

Answer 1

I know this is super late but I would do it like this:
x + y $\geq{xy}$
x $\geq{xy-y}$
x $\geq{y(x-1)}$
$\frac{x}{x-1}$ $\geq{y}$ or $y \leq{\frac{x}{x-1}}$
So y would be the shaded region under the graph $g(x) = \frac{x}{x-1}$

Edit: It would also contain the line