# How do I test input values with imaginary numbers for rational inequalities?

The goal is to find all solutions (if any exist) for

$$\frac{x^2+1}{y}\le0$$

Now, I set my constraints, {$$y|y\ne0$$}, and I set the expression in the numerator and denominator equal to zero to know which values I need to use to determine test points, getting me $$y=0$$ and $$x=\pm i$$

Normally, I'd use a real number line to determine test points. For example, if I happened to have $$x=-3, y=5$$ then I would test values within the sets $$(-\infty, -3), (-\infty, 5), (5, \infty)$$ then determine if they are $$\le 0$$, finally including the true solutions in interval notation.

In this case, however, I can't use the real number line because $$x$$ contains imaginary numbers, so I think I'll have to use a two dimensional plane to work this out, but I'm not sure how to do something like that. All I know for the problem above is to test y values less than and greater than 0, but I don't know what to do for x values. I've only done single-variable rational inequalities prior to this.

Any ideas what I can do here?

• In this particular case, the numerator is always positive, so the sign of the expression will be the sign of the denominator. When a factor of a rational expression has no real roots, then it will always have the same sign. – John Wayland Bales Mar 20 at 1:00