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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?

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  • $\begingroup$ 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. $\endgroup$ – John Wayland Bales Mar 20 at 1:00

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