Signs of a Fraction

A fraction is negative exactly when its top and bottom have opposite signs: a positive over a negative, or a negative over a positive. Take $\frac{x-2}{x+5} < 0$. You cannot multiply both sides by $x+5$ as you would by a number, because its sign is unknown, and multiplying by a negative flips the inequality.

So split into sign cases instead. Either $x-2 > 0$ and $x+5 < 0$, or $x-2 < 0$ and $x+5 > 0$. The first case asks for x above 2 and below -5 at once. No number is both, so that case is empty. The second asks for x below 2 and above -5, which is $-5 < x < 2$.

Test a number to be sure. x = 0 gives -2 over 5, which is negative, so it belongs. x = 3 gives 1 over 8, positive, so it does not. The answer sits between -5 and 2, the two numbers where the top or the bottom is 0.