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 (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.