Rational Inequalities and Sign Charts

A rational function can only switch between positive and negative at two kinds of places: where its numerator equals zero, and where its denominator equals zero. Everywhere else, the function is a continuous, unbroken curve, so it cannot sneak from positive to negative without crossing zero or blowing up at an asymptote. That's why those two sets of points are the only candidates for a sign change, and why solving a rational inequality really just means locating them.

To build a sign chart, factor the numerator and denominator completely, then mark every zero and every vertical asymptote on a number line. These points cut the line into intervals. Pick one test value inside each interval, plug it into the factored expression, and record whether the result is positive or negative. Once every interval has a sign, read off the ones that match the inequality. Zeros of the numerator can be included when the inequality allows equality, but zeros of the denominator are always excluded, since the function is undefined there no matter what the inequality symbol says.

This builds directly on the zero and asymptote behavior from earlier in the unit: factors with odd multiplicity flip the sign as you cross them, while factors with even multiplicity, like a squared term, touch zero but don't change sign. Recognizing multiplicity turns the sign chart from a guessing game into a quick, reliable prediction, and sets up later work where end behavior and local behavior near asymptotes both come from the same factored form.