Rational Inequalities

Algebra 2 · Axiom Academy

Solve with sign analysis — critical points split the number line, then test the sign in each interval. 1. Finding the Critical Points To solve , first find the critical points — the x -values where the expression can switch sign. They come from the two parts of the fraction. Zeros — where the numerator is 0 . The whole fraction equals 0 here. Undefined points — where the denominator is 0 . The fraction is undefined here (a vertical asymptote), so it can never be part of a solution. 2. Split the Line into Intervals Plot the critical points on a number line. They cut it into intervals, and inside each interval the sign of the expression cannot change. The dot style records whether a point is allowed in the solution. The numerator is 0 , so the fraction is 0 . Includable only when the inequality is or . The denominator is 0 , so the fraction is undefined. Never part of the solution. 3. Test the Sign in Each Interval Pick any test value inside an interval and check the sign of the expression. A fraction's sign is the sign of the numerator over the sign of the denominator, so just track each factor. The sign can only change at a critical point — but a zero and an asymptote flip it for different reasons. The graph of shows both at once. The graph crosses the axis. A simple zero flips the sign as you pass through it. The graph jumps from to . The expression is undefined, and the sign flips across the break.

This is the written version of the interactive lesson above. See the full Algebra 2 course.