AHL 2.15 (HL)—Inequalities between functions

Solve an inequality by comparing graphs or sign intervals.

To solve g(x)≥f(x), find where h(x)=g(x)−f(x) is non-negative. Intersections split the number line into intervals whose signs must be checked.

Worked example

For x²≥2x, x(x−2)≥0, so x≤0 or x≥2. The graph gives the same result because the parabola lies above the line outside the intersections.

Include equality at roots for ≥ or ≤, and exclude points where an expression is undefined.

The intersection points are boundaries; they are not automatically the only solutions.