Quadratic inequalities require determining the sign of $ax^2$ + bx + c over specified intervals. The wavy curve (sign chart) method is the systematic approach.

Step 1: Find roots of $ax^2$ + bx + c = 0. If D < 0, sign is constant (same as sign of a) for all x. If D >= 0, roots are alpha <= beta.

Step 2: Factor as a(x - alpha)(x - beta). The sign alternates across roots for distinct roots. At x < alpha: sign of a (both factors have same sign). Between alpha and beta: opposite sign of a. At x > beta: sign of a again.

Step 3: Include or exclude endpoints based on strict/non-strict inequality.

For rational inequalities P$\frac{x}{Q}$(x) >= 0: (1) Find all roots of numerator and denominator. (2) Mark on number line with open circles for denominator roots (never included) and closed/open for numerator roots based on inequality type. (3) Apply alternating signs starting from the rightmost interval (positive for even multiplicity factor at boundary, sign flip for odd multiplicity).

Systems of quadratic inequalities: Solve each inequality separately and take intersection of solution sets. For union problems, take the union.

Parametric inequalities: "Find k such that f(x, k) > 0 for all x" requires: if degree in x is 2, need leading coefficient > 0 and D < 0. "Find k such that f(x, k) > 0 for some x" requires: if a > 0, always true; if a < 0, need minimum > 0, which is impossible, or maximum > 0, giving D > 0.