system of inequalities

Where a system of equations seeks exact points, a system of inequalities seeks regions. Think of stacking several rules of the form at least this much or at most that much, and asking which points obey all of them at once. The answer is usually not a lone point but a whole shaded zone.

A system of inequalities is two or more inequalities in the same variables, solved together. A solution is any point that satisfies every inequality simultaneously, and the collection of all such points forms the solution region. On a coordinate plane, each inequality shades a half-plane (or a curved region), and the system's solution is the overlap of those shaded areas.

Boundaries matter: a strict inequality like y > x excludes its boundary line (drawn dashed), while y ≥ x includes it (drawn solid). The overlap region — often called the feasible region — may be bounded or unbounded, and it can even be empty if the inequalities leave no common ground. Such systems are the backbone of linear programming, where you optimize within the feasible region.