feasible region

If a system of inequalities is a list of restrictions — your budget, your time, your space — the feasible region is the menu of everything you are actually allowed to choose. It is the set of all points that satisfy every inequality in the system simultaneously, the common ground where no rule is broken.

On a coordinate plane, you find it by shading each inequality's allowed half-plane and keeping only the overlap. The result is typically a polygon-shaped area (for linear inequalities) with edges along the boundary lines and corners where boundaries meet. These corner points, called vertices, are especially important.

The term comes from optimization and linear programming, where you maximize or minimize a quantity over the feasible region. A key fact there: for a linear objective, the best value always occurs at a vertex of a bounded feasible region — so you can check the corners rather than every interior point. The region may be bounded, unbounded, or empty when the constraints conflict.