Systems of Inequalities

Graph each inequality on the same axes, then look for the region shaded by *both*. That overlap is the solution.

Every point in the overlap satisfies both inequalities at once. Points shaded by only one do not count.

Shade lightly, or use different directions of hatching, so the overlap stays visible. A common trick is to shade only lightly at first and then outline the final region clearly.

To check, pick any point inside your region and test it in both inequalities. Both must be true.

Some systems have no overlap, which means no solution.

The solution set is the intersection of the individual half-planes, a convex region — possibly unbounded, possibly empty. Convexity follows because each half-plane is convex and intersections of convex sets are convex.

Boundary treatment carries over: dashed edges are excluded, solid edges included, and a corner formed by one dashed and one solid edge belongs to the region only if it satisfies every inequality.

This is the setup for linear programming. When the region is bounded, a linear objective P = ax + by attains its maximum and minimum at vertices of the feasible region, so optimisation reduces to finding the corner points — each the intersection of two boundary lines, found by solving a 2×2 system — and evaluating the objective at each.

That connection is why systems of inequalities are worth the graphing effort: they are the standard model for problems with resource constraints.