Systems of Inequalities
Turn each linear inequality into a boundary plus a half-plane, then keep only the coordinate-plane region that satisfies every constraint at once.
How does an inequality become an area instead of a line?
The related equation draws the boundary. The inequality chooses one side of that boundary. A system keeps only the points that survive every half-plane constraint.
Strict inequalities exclude the boundary line. Inclusive inequalities keep it.
Boundary first, test point second, shading third.
For y ≥ x + 1, first draw y = x + 1 as a solid line. A test point such as (0, 2) makes 2 ≥ 1 true, so the half-plane containing that point is allowed.
Because ≥ includes equality, the boundary is solid.
The test point satisfies the inequality, so its side is allowed.
Every point on the accepted side satisfies the same constraint.
The allowed y-values are greater than the boundary value for each x.
The allowed y-values are less than the boundary value for each x.
Keep only points that survive every shaded constraint.
Build each half-plane, then intersect them.
Which side of the boundary satisfies this inequality?