Graphing Linear Inequalities

Graphing an inequality in two variables gives a region, not a line.

1. Graph the boundary as if it were an equation.
2. Make it dashed for < or >, solid for ≤ or ≥.
3. Pick a test point not on the line — (0,0) is easiest — and check it in the inequality.
4. If it works, shade that side. If not, shade the other.

For y < 2x + 1: dashed line, test (0,0): is 0 < 1? Yes, so shade the side containing the origin.

The shaded region contains every point that makes the inequality true.

A linear inequality in two variables partitions the plane into two half-planes separated by the boundary line Ax + By = C. Strict inequalities exclude the boundary (dashed); non-strict inequalities include it (solid).

The test-point method works because the sign of Ax + By − C is constant on each side of the line, so one point determines the whole half-plane. Any point off the line serves; the origin is preferred purely for arithmetic ease, and must be replaced when the boundary passes through it.

Solving for y allows shading without testing: y > mx + b shades above and y < mx + b shades below. That shortcut is only reliable once y is genuinely isolated — and if the isolating step divided by a negative, the inequality has already flipped.

Systems of such inequalities intersect their half-planes into a feasible region, whose corner points are where the optimum of a linear objective must occur. That result is the basis of linear programming.