Linear Inequalities in Two Variables

Boundary line, solid or dashed, then shade the right half-plane.

The idea

A linear inequality in two variables is satisfied by a whole region of the plane — every point on one side of a line — called a half-plane. Graphing the solution takes three steps.

Draw the boundary. Replace the inequality sign with $=$ and graph the resulting line. The boundary separates the points that satisfy the inequality from the points that fail it.

Decide solid or dashed. When the sign is $\le$ or $\ge$, the points of the boundary satisfy the inequality, so draw the line solid; when the sign is strict they do not, so draw it dashed.

Shade the correct side. Choose a test point off the boundary and substitute its coordinates. If the resulting statement is true, shade the side containing the test point; if it is false, shade the other side. One test point decides the whole region, because the inequality holds on all of one side and fails on all of the other. The origin is the most convenient test point whenever the boundary misses it.

Ways to work on it

• Walkthrough. Boundary line, solid vs. dashed, and a test point to shade.
• Practice. Decide whether a point lies in the solution half-plane.
• Hardest. A boundary through the origin: choose a valid test point and shade.

Not sure where to start? Take the ten-question placement test.