Graphing linear inequalities in two variables

The related equation creates the boundary

Replace the inequality sign with an equals sign and graph that line first. Every point on the line makes the two sides equal.

Use a solid boundary for ≤ or ≥ because equality is allowed. Use a dashed boundary for < or > because boundary points are excluded.

y\ge2x-1\Rightarrow y=2x-1\text{ is solid}

A test point chooses the side

Select a point not on the boundary—often (0, 0)—and substitute it into the original inequality. If the statement is true, shade the side containing that point.

If the boundary passes through the origin, choose another simple point. The test must distinguish one side from the other.

(0,0):\quad0>-2(0)+3\text{ is false}

Reversing the inequality matters when solving for y

Multiplying or dividing both sides by a negative reverses the order relation. Missing that reversal shades exactly the wrong region.

You can avoid solving for y by graphing the boundary in standard form and testing a point directly, but the solid-or-dashed decision still comes from the original symbol.

-2y<4x-6\Rightarrow y>-2x+3

Worked example

Common mistakes

Using a solid line for a strict inequality.
Testing a point that lies on the boundary.
Forgetting to reverse the inequality after division by a negative.

Keep these ideas

Graph the related equation first.
Boundary style records equality.
One valid test point identifies the region.