Linear Inequalities in Two Variables — Mathematics Knowledge System

Overview

A linear inequality in two variables, such as y < 2x + 3, has a solution set that is an entire region of the coordinate plane rather than a single line. The boundary line (from the corresponding equation) divides the plane into two half-planes, and the inequality's solution is one of those half-planes, either including or excluding the boundary line itself.

Formal Definition

Definition

A linear inequality in two variables takes the form:

Ax+By<C \text{ (or } \le,\ >,\ \ge\text{)}

General form

Properties

Boundary line

\text{Replace the inequality with } = \text{ to find the boundary line}

Solid vs. dashed boundary

\le, \ge \Rightarrow \text{solid line (included)}; \quad <,> \Rightarrow \text{dashed line (excluded)}

Test point method

\text{Substitute a point not on the line (e.g. (0,0)) to determine which half-plane satisfies the inequality}

Worked Examples

Graph the boundary line y = 2x - 1 as a SOLID line (since ≤ includes equality).
$y=2x-1$
Test the point (0,0): is 0 ≤ 2(0)-1 = -1? No, 0 is not ≤ -1.
$0 \le -1 \text{ is false}$
Since (0,0) fails, shade the half-plane NOT containing (0,0) — below the line.

Answer: Solid boundary line y = 2x - 1, shaded below the line

Practice Problems

Difficulty 3/10

Describe how to graph y > -x + 4, including whether the boundary is solid or dashed.

Common Mistakes

Common Mistake

Using a solid boundary line for strict inequalities (< or >).

Strict inequalities (< or >) always use a DASHED boundary line, since points exactly on the line do not satisfy the inequality. Only ≤ and ≥ use a solid line.

Summary

A linear inequality in two variables has a solution set that is a half-plane.
The boundary line is solid for ≤/≥ and dashed for </>.
Use a test point (commonly the origin) to determine which half-plane to shade.

References

WebsiteWikipedia — Linear inequality