Graphing Inequalities on the Number Line

Graphing Two-Variable Inequalities

By the end of this lesson, you will be able to graph linear inequalities on the coordinate plane and identify solutions in the shaded region.

From Equation to Inequality

A linear equation ($y = mx + b$) has solutions on a line. A linear inequality ($y > mx + b$, etc.) has solutions in a half-plane — all points on one side of the boundary line.

Steps to Graph a Linear Inequality

1. Graph the boundary line as if it were an equation ($y = mx + b$)

   • Use a dashed line for $<$ or $>$ (boundary not included)
   • Use a solid line for $\leq$ or $\geq$ (boundary included)

2. Test a point not on the line (usually the origin $(0, 0)$ if not on the line)

   • Substitute into the inequality
   • If TRUE: shade the side containing that point
   • If FALSE: shade the other side

3. Shade the appropriate half-plane

Example

Graph $y < 2x - 1$:

Step 1: Draw $y = 2x - 1$ (dashed, since $<$)

Step 2: Test $(0, 0)$: Is $0 < 2(0) - 1 = -1$? Is $0 < -1$? FALSE

Step 3: Shade the side NOT containing the origin (below the line)

Checking Solutions

A point is a solution if it makes the inequality true.

Is $(3, 2)$ a solution to $y \leq x + 1$?

$2 \leq 3 + 1 = 4$? Yes, $2 \leq 4$ ✓ — it's a solution (and will be in the shaded region)

Summary

Boundary line: dashed ($<$, $>$), solid ($\leq$, $\geq$).

Test a point: substitute into the inequality; shade the side that makes it true.

Solutions: all points in the shaded region (plus the boundary line if solid).