Inequalities in Two Variables

An equation like $2x + y = 4$ describes a line — a thin, one-dimensional object. But what happens when we replace the equals sign with an inequality? Instead of a line, we get an entire region of the coordinate plane.

Half-Planes

📐Definition — Inequality in Two Variables

An inequality in two variables has the form $ax + by \leq c$ (or $\geq$ , $<$ , $>$ ). Its solution set is the set of all points $(x, y)$ that satisfy the inequality.

The boundary line $ax + by = c$ divides the plane into two half-planes. Every point in one half-plane satisfies the inequality; every point in the other does not.

The boundary line itself is part of the solution when the inequality uses $\leq$ or $\geq$ (draw it solid), and excluded when the inequality uses $<$ or $>$ (draw it dashed).

The Test Point Method

⚡Theorem — Test Point Method

To graph $ax + by \leq c$ (or any variant):

Draw the boundary line $ax + by = c$ . Use a solid line for $\leq$ or $\geq$ ; dashed for $<$ or $>$ .
Pick a test point not on the line — $(0,0)$ is the easiest choice when the line does not pass through the origin.
Substitute the test point into the inequality. If it satisfies the inequality, shade the side of the line containing the test point. Otherwise, shade the opposite side.

For the common case $ax + by \leq c$ : testing $(0,0)$ gives $0 \leq c$ . So if $c \geq 0$ , the origin is in the solution region.

Worked Examples

✎Example — Example 1 — Non-strict Inequality

Graph $2x + y \leq 4$ .

Solution.

Draw the boundary line $2x + y = 4$ . Find intercepts: $(2, 0)$ and $(0, 4)$ . Draw a solid line (since $\leq$ ).
Test $(0, 0)$ : $2(0) + 0 = 0 \leq 4$ — true.
Shade the side containing the origin (below and to the left of the line).

The solution set is the half-plane below the solid line $2x + y = 4$ , including the line itself.

✎Example — Example 2 — Strict Inequality

Graph $x - 2y > 3$ .

Solution.

Draw the boundary line $x - 2y = 3$ . Intercepts: $(3, 0)$ and $(0, -\tfrac{3}{2})$ . Draw a dashed line (since $>$ , strict inequality).
Test $(0, 0)$ : $0 - 2(0) = 0 > 3$ ? — false.
Shade the side opposite to the origin (below and to the right of the line).

The solution set is the open half-plane on the far side of the dashed line from the origin.

Interactive Explorer

Use the explorer to visualize the half-plane for different coefficients.

Inequality 1: 1x + 1y <= 3

a=b=c=

Inequality 2: -1x + 2y >= 2

a=b=c=

Show second inequality

Special Cases

ℹNote — Horizontal and Vertical Half-Planes

Some inequalities involve only one variable:

$x > 2$ defines a vertical half-plane — everything to the right of the vertical line $x = 2$ (dashed).
$y \leq -1$ defines a horizontal half-plane — everything on or below the horizontal line $y = -1$ (solid).

These still follow the same test-point procedure, but the boundary lines are vertical or horizontal.

Practice

✏Exercise — Graph an Inequality

Graph the inequality $-x + 2y \geq 0$ .

Hint: The boundary line passes through the origin, so you cannot use $(0,0)$ as the test point. Try $(1, 0)$ or $(0, 1)$ instead.