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Half-Plane — Definition, Formula & Examples

A half-plane is one of the two regions created when a line divides the coordinate plane. Every linear inequality in two variables, such as y>2x+1y > 2x + 1, has a solution set that forms a half-plane.

Given a line ℓ\ell defined by ax+by=cax + by = c in R2\mathbb{R}^2, the two half-planes determined by ℓ\ell are the sets {(x,y)∣ax+by>c}\{(x,y) \mid ax + by > c\} and {(x,y)∣ax+by<c}\{(x,y) \mid ax + by < c\}. The line ℓ\ell itself is called the boundary and belongs to neither open half-plane. If the inequality is non-strict (≤\leq or ≥\geq), the boundary is included, forming a closed half-plane.

How It Works

To identify a half-plane, first graph the boundary line. Use a dashed line for strict inequalities (<< or >>) and a solid line for non-strict inequalities (≤\leq or ≥\geq). Then pick a test point not on the line — the origin (0,0)(0,0) is usually the easiest choice. Substitute the test point into the inequality: if it makes the inequality true, shade the side containing that point; otherwise, shade the opposite side. The shaded region is the half-plane representing all solutions to the inequality.

Worked Example

Problem: Graph the half-plane defined by y≤−x+4y \leq -x + 4.
Draw the boundary: Graph the line y=−x+4y = -x + 4. Since the inequality uses ≤\leq, draw a solid line through (0,4)(0, 4) and (4,0)(4, 0).
Test a point: Substitute the origin (0,0)(0, 0) into the inequality.
0≤−0+4  ⟹  0≤4True0 \leq -0 + 4 \implies 0 \leq 4 \quad \text{True}
Shade the correct side: Because the test point satisfies the inequality, shade the side of the line that contains (0,0)(0,0) — the region below and to the left of the line.
Answer: The half-plane is the solid line y=−x+4y = -x + 4 together with the entire region below it.

Why It Matters

In linear programming, feasible regions are formed by the intersection of several half-planes. Understanding half-planes is essential for solving optimization problems in algebra 2, precalculus, and introductory economics or operations research courses.

Common Mistakes

Mistake: Using a solid boundary line for a strict inequality like y>2xy > 2x.
Correction: Strict inequalities (<<, >>) exclude the boundary, so always use a dashed line. A solid line means points on the line are included, which only applies to ≤\leq or ≥\geq.