Graph linear inequalities in two variables as half-planes. | Linear inequalities in one or two variables | SAT Math

Graph a two-variable inequality with a boundary line and shaded half-plane.

Core Idea

Graph the boundary line first (dashed for strict, solid for inclusive), then shade the half-plane that satisfies the inequality.

Understanding

A linear inequality in two variables, like $𝑦 \leq 2 𝑥 + 1$ , describes a region, not a line. Start by graphing the boundary line $𝑦 = 2 𝑥 + 1$ as if it were an equation.

The boundary line style tells you whether points on the line itself count. Use a solid line for $\leq$ or $\geq$ (line included) and a dashed line for $<$ or $>$ (line not included).

To decide which side to shade, pick any test point not on the line — $(0, 0)$ is the easiest if the line doesn't pass through the origin. Plug it in: if the inequality is true, shade the side containing that point; if false, shade the other side.

Step by Step

Rewrite the inequality in slope-intercept form $𝑦 \leq 𝑚 𝑥 + 𝑏$ (or similar) if needed.
Graph the boundary line: solid for $\leq$ / $\geq$ , dashed for $<$ / $>$ .
Choose a test point not on the line ( $(0, 0)$ is simplest).
Substitute the test point into the inequality — if true, shade that side; if false, shade the opposite side.

Misconceptions

Using a solid line for a strict inequality (< or >) — solid is only for ≤ or ≥.
Shading the wrong side because of an error when testing the point — always double-check the substitution.
Forgetting to isolate y before deciding shade direction — when the inequality is in standard form (like $2 𝑥 + 3 𝑦 > 6$ ), you must test a point or rearrange.

Question

Worked Example

Which of the following describes the graph of $𝑦 > - 𝑥 + 3$ ?

Select an answer to see the explanation