Graph the solution set of a linear inequality in two variables as a half-plane, using a solid or dashed boundary and a test point to choose the shaded side (Ohio A-REI.12, A-REI.11).

What this topic is asking

Ohio standard A-REI.12 asks you to graph a linear inequality in two variables. The solution is not a line but a half-plane, the whole region on one side of a boundary line. You decide two things: whether the boundary is solid or dashed, and which side to shade. This skill sits in the Expressions and Equations category and is the building block for systems of inequalities.

Step 1: the boundary line, solid or dashed

Replace the inequality symbol with $=$ and graph that line. The symbol tells you the line style.

For $y \geq -x + 4$, draw the line $y = -x + 4$ solid. For $y < 3x$, draw $y = 3x$ dashed.

Step 2: shade the correct half-plane

The boundary splits the plane into two halves; exactly one half is the solution set. The fastest way to pick it is a test point.

The origin is the easiest test point whenever the line does not pass through it. If the line does pass through $(0, 0)$, choose another point, like $(1, 0)$.

Reading and solving-for-y shortcut

If you first solve the inequality for $y$, the symbol points to the shading: $y > \ldots$ or $y \geq \ldots$ shades above the line, and $y < \ldots$ or $y \leq \ldots$ shades below. The test-point method always works and is the safe check, especially when the inequality is in standard form $ax + by \leq c$ and not yet solved for $y$.

How Ohio examines this topic

• Graphing. Draw the boundary with the correct line style and shade the right half-plane on the online grid.
• Multiple choice and multiple-select. Match an inequality to its graph, or pick which graphs use a dashed versus solid line.
• Equation response. State whether a given point is a solution by substituting it.

Why the solution is a whole region

A two-variable inequality is satisfied by every point whose coordinates make it true, and there are infinitely many such points filling an entire side of the line. That is why the solution is a region (a half-plane), not the handful of points a one-variable inequality marks on a number line. The boundary line is where the two sides equal each other; crossing it flips the inequality from true to false, which is why exactly one side works. Understanding that every shaded point is a genuine solution, and you can verify any of them by substitution, is what makes the picture meaningful rather than a memorized shading rule.

Why the line style matters

The solid-versus-dashed choice encodes whether the boundary itself is part of the answer. With $\leq$ or $\geq$, a point on the line gives equality, which the inequality allows, so those points count and the line is solid. With strict $<$ or $>$, equality is not allowed, so the boundary points fail and the line is dashed to show they are excluded. Getting this right matters on exact-match graphing items, where a dashed line drawn solid (or the reverse) is a different graph and loses the point.

Try this

Q1. Is the line for $y \leq 2x + 1$ solid or dashed? [1 point]

• Cue. Inclusive $\leq$, so solid.

Q2. For $y > x - 3$, test $(0, 0)$. Do you shade the side with the origin? [2 points]

• Cue. $0 > -3$ is true, so yes, shade the side containing $(0, 0)$ (above the line).