How Do You Graph Inequalities on a Coordinate Plane?

To graph an inequality on a coordinate plane, first plot its boundary line, then use a test point to decide which side of that line contains the solutions, and finally shade that region. The boundary is drawn as a dashed line for strict symbols (<, >) and a solid line for non-strict symbols (≤, ≥). Everything rests on one definition: an inequality is a statement that two expressions are not equal, and its solution is the entire set of points that make the statement true, not a single line of points.

Graph the Boundary Line

The boundary line is the equation you get by temporarily replacing the inequality symbol with an equals sign. If you are given y > 2x + 1, the related equation is y = 2x + 1. That line divides the coordinate plane into two half-planes, and the actual solution set lies entirely on one side of it (or on the line itself, depending on the symbol).

Graph that equation using whatever method you prefer — slope-intercept form, a table of values, or intercepts. In the example, the line has a y-intercept at (0, 1) and a slope of 2, so from (0, 1) you move up 2 and right 1 to reach (1, 3). Plot those points and draw a straight line through them.

The inequality symbol decides how the boundary itself is drawn:

• Strict inequalities (< and >) exclude the boundary, so use a dashed line. Points on a dashed line are not solutions.
• Non-strict inequalities (≤ and ≥) include the boundary, so use a solid line. Points on a solid line are solutions.

This dashed-versus-solid distinction is one of the most commonly tested details, and Khan Academy treats it as a required first check before any shading begins. A useful habit: circle the symbol before you start, so you do not lose track of whether the line should be dashed or solid.

Test a Point

Once the boundary is drawn, you need to know which half-plane to shade. The fastest way is to pick any point that is not on the line, substitute its coordinates into the original inequality, and check whether the result is true.

The origin, (0, 0), is the convenient default because substitution is trivial. Only avoid it when the line passes through the origin — then choose something simple like (1, 0) or (0, 1) instead.

Work through the example, y > 2x + 1, with the test point (0, 0):

• Substitute: 0 > 2(0) + 1
• Simplify: 0 > 1

That statement is false, so the origin is not part of the solution set. The shading must go on the opposite side of the line from (0, 0) — that is, the side below the line, since the origin sits below it here. If the substitution had produced a true statement, you would shade the side containing the test point.

Shade the Solution Region

Shading is what turns a line on a grid into a picture of an entire solution set. Every point inside the shaded region makes the inequality true, and every point outside it makes the inequality false. The label “above” or “below” always refers to the boundary line’s direction on the plane, not to where the shading physically sits on your paper.

Two reliable shortcuts after practice:

When the inequality is solved for y, “greater than” always means above the line and “less than” always means below it, because you are comparing y-values. A vertical line such as x > 2 is different: shade to the right of it, since you are comparing x-values. In every case, the test point is the safety net when you are unsure. Math is Fun frames the shaded area as the “wanted region,” which is a helpful mental image for keeping track of what the shading represents.

Plot Multiple Inequalities

A system of inequalities has more than one constraint, and its solution is the set of points that satisfy all of them at once. Graph each inequality separately on the same coordinate plane — boundary line, test point, and shading for each — then look at where the shaded regions overlap.

That overlap is the solution to the system. Points in only one shaded region satisfy just one inequality; points in the overlap satisfy every inequality. A good way to confirm your answer is to test a point inside the overlap in each original inequality and verify that all of them come out true.

A common trap is shading a region that looks plausible visually but fails one test point; when in doubt, substitute a specific point in the overlap back into every inequality. With three or more inequalities, the same logic holds: shade each region and keep the intersection.

Key Takeaways

• Replace the inequality symbol with an equals sign to get the boundary, then graph that line.
• Use a dashed line for < and > and a solid line for ≤ and ≥.
• Pick a test point (usually the origin) and substitute it; shade the side that makes the statement true.
• For y-isolated inequalities, “greater than” means above and “less than” means below the line.
• For systems, shade every inequality and identify the overlapping region — that overlap is the solution set.
• When the line passes through the origin, choose a different test point such as (1, 0).