How to graph and read inequalities on Desmos

Q: What do dashed and solid lines mean on a Desmos inequality?

A dashed boundary appears for strict inequalities ( ) and means the line itself is not part of the solution. A solid boundary appears for non-strict inequalities (≤ or ≥) and means the points on the line are included.

Q: Why isn't my inequality shading in Desmos?

Make sure you typed an inequality sign ( , ≤ or ≥), not an equals sign, and that the expression is in terms of x and y so Desmos knows which region to shade. If two regions overlap, look for the darker band where they cross. Calculator behavior can change, so confirm details on the official sites.

Inequality and "system of inequalities" questions look fiddly on paper but are nearly free with Desmos: type the inequality, and it shades every point that works. For a system, the answer is wherever the shadings overlap.

Type an inequality straight onto a line — y < 2x + 1 or y ≥ x^2 − 4 — and Desmos shades the solution region. A strict sign (<, >) gives a dashed boundary; ≤ or ≥ gives a solid one. For a system, enter each inequality on its own line; the overlap of the shading is the solution. To check a point, see whether it sits inside the shaded region — or plug its coordinates into each inequality and confirm every statement is true.

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two shaded regions overlapping: Make the mathematical object in this brief the hero: a Desmos-style coordinate plane: one half-plane shaded with a dashed boundary line and a second region with a solid boundary, their overlap drawn as a darker accent-colored wedge with a single test point dotted inside it

Auto-shadingType the inequality and Desmos shades every point that satisfies it.

Boundary tells you ≤ vs <Dashed = not included; solid = included.

Overlap = solutionFor a system, the answer is where the regions cross.

Two ways to test a pointRead the shading, or substitute and check.

Step by step

From an inequality to the right region

Eight steps covering a single inequality, a full system, and both ways to verify a point. The core move is simple: type it, and read the shading.

Type the inequality

On an expression line, enter it just as written — y < 2x + 1. Pull the <, >, ≤ and ≥ signs from the on-screen keypad if you can't find them on the keyboard.

Watch it shade

Desmos immediately fills in every point that makes the statement true. That shaded area is the solution set — you're reading the answer, not computing it.

Check the boundary style

A dashed line means a strict inequality (< or >): points on the line don't count. A solid line (≤ or ≥) means the boundary is included.

Handle curves the same way

Nonlinear inequalities work identically: y ≥ x^2 − 4 shades everything on or above the parabola. The boundary curve follows the same dashed/solid rule.

Add the second inequality

For a system, type each inequality on its own line. Desmos shades both regions in different tints, layered on the same plane.

Find the overlap

Look for the darker band where the two shadings cross. Every point in that overlap satisfies both inequalities — that region is the system's solution.

Test a point by eye

To check a candidate point, glance at the graph: if it sits inside the overlap, it's a solution. You can type the point as (3, 4) on its own line to plot it and see exactly where it lands.

Or verify by plugging in

Prefer algebra? Substitute the coordinates into each inequality. If every statement is true, it's a solution; one false statement rules it out. The graph and the substitution always agree.

Worked example

Which point satisfies both?

A standard SAT system-of-inequalities question: two inequalities, four candidate points, and one that lands in the overlap.

SAT-style problem

A system is defined by y > 2x − 3 and y ≤ −x + 6. Which of the following points is a solution to the system: (5, 1), (4, 2), (0, 7), or (2, 3)?

Graph both inequalities. Type y > 2x − 3 on one line and y ≤ −x + 6 on the next. The first shades above a dashed line; the second shades on and below a solid line. The overlap is a wedge opening to the left.

Drop in the candidates. Type (5,1), (4,2), (0,7) and (2,3) as four points. Reading the graph, only (2, 3) sits inside the overlapping region; the others fall outside one shading or the other.

Confirm by substitution. For (2, 3): is 3 > 2(2) − 3 = 1? Yes. Is 3 ≤ −2 + 6 = 4? Yes. Both hold, so the answer is (2, 3). Spot-check a reject — (0, 7) gives 7 ≤ 6, which is false, so it's out.

Reading the overlap with confidence

When the two tints stack, the solution is only the darker region where they coincide — not either single-shaded area. Plotting the candidate points right on top makes the answer unmistakable: the correct one is visibly inside that darker wedge, and the distractors sit just outside it.

Boundary gotcha: the dashed line from y > 2x − 3 means points on that line don't qualify, while the solid line from y ≤ −x + 6 means points on it do. A candidate sitting exactly on a boundary can be a trap — check the sign, strict versus non-strict, before you commit.

When to use / pitfalls

Where shading wins — and what to double-check

Use it

Systems and "which point" questions.

Anytime you must find a solution region, test candidate points, or pin down where constraints overlap, graphing beats hand-shading and guessing — the overlap is right there to read.

Watch for

Strict vs non-strict, and the right overlap.

Mixing up dashed and solid boundaries flips whether a point on the line counts. And for a system, the answer is only the shared region — don't grab a point that's in just one shading.

Skip it

For a one-line check you can do in your head.

If a question gives one simple inequality and one point, plugging in is instant. Save the graph for systems, curved boundaries, or when several points need testing at once.

Drill systems on SAT-style questions.

Reading overlaps becomes automatic only when you practice on real questions with the built-in calculator.

Start free practice

FAQ

Desmos inequalities on the SAT, answered

How do I graph an inequality in Desmos on the SAT?

Type the inequality straight onto an expression line, for example y < 2x + 1 or y ≥ x^2 − 4, using the inequality signs from the on-screen keypad. Desmos automatically shades every point that makes the statement true.

What do dashed and solid lines mean on a Desmos inequality?

A dashed boundary appears for strict inequalities (< or >) and means the line itself is not part of the solution. A solid boundary appears for non-strict inequalities (≤ or ≥) and means the points on the line are included.

How do I solve a system of inequalities in Desmos?

Enter each inequality on its own line. Desmos shades both regions, and the area where the two shadings overlap is the solution set for the system. Any point inside that overlap satisfies all of the inequalities at once.

How can I tell if a point satisfies an inequality in Desmos?

Two ways. Read it off the graph: if the point lies inside the shaded region (the overlap, for a system) it's a solution — you can type it as (3, 4) to plot it. Or plug the point's coordinates into each inequality by hand; if every statement comes out true, the point works.

Why isn't my inequality shading in Desmos?

Make sure you typed an inequality sign (<, >, ≤ or ≥), not an equals sign, and that the expression is in terms of x and y so Desmos knows which region to shade. If two regions overlap, look for the darker band where they cross. Calculator behavior can change, so confirm details on the official sites.

Turn shaded regions into easy points.

Start free and practice SAT-style inequality and system questions with the built-in calculator, until reading the overlap is automatic.

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