What is a linear inequality on the SAT?

It is a linear equation with an inequality sign instead of an equals sign — less than, greater than, or the same with an 'or equal to.' Instead of one answer, you get a range of answers: 2x + 3 = 11 has one solution, but 2x + 3 < 11 has a whole set. It is an Algebra skill on the digital SAT, one of the two largest math domains at about 13 to 15 of the 44 questions, roughly a third of the test.

How do you solve a one-variable linear inequality?

Solve it like an equation — add, subtract, multiply, and divide to get the variable alone. There is one extra rule: when you multiply or divide both sides by a negative number, flip the inequality sign. For example, −2x −3 after dividing by −2. Forgetting to flip the sign is the most common mistake, and the wrong-direction answer is usually one of the options.

How do you graph a two-variable linear inequality?

Graph the boundary line, then shade the side that makes the inequality true. Use a solid line when the symbol includes 'or equal to' and a dashed line for strict less-than or greater-than. To pick the side, test a point that is not on the line — (0, 0) is easiest — and shade that side if it satisfies the inequality, otherwise shade the other side.

SAT Linear Inequalities

Q: How do you solve a one-variable linear inequality?

Solve it like an equation — add, subtract, multiply, and divide to get the variable alone. There is one extra rule: when you multiply or divide both sides by a negative number, flip the inequality sign. For example, −2x −3 after dividing by −2. Forgetting to flip the sign is the most common mistake, and the wrong-direction answer is usually one of the options.

Q: How do you graph a two-variable linear inequality?

Graph the boundary line, then shade the side that makes the inequality true. Use a solid line when the symbol includes 'or equal to' and a dashed line for strict less-than or greater-than. To pick the side, test a point that is not on the line — (0, 0) is easiest — and shade that side if it satisfies the inequality, otherwise shade the other side.

Q: What is the solution to a system of linear inequalities?

It is the region where the shaded regions overlap. Graph each inequality and shade it; any point in the overlap satisfies every inequality in the system. When a question asks which point is a solution, test the options — the right one makes all the inequalities true at once. The built-in Desmos calculator shades the regions and the overlap for you.

Last updated: July 3, 2026

Linear inequalities are an Algebra skill on the digital SAT, and Algebra is one of the two biggest math domains — about a third of the section. You solve them almost exactly like linear equations, with one rule that trips people up: flip the inequality sign when you multiply or divide by a negative. The rest is knowing when the answer is a range, a shaded region, or the overlap of two regions.

What is a linear inequality?

It's a linear equation with an inequality sign instead of an equals sign — <, >, ≤, or ≥. Instead of one answer, you get a range of answers.

2x + 3 = 11 has one solution: x = 4. But 2x + 3 < 11 has a whole set of them: every x less than 4. Same algebra to solve, different kind of answer.

This is a skill in the Algebra content domain, the same block as linear equations, linear functions, and systems. Algebra is one of the two largest domains on the math section, about 13–15 of the 44 questions — roughly a third of the test. Inequalities show up on their own and buried inside word problems, so it's worth locking down.

How do you solve a one-variable inequality?

Solve it like an equation — add, subtract, multiply, divide to get the variable alone. There's exactly one extra rule:

When you multiply or divide both sides by a negative number, flip the inequality sign.

Say you have −2x < 6. Divide both sides by −2 and the sign flips: x > −3. Miss the flip and you get x < −3, which is the opposite set of answers. This is the single most common mistake on these questions, and the test knows it — wrong-direction answers are almost always sitting there in the options.

Everything else is identical to solving a linear equation. If solving 2x + 3 = 11 is automatic for you, so is the inequality version — just watch the sign.

What about two-variable inequalities?

Now the answer is a region on the graph, not a range on a number line.

y > 2x + 1 is every point above the line y = 2x + 1. You graph the boundary line, then shade the side that makes the inequality true. Two details the test checks:

Solid or dashed line. Use a solid line for ≤ or ≥ (the boundary counts) and a dashed line for < or > (the boundary doesn't).
Which side to shade. Pick any point not on the line — (0, 0) is easiest — plug it in, and shade that side if it makes the inequality true. If it doesn't, shade the other side.

A common SAT question gives you a point and asks whether it's a solution. You don't need the graph for that — just plug the point in and check whether the inequality holds.

How do you handle a system of inequalities?

Same idea as a system of equations, but the answer is where the shaded regions overlap.

Graph each inequality, shade each one, and the solution is the area covered by both. Any point in that overlap satisfies every inequality in the system. When a question asks which point is a solution to the system, test the options — the right one makes all the inequalities true at once.

You have the Desmos graphing calculator on every math question. Type the inequalities in and Desmos shades the regions for you, overlap included. Solve the one-variable ones by hand for speed, and lean on Desmos for the two-variable and system questions where seeing the region is faster than reasoning it out.

Practice routine

Work a set of 10–15 questions mixing all three types — one-variable, two-variable, and systems. After each one:

On one-variable problems, check whether you multiplied or divided by a negative — if you did, confirm you flipped the sign
On two-variable problems, name why the line is solid or dashed and which side you shaded
When you miss one, name whether it was the sign flip, the shading direction, or an arithmetic slip — each is a different fix

This skill sits right on top of linear equations and systems, so if those still feel slow, tighten them first. If you want to know whether inequalities are solid or still costing you points, join the HIROSCORE beta and get a breakdown of exactly which skills are leaking the most.

HIROSCORE tracks your accuracy on this skill separately from the rest of Algebra, so you can see it clearly instead of guessing. The GPS for your SAT score.

References