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SAT Math Inequalities: The Complete Guide
Everything the SAT tests about inequalities in one place: solving and the flip rule, compound inequalities, graphing on a number line, two-variable inequalities, and systems of inequalities — with step-by-step examples and 3 free practice quizzes.
Practice these quizzes
Free · No signupWhat is a linear inequality?
A linear inequality looks almost exactly like a linear equation, except the equals sign is replaced with one of four comparison symbols: <, >, ≤, ≥. Instead of one exact solution, an inequality describes a whole range of values that make the statement true.
| Symbol | Meaning | Boundary included? |
|---|---|---|
| < | less than | No — open circle |
| > | greater than | No — open circle |
| ≤ | less than or equal to | Yes — closed circle |
| ≥ | greater than or equal to | Yes — closed circle |
Where a linear equation like x = 3 has exactly one solution, the inequality x > 3 has infinitely many — every real number greater than 3. That's the core shift in thinking: you're solving for a region, not a single point.
The SAT tests inequalities three ways: as pure algebra (solve for x), as a graph on a number line or coordinate plane, and as word problems where you build the inequality from a real-world constraint like a budget or a weight limit.
Solving & the flip rule
Solving a linear inequality works exactly like solving a linear equation — same moves, same order — with a single exception that the SAT loves to test.
Solve for x: −3x + 7 ≤ 22
The most frequent inequality error on the SAT: forgetting to flip the sign when dividing by a negative, or flipping it when the divisor was actually positive. After every division step, check the sign of the number you divided by.
Compound inequalities
A compound inequality combines two conditions on the same variable, joined by AND or OR. The SAT usually writes the AND case as a single three-part inequality.
AND (between)
Written as a < x < b. Solve all three parts together, applying the same operation across all three sections at once.
OR (either / outside)
Two separate inequalities that don't overlap — solve each independently, then combine with "or."
Solve for x: −4 ≤ 2x + 6 < 10
AND compounds graph as one continuous segment between two points. OR compounds graph as two separate rays pointing away from each other.
Graphing on a number line
Every one-variable inequality solution can be drawn on a number line in four moves: locate the boundary point, choose open or closed, shade the correct direction, and add an arrow if the region is unbounded.
- Closed circle: used with ≤ or ≥ — the boundary value is included.
- Open circle: used with < or > — the boundary value is excluded.
- Shading direction: shade toward larger values for > or ≥, toward smaller values for < or ≤.
- Reading a graph backward: if the SAT shows a number line and asks for the inequality, read the circle type first, then the shading direction.
Two-variable inequalities
A two-variable inequality like y < 2x + 1 doesn't describe a line — it describes an entire half-plane, one whole side of the boundary line.
Step 1 — graph the boundary
Treat the inequality as an equation and graph that line using slope and y-intercept.
Step 2 — solid or dashed
≤ or ≥ gets a solid line. < or > gets a dashed line.
Step 3 — shade a side
Test the point (0, 0). If true, shade the side with the origin; if false, shade the other side.
Which side of y = −x + 4 should be shaded for y ≤ −x + 4?
If the boundary line passes through the origin, (0, 0) can't be used as a test point. Pick a different easy point instead, such as (1, 0) or (0, 1).
Systems of inequalities
A system of inequalities asks for the region where every inequality is simultaneously satisfied — the overlap of all the shaded half-planes.
Is (2, 1) a solution to: y < x + 2 and y ≥ 2x − 5 ?
Inequality word problems
The SAT's most common inequality application is a constraint problem — a budget, a weight limit, a minimum score — where you translate a sentence into an inequality before solving anything.
| Phrase | Symbol |
|---|---|
| at least, no less than, minimum | ≥ |
| at most, no more than, maximum | ≤ |
| more than, exceeds | > |
| fewer than, less than | < |
A caterer charges a $150 flat fee plus $12 per guest. A client's budget is at most $1,000. What is the maximum number of guests?
When a real-world quantity must be a whole number, never use standard rounding. For a maximum under a ≤ constraint, round down even if the decimal is .83 or higher.
Test day strategy for inequalities
| Question signal | Fastest approach |
|---|---|
| Solving for a variable | Isolate like an equation; flip only when multiplying/dividing by a negative |
| Three-part inequality (AND) | Apply the same operation to all three sections at once |
| "At least / at most" word problem | Translate the phrase to ≥ or ≤ before writing anything else |
| Whole-number answer required | Round down for a maximum, round up for a minimum |
| System of inequalities | Test the point against each inequality; one failure disqualifies it |
Now put it to work
Three quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Inequalities
Solving, the flip rule, and compound inequalities.
Start quiz → Quiz 2Inequalities 2
Graphing on a number line and two-variable inequalities.
Start quiz → Quiz 3Inequalities 3
Systems of inequalities and word problems.
Start quiz →