All articles / PSAT/NMSQT Math / Heart of Algebra
Linear Inequalities for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about linear inequalities in one place: solving inequalities, the flip rule, compound inequalities, graphing on a number line, and word problems — with step-by-step examples, worked problems, and a free practice quiz.
Practice this quiz
Free · No signupLinear Inequalities
Solving, the flip rule, compound inequalities, and graphing.
What 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, it 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 |
Solving & the flip rule
Solving a linear inequality works exactly like solving a linear equation — same moves, same order — with one key exception.
Solve for x: −4x + 3 ≤ 19
The most frequent inequality error: forgetting to flip the sign when dividing by a negative, or flipping it when the divisor was actually positive. Always check the sign of the number you're dividing by.
Compound inequalities
A compound inequality combines two conditions on the same variable, joined by AND or OR.
Solve for x: −6 ≤ 3x + 3 < 15
In a three-part compound inequality, apply every operation to all three sections at the same time — the left side, the middle expression, and the right side all move together.
Graphing on a number line
- 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 ≤.
If the PSAT shows a number line and asks for the matching inequality, read the circle type first (open/closed determines </≤), then the shading direction (determines which variable is larger).
Inequality word problems
| Phrase | Symbol |
|---|---|
| at least, no less than, minimum | ≥ |
| at most, no more than, maximum | ≤ |
| more than, exceeds | > |
| fewer than, less than | < |
A rental company charges a $30 base fee plus $8 per hour. A customer's budget is at most $100. What is the maximum number of hours they can rent?
When a real-world quantity must be a whole number, don't use standard rounding. For a maximum under a ≤ constraint, round down even if the decimal is .75 or higher.
Common PSAT traps
- Forgetting to flip the sign: only triggered by multiplying or dividing by a negative — never by adding or subtracting.
- Solving only part of a three-part compound inequality: every operation must apply to all three sections.
- Reversing "at least" and "at most": at least means ≥, at most means ≤ — these are easy to swap under time pressure.
- Standard rounding on whole-number answers: always round in the direction that keeps the answer valid, not to the nearest whole number.
Test day strategy for linear inequalities
| Question signal | Fastest approach |
|---|---|
| Solving for a variable | Isolate like an equation; flip the sign 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 — never to the nearest |
| Shaded number line given | Identify open/closed circle first, then shading direction |
Now put it to work
One comprehensive quiz covering the full topic — solving inequalities, the flip rule, compound inequalities, and graphing.
Linear Inequalities
Solving, the flip rule, compound inequalities, and graphing.