PSAT/NMSQT Math: Linear Inequalities
Linear inequalities work almost exactly like linear equations, with one critical exception: multiplying or dividing by a negative number flips the direction of the inequality sign. This free, complete lesson covers solving one-step and multi-step inequalities, the sign-flip rule, inequalities involving fractions, and word problems that ask you to interpret a minimum or maximum value. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Solving One-Step Inequalities
A linear inequality compares two expressions using <, ≤, >, or ≥ instead of an equal sign. Solving one works just like solving an equation: use inverse operations to isolate the variable, and the solution is not a single value but an entire range of values.
Worked Example: Solve x + 6 > 10.
x + 6 > 10 → x > 4
1. Solve for x: x − 5 ≤ 8
Explanation: Adding 5 to both sides: x ≤ 13.
2. Solve for x: 3x > 21
Explanation: Dividing both sides by a positive number (3) keeps the direction of the inequality the same: x > 7.
3. Which value of x satisfies x + 2 ≥ 9?
Explanation: x + 2 ≥ 9 means x ≥ 7. Of the choices, only x = 8 satisfies this.
2. Multi-Step Inequalities and the Sign-Flip Rule
Multi-step inequalities need several operations, just like multi-step equations. The one rule that has no equation counterpart: whenever you multiply or divide both sides by a negative number, you must flip the direction of the inequality symbol.
Worked Example: Solve −3x + 5 < 20.
Subtract 5: −3x < 15. Divide by −3 and flip the symbol:
x > −5
1. Solve for x: −4x + 3 ≥ 15
Explanation: −4x ≥ 12. Dividing by −4 flips the inequality: x ≤ −3.
2. Solve for x: 2(x − 3) < 4x + 6
Explanation: 2x − 6 < 4x + 6 → −12 < 2x → dividing by a positive 2 keeps the direction: x > −6.
3. Solve for x: 5 − 2x ≤ 17
Explanation: −2x ≤ 12. Dividing by −2 flips the inequality: x ≥ −6.
4. A student solves −5x > 25 and writes x > −5. What error did they make?
Explanation: Dividing −5x > 25 by −5 requires flipping the sign, giving x < −5, not x > −5.
3. Inequalities with Fractions and the Distributive Property
Just like with equations, clearing fractions by multiplying every term by a common denominator, and distributing across parentheses first, makes an inequality much easier to solve. The sign-flip rule still applies any time you multiply or divide by a negative value.
Worked Example: Solve x3 − 2 ≥ 1.
Add 2: x/3 ≥ 3. Multiply both sides by 3 (positive, no flip):
x ≥ 9
1. Solve for x: x4 + 1 < 6
Explanation: x/4 < 5 → multiply both sides by 4 (positive, no flip): x < 20.
2. Solve for x: −x2 + 3 ≤ 7
Explanation: −x/2 ≤ 4 → multiply both sides by −2 and flip the sign: x ≥ −8.
3. Solve for x: 3(x + 2) ≥ 2(x + 5)
Explanation: 3x + 6 ≥ 2x + 10 → x ≥ 4 (only a positive coefficient subtraction occurs, so no flip is needed).
4. Word Problems and Interpreting Minimum/Maximum Values
Word problems involving inequalities usually describe a constraint, such as a budget, weight limit, or minimum score, using phrases like "at least," "at most," "no more than," or "greater than." Translate these phrases carefully into the correct inequality symbol, then solve or interpret the boundary value.
Worked Example: A delivery van can carry at most 2,000 pounds. Each box weighs 40 pounds. Write and solve an inequality for the maximum number of boxes, b, the van can carry.
40b ≤ 2000 → b ≤ 50, so the van can carry at most 50 boxes
1. To earn a B or higher, a student needs an average of at least 80 across 4 tests. Their first three scores are 75, 82, and 79. What is the minimum score, s, needed on the fourth test?
Explanation: (75 + 82 + 79 + s)/4 ≥ 80 → 236 + s ≥ 320 → s ≥ 84.
2. A phrase in a word problem reads "the temperature will not exceed 90°F." Which inequality correctly models this, using T for temperature?
Explanation: "Will not exceed" means the value can reach 90 but never go above it, so T ≤ 90 (using ≤, not strict <, since 90 itself is still allowed).
3. A gym membership costs $25 to join plus $15 per month. A customer has budgeted no more than $130 total. Write an inequality for the maximum number of months, m, they can attend.
Explanation: The fixed join fee is 25, and 15 per month is multiplied by m; the total must be no more than (≤) 130.
4. Using 25 + 15m ≤ 130 from the previous problem, what is the greatest number of whole months the customer can attend?
Explanation: 15m ≤ 105 → m ≤ 7. Since m represents whole months, the greatest possible value is 7.
Common Mistakes to Avoid
- Forgetting to flip the inequality symbol when multiplying or dividing both sides by a negative number. This is the single most common error on inequality problems.
- Mistranslating word phrases into the wrong symbol. "At least" means ≥, "at most" or "no more than" means ≤, and "more than" or "fewer than" (without "at") mean strict > or <.
- Ignoring real-world constraints on the answer, such as requiring a whole number of months, tickets, or boxes, even when the algebra alone gives a decimal boundary.
- Distributing incorrectly across parentheses before applying the sign-flip rule, which can lead to solving the wrong inequality entirely.
- Treating a compound or boundary condition carelessly, forgetting whether the boundary value itself is included (≤/≥) or excluded (</>).
Frequently Asked Questions
Why do you flip the inequality sign when multiplying or dividing by a negative number?
Multiplying or dividing by a negative number reverses the relative order of numbers on the number line, so what was "less than" becomes "greater than" to keep the statement true. This has no equivalent rule for equations, since equality doesn't depend on direction.
How do I know if a word problem needs ≤/≥ or strict </>?
Phrases like "at least," "at most," "no more than," and "no less than" all include the boundary value itself, so use ≤ or ≥. Phrases like "more than" or "fewer than" alone (without "at") exclude the boundary, so use strict < or >.
Where can I practice more problems like these?
The Linear Inequalities quiz in the PSAT/NMSQT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.