Overview
Linear inequalities in one variable appear on every digital SAT exam and are tested throughout the Algebra domain (approximately 13–15 of 44 math questions). The SAT tests inequality solving, writing inequalities from real-world constraints, interpreting solution sets on a number line, and working with compound inequalities (AND/OR). The critical rule that separates inequalities from equations is that multiplying or dividing by a negative number reverses the inequality direction. Approximately 30% of inequality questions appear in real-world context requiring translation of verbal constraints into mathematical inequalities.
Key Points
1. Solving Linear Inequalities — The Process
Solving a linear inequality follows the same steps as a linear equation, with one critical addition:
Step 1: Distribute parentheses
Step 2: Combine like terms on each side
Step 3: Move variable terms to one side
Step 4: Isolate the variable
Step 5: *** If you multiply or divide by a NEGATIVE, FLIP the inequality sign ***
Step 6: Verify — plug a value from your solution range into the original inequality
Examples:
Positive coefficient (sign stays):
2x - 3 < 7
2x < 10
x < 5 ← sign unchanged
Negative coefficient (sign flips):
-3x + 4 > 1
-3x > -3
x < 1 ← sign FLIPPED when dividing by -3
2. Compound Inequalities — AND vs. OR
| Type | Notation | Meaning | Number Line |
|---|---|---|---|
| AND (intersection) | a < x < b | x satisfies BOTH | Segment between a and b |
| OR (union) | x < a OR x > b | x satisfies EITHER | Two rays pointing outward |
AND example: -2 < x ≤ 5 (x is between -2 and 5, not including -2)
OR example: x < -2 OR x > 5 (x is outside the interval)
Key mnemonic: “Less than = IN between (AND), Greater than = split APART (OR)” — especially useful for absolute value inequalities that convert into compound form.
3. Writing Inequalities from Real-World Constraints
The SAT frequently presents budgets, capacities, or quantity limits that must be modeled with an inequality.
| English phrase | Mathematical symbol |
|---|---|
| at least / no less than / minimum | ≥ |
| at most / no more than / maximum | ≤ |
| more than / greater than / exceeds | > |
| fewer than / less than | < |
Translation framework:
Step 1: Identify the constraint (what is being limited)
Step 2: Identify "what you have" and "what is allowed"
Step 3: Write: [expression for what you have] [symbol] [limit]
Step 4: Solve and interpret in context
Example: “A student can spend at most 8. How many books b can they buy?“
8b ≤ 50
b ≤ 6.25
b ≤ 6 (since b must be a whole number)
4. Graphing on a Number Line
Open circle ◦ ← use for strict inequalities < or > (endpoint NOT included)
Closed circle ● ← use for non-strict inequalities ≤ or ≥ (endpoint IS included)
x > 3: ──────◦════════>
x ≥ 3: ──────●════════>
x < 3: <════════◦──────
x ≤ 3: <════════●──────
5. SAT-Specific Patterns
- Backsolving on multiple-choice: Test each answer choice directly in the inequality — often faster than algebraic solving.
- Checking solution sets: After solving, plug a value from your solution range into the original inequality to confirm it is satisfied.
- Avoid testing 0, -1, or 1 when verifying — these values can behave atypically and mask errors.
- Desmos: Can graph inequalities and shade automatically; useful for compound and two-variable inequalities.
- 30% of questions are in context — know the key phrase translations cold.
Pitfalls and Common Mistakes
Mistake 1: Forgetting to flip the inequality sign when dividing by a negative
- Wrong approach: -2x > 6 → x > -3
- Correct approach: -2x > 6 → x < -3 (flip the sign when dividing by -2)
- Fix: Any time you divide or multiply both sides by a negative number, immediately reverse the inequality direction as a habit.
Mistake 2: Confusing AND vs. OR in compound inequalities
- Wrong approach: x < -2 OR x > 5 is written as -2 < x < 5
- Correct approach: -2 < x < 5 is AND (intersection); two separate pieces x < -2 OR x > 5 is OR (union)
- Fix: AND means the two conditions must be true simultaneously — it’s a single connected interval. OR means the solution is in either region — two separate pieces.
Mistake 3: Using the wrong inequality symbol for word problems
- Wrong approach: “At least 10 students” → x < 10
- Correct approach: “At least 10 students” → x ≥ 10
- Fix: Memorize the key phrase table. “At least” always means ≥ (the stated number is the minimum allowed).
Mistake 4: Treating the solution as a single point instead of a range
- Wrong approach: x < 5, so the answer is x = 4
- Correct approach: x < 5 means ALL values less than 5 are solutions; the question may ask for a specific interpretation or the largest integer, etc.
- Fix: Re-read what the question asks for — sometimes it wants the maximum integer value, sometimes it wants the inequality itself.
Related Entries
- Linear_Equations — Solving equations (the foundation before inequalities)
- Systems_Linear_Inequalities — Two-variable extension with graphical feasible regions
- Absolute_Value — Absolute value inequalities convert to compound inequalities
- Linear_Functions_Graphs — Graphical representation of linear relationships
- Systems_Linear_Equations — Systems analogy for inequalities
Quick Reference Card
| Concept | Rule |
|---|---|
| Multiply/divide by negative | Flip the inequality sign |
| AND compound inequality | a < x < b — single connected interval |
| OR compound inequality | x < a OR x > b — two separate pieces |
| ”at least” | ≥ |
| “at most” | ≤ |
| “more than” / “exceeds” | > |
| “fewer/less than” | < |
| Open circle on number line | Strict inequality (< or >) |
| Closed circle on number line | Non-strict inequality (≤ or ≥) |
| Verify solution | Plug a value from the solution range into the original inequality |
| Backsolving | Test answer choices directly in the inequality (often fastest on multiple choice) |