Linear Inequalities — SAT Math Explained
A mathematical statement that compares a linear expression to a value using inequality signs (<, >, ≤, ≥), representing a range of solutions rather than a single value.
The Core Idea
Unlike equations that have one answer, inequalities have infinitely many — an entire region or interval of solutions. The critical flip rule makes inequalities unique: multiplying or dividing by a negative number reverses the inequality sign.
Key Vocabulary
< or > — the boundary value is NOT included (open circle or dashed line)
≤ or ≥ — the boundary IS included (closed circle or solid line)
All values of the variable that make the inequality true
A visual representation of the solution set using arrows and circles
Solving Process
1. Solve exactly like an equation — isolate the variable using inverse operations
2. CRITICAL: If you multiply or divide both sides by a NEGATIVE number, FLIP the inequality sign
3. Graph the solution on a number line: open circle for strict (<, >), closed for non-strict (≤, ≥)
4. Shade the correct direction — test a point to confirm
The Critical Rule
Multiplying or dividing by a negative flips the sign because negative numbers reverse order on the number line. Example: 5 > 3, but -5 < -3.
Writing Solutions Three Ways
x > 4
(4, ∞)
Open circle at 4, arrow pointing right
Common Errors to Avoid
Forgetting to flip the sign when dividing by a negative
Using a closed circle when the sign is strict (< or >)
Shading the wrong direction on the number line
Practice: Linear Inequalities
5 SAT-style questions. Select your answer and get an instant explanation.
Solve: x + 5 > 12
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