Inequalities: Solving & Graphing
SAT Math — Complete Mastery Guide
Algebra
Number Line Graphs
Coordinate Plane
Absolute Value & Quadratic
1. Introduction to Inequalities
What Is an Inequality?
An inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike an equation (which says two things are equal), an inequality says one side is greater than, less than, or not equal to the other.
| Symbol | Meaning | Example | Read As |
|---|---|---|---|
| < | Less than | $x < 5$ | "x is less than 5" |
| > | Greater than | $x > 3$ | "x is greater than 3" |
| ≤ | Less than or equal to | $x \le 5$ | "x is less than or equal to 5" |
| ≥ | Greater than or equal to | $x \ge 3$ | "x is greater than or equal to 3" |
| ≠ | Not equal to | $x \ne 0$ | "x is not equal to 0" |
Key Difference — Equation vs. Inequality:
• $x = 5$ → exactly one value (5).
• $x < 5$ → infinitely many values (all real numbers less than 5).
The set of all values that make an inequality true is called the solution set.
• $x = 5$ → exactly one value (5).
• $x < 5$ → infinitely many values (all real numbers less than 5).
The set of all values that make an inequality true is called the solution set.
Reading Tip
Always read from the variable toward the number: $x < 5$ → "x is less than 5." You can also read right-to-left: "5 is greater than x."
2. Properties of Inequalities
You solve inequalities almost exactly like equations — with one critical exception.
Direction STAYS the Same
- Addition: If $a < b$, then $a + c < b + c$
- Subtraction: If $a < b$, then $a - c < b - c$
- Multiply by positive: If $a < b$ and $c > 0$, then $ac < bc$
- Divide by positive: If $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$
Direction REVERSES ⚠️
- Multiply by negative: If $a < b$ and $c < 0$, then $ac \;\mathbf{>}\; bc$
- Divide by negative: If $a < b$ and $c < 0$, then $\frac{a}{c} \;\mathbf{>}\; \frac{b}{c}$
THE #1 RULE: Multiplying or dividing both sides by a negative number FLIPS the inequality sign. This is the most common source of errors on the SAT!
Quick Proof of the Flip Rule
We know $2 < 5$. Multiply both sides by $-1$: $-2$ and $-5$. On the number line $-2$ is to the right of $-5$, so $-2 > -5$. The inequality flipped! ✓