Introduction to Inequalities

Not every mathematical relationship is about equality. When we say "you must be at least 48 inches tall to ride," we are describing an inequality -- a statement that one value is greater than, less than, or not equal to another.

Inequality Symbols

Symbol	Meaning	Example
<	less than	3 < 7
>	greater than	10 > 4
≤	less than or equal to	x ≤ 5 (x can be 5)
≥	greater than or equal to	x ≥ -2 (x can be -2)

Reading the Symbol

The symbol always "points" to the smaller value. Think of the arrow opening toward the larger side: 3 < 7 means 3 is smaller, 7 is larger.

Solving One-Step Inequalities

Solving inequalities works almost exactly like solving equations. You can add, subtract, multiply, or divide both sides -- with one critical exception we will cover shortly.

Worked Example 1: Addition/Subtraction

Solve x + 4 > 11

Subtract 4 from both sides: x > 11 - 4
Simplify: x > 7
Solution: all numbers greater than 7 (not including 7)

On a number line, use an open circle at 7 and shade to the right.

Solving Two-Step Inequalities

Worked Example 2: Two Steps

Solve 3x - 5 ≤ 10

Add 5 to both sides: 3x ≤ 15
Divide both sides by 3: x ≤ 5
Solution: all numbers less than or equal to 5

On a number line, use a closed (filled) circle at 5 and shade to the left.

The Critical Rule: Flipping the Sign

When you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

Why? Consider this: 2 < 5 is true. Multiply both sides by -1: -2 and -5. But -2 > -5. The order flipped!

Worked Example 3: Flipping the Sign

Solve -4x > 20

Divide both sides by -4. Since we are dividing by a negative, flip the inequality.
x < -5
Solution: all numbers less than -5

The #1 Inequality Mistake

Forgetting to flip the inequality when multiplying or dividing by a negative is the most common error. Always ask yourself: "Am I multiplying or dividing by a negative?" If yes, reverse the sign.

Number Line Representation

Open circle (○): the endpoint is NOT included

Used with < and >

Closed circle (●): the endpoint IS included

Used with ≤ and ≥

Practice Problems

1. Solve: x - 3 ≥ 8

Show Solution

Add 3 to both sides: x ≥ 11. Closed circle at 11, shade right.

2. Solve: 2x + 1 < 9

Show Solution

Subtract 1: 2x < 8. Divide by 2: x < 4. Open circle at 4, shade left.

3. Solve: -5x ≤ 30

Show Solution

Divide by -5 and flip the sign: x ≥ -6. Closed circle at -6, shade right.

4. Solve: 7 - 2x > 3

Show Solution

Subtract 7: -2x > -4. Divide by -2, flip: x < 2. Open circle at 2, shade left.

5. A parking garage charges $3 plus $2 per hour. You have $15. Write and solve an inequality for the number of hours you can park.

Show Solution

3 + 2h ≤ 15. Subtract 3: 2h ≤ 12. Divide by 2: h ≤ 6. You can park for at most 6 hours.

Lesson Summary

Inequalities use <, >, ≤, ≥ to compare values.
Solve them like equations, but flip the inequality sign when multiplying or dividing by a negative.
Open circles for strict inequalities (<, >); closed circles for inclusive (≤, ≥).
The solution is a range of values, not a single number.