How to Solve Inequalities Step-by-Step

What Is an Inequality?

An inequality is like an equation, but instead of saying two things are equal, it says one is greater than, less than, or not equal to the other.

While an equation like x + 3 = 7 has one solution (x = 4), an inequality like x + 3 > 7 has many solutions (x > 4 — any number greater than 4).

Inequality Symbols

| Symbol | Meaning | Example | |---|---|---| | > | Greater than | x > 5 | | < | Less than | x < 3 | | ≥ | Greater than or equal to | x ≥ 2 | | ≤ | Less than or equal to | x ≤ 10 | | ≠ | Not equal to | x ≠ 0 |

Tip: The symbol always "opens" toward the larger value. Think of it as a hungry alligator that eats the bigger number.

Solving One-Step Inequalities

Solve them just like equations — do the same operation on both sides.

Example 1: x + 5 > 12

Subtract 5 from both sides: x > 7

Example 2: x − 3 ≤ 8

Add 3 to both sides: x ≤ 11

Example 3: 4x ≥ 20

Divide both sides by 4: x ≥ 5

Example 4: x/2 < 6

Multiply both sides by 2: x < 12

Solving Multi-Step Inequalities

Follow the same process as multi-step equations:

Example: 3x + 7 > 16

Step 1: Subtract 7 from both sides: 3x > 9

Step 2: Divide by 3: x > 3

Example: 2(x − 4) ≤ 10

Step 1: Distribute: 2x − 8 ≤ 10

Step 2: Add 8: 2x ≤ 18

Step 3: Divide by 2: x ≤ 9

Example: (x + 5)/3 ≥ 4

Step 1: Multiply by 3: x + 5 ≥ 12

Step 2: Subtract 5: x ≥ 7

The Flip Rule: Multiplying or Dividing by Negatives

This is the most important rule in inequalities and the one students forget most:

When you multiply or divide both sides by a NEGATIVE number, you MUST FLIP the inequality sign.

Example: −2x > 8

Divide by −2 AND flip the sign: x < −4

Why does the sign flip?

Think about it: 3 > 1 is true. But if you multiply both sides by −1: −3 > −1? No! −3 < −1.

Multiplying by a negative reverses the order of numbers, so you must reverse the inequality.

Example: −x/4 ≤ 3

Multiply by −4 AND flip: x ≥ −12

Example: 5 − 3x > 14

Subtract 5: −3x > 9 Divide by −3 AND flip: x < −3

Graphing Inequalities on a Number Line

Visualizing solutions on a number line helps you understand them:

• Open circle (○) for > or < (the endpoint is NOT included)
• Closed circle (●) for ≥ or ≤ (the endpoint IS included)
• Arrow pointing in the direction of the solutions

Examples:

• x > 3 → open circle at 3, arrow pointing right
• x ≤ −2 → closed circle at −2, arrow pointing left
• x ≥ 0 → closed circle at 0, arrow pointing right

Compound Inequalities

A compound inequality combines two inequalities. There are two types:

"And" compound inequalities

Both conditions must be true simultaneously.

Example: 2 < x ≤ 7

This means x is greater than 2 AND less than or equal to 7.

Solving example: −3 < 2x + 1 ≤ 9

Subtract 1 from all three parts: −4 < 2x ≤ 8

Divide all parts by 2: −2 < x ≤ 4

"Or" compound inequalities

At least one condition must be true.

Example: x < −1 OR x > 5

This means x is either less than −1 or greater than 5.

Practice Problems

Solve each inequality:

1. x + 8 > 15
2. 3x − 4 ≤ 11
3. −5x > 25
4. 2(x + 3) ≥ 14
5. Solve and describe: −1 ≤ 3x + 2 < 11

Answers: (1) x > 7 (2) x ≤ 5 (3) x < −5 (4) x ≥ 4 (5) −1 ≤ x < 3

Need More Help?

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