How to Solve Inequalities Step by Step

Inequalities are like equations, with one twist: multiplying or dividing by a negative flips the sign.

📐 Algebra⏱️ ~12 min read📊 Intermediate

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What You'll Learn

✓The symbols: <, >, ≤, ≥ and what they mean
✓Why and when to flip the inequality sign
✓Graphing solutions on a number line
✓Solving compound inequalities (like 2 < x < 7)

Equations vs. Inequalities

An equation says two things are equal. An inequality says one side is greater or less than the other. The solving process is almost identical—with one critical difference.

Inequality Symbols

< means "less than"
> means "greater than"
≤ means "less than or equal to"
≥ means "greater than or equal to"

⚠️ The Critical Rule

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

For a visual explanation of why the sign flips, see Khan Academy's inequality video.

Step-by-Step Method

Step 1: Simplify Each Side

Distribute and combine like terms, just like with equations.

Step 2: Move Variable Terms to One Side

Add or subtract to get all x terms on one side. This does NOT flip the sign.

Step 3: Move Constants to the Other Side

Add or subtract to isolate the variable term. Still no sign flip.

Step 4: Divide or Multiply to Solve

Isolate x by dividing/multiplying. If you divide or multiply by a negative, FLIP THE SIGN!

-3x > 12 → divide by -3 → x < -4

Notice: > became <

Step 5: Write the Solution

Express as an inequality (x > 3), interval notation ((3, ∞)), or graph it.

Worked Examples

Example 1: No sign flip needed

Solve: 2x + 5 > 11

2x + 5 > 11

2x > 11 - 5

2x > 6

x > 3

Test x=4: 2(4)+5 = 13 > 11 ✓

Example 2: Sign flip required!

Solve: -4x + 2 ≤ 14

-4x + 2 ≤ 14

-4x ≤ 14 - 2

-4x ≤ 12

Divide by -4 and FLIP:

x ≥ -3

Test x=0: -4(0)+2 = 2 ≤ 14 ✓

Example 3: Variables on both sides

Solve: 5x - 3 < 2x + 9

5x - 3 < 2x + 9

5x - 2x < 9 + 3

3x < 12

x < 4

Example 4: Compound inequality

Solve: -2 < 3x + 1 ≤ 10

-2 < 3x + 1 ≤ 10

Subtract 1 from ALL parts:

-3 < 3x ≤ 9

Divide ALL parts by 3:

-1 < x ≤ 3

x is between -1 (not included) and 3 (included)

Graphing on a Number Line

Open vs Closed Circles

< or > → Use an open circle (the endpoint is NOT included)
≤ or ≥ → Use a closed/filled circle (the endpoint IS included)

Arrow Direction

x > 3 → Arrow points RIGHT (toward bigger numbers)
x < 3 → Arrow points LEFT (toward smaller numbers)

Common Mistakes to Avoid

⚠️

Forgetting to flip when dividing by negative

This is the #1 mistake. -2x < 6 becomes x > -3 (not x < -3).

⚠️

Flipping when you shouldn't

Adding/subtracting (even negative numbers) does NOT flip the sign. Only multiplying/dividing by negatives does.

⚠️

Wrong circle type on graph

≤ and ≥ use filled circles. < and > use open circles.

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