What are multi-step inequalities?

A multi-step inequality is an inequality that requires more than two steps to solve.

• Usually involves addition/subtraction, multiplication/division, and combining like terms.

• Often appears in word problems or complex algebra expressions.

Goal: Solve for the variable while keeping the inequality true.

Symbols reminder

Steps to solve multi-step inequalities

1. Simplify both sides (combine like terms, distribute)

2. Move variable terms to one side

3. Move constants to the other side

4. Divide or multiply to isolate the variable

5. Flip the inequality sign if multiplying or dividing by a negative number

Examples

Example 1: Solve 3x − 5 + 2x ≤ 10 + 4

Step 1: Combine like terms

Step 2: Add 5 to both sides

Step 3: Divide both sides by 5

✅ Solution: all numbers less than or equal to 19/5

Example 2: Solve −2(x − 3) + 5 > 7

Step 1: Distribute −2

Step 2: Subtract 11

Step 3: Divide by −2 → flip inequality

✅ Solution: all numbers less than 2

Example 3: Solve 4x + 3 < 2x + 11

Step 1: Subtract 2x from both sides

Step 2: Subtract 3 from both sides

Step 3: Divide by 2

✅ Solution: all numbers less than 4

Graphing multi-step inequalities

• < or > → open circle

• ≤ or ≥ → closed circle

• Shade left for smaller numbers

• Shade right for larger numbers

Example: x ≤ 19/5

• Closed circle at 19/5

• Shade to the left

Common beginner mistakes

1. ❌ Forgetting to distribute negative signs

2. ❌ Forgetting to flip the inequality when multiplying/dividing by a negative

3. ❌ Forgetting combine like terms first

4. ❌ Confusing shading on the number line

Why multi-step inequalities matter

• Used in real-world problems like budgets, speed limits, distances

• Foundation for compound inequalities and systems of inequalities

• Helps develop critical algebra problem-solving skills