What is a two-step inequality?

A two-step inequality is an inequality that requires two operations to solve — usually a combination of:

• Addition or subtraction

• Multiplication or division

Think of it as a slightly longer version of one-step inequalities.

Goal: Solve for the variable by undoing operations in reverse order.

Step-by-step solving rules

1. Undo addition or subtraction first

   • Move numbers across the inequality using inverse operations

2. Undo multiplication or division next

   • Isolate the variable

3. Important: If you multiply or divide by a negative number, flip the inequality sign.

Symbols reminder

Examples

Example 1: Solve 2x + 3 > 7

Step 1: Subtract 3 from both sides

Step 2: Divide both sides by 2

✅ Solution: all numbers greater than 2

Example 2: Solve −3y + 5 ≤ 11

Step 1: Subtract 5 from both sides

Step 2: Divide both sides by −3 → flip inequality

✅ Solution: all numbers greater than or equal to −2

Example 3: Solve 4x − 7 < 9

Step 1: Add 7 to both sides

Step 2: Divide by 4

✅ Solution: all numbers less than 4

Graphing two-step inequalities on a number line

• < or > → open circle

• ≤ or ≥ → closed circle

• Shade left for smaller numbers

• Shade right for larger numbers

Example: x ≥ −2

• Closed circle at −2

• Shade to the right

Common beginner mistakes

1. ❌ Forgetting to flip the inequality when dividing by a negative number

2. ❌ Forgetting two steps are needed

3. ❌ Using wrong shading or circle type on the number line

Why two-step inequalities matter

• They are a bridge between simple and complex inequalities

• Used in real-world problems like budgets, distances, and temperatures

• Foundation for multi-step inequalities and systems of inequalities