What are compound inequalities?

A compound inequality combines two inequalities into one statement using either “and” or “or”.

• “And” → both conditions must be true

• “Or” → at least one condition must be true

Think of it as a way to narrow or widen the range of solutions.

Symbols reminder

Types of compound inequalities

1️⃣ “And” inequalities (intersection)

• Written as:

• Solution: values of x that satisfy both inequalities at the same time

• Graph: overlap of two solution sets

Example:

• x must be greater than 1 AND less than or equal to 5

• Solution: 1 < x ≤ 5

2️⃣ “Or” inequalities (union)

• Written as:

• Solution: values of x that satisfy either inequality

• Graph: union of two solution sets

Example:

• x can be less than or equal to −2 OR greater than or equal to 4

• Graph: shade both ends

How to solve compound inequalities

Step 1: Solve each inequality separately

• Treat like one-step, two-step, or multi-step inequality

Step 2: Combine solutions

• “And” → intersection (overlap)

• “Or” → union (combine all)

Example 1: “And” compound inequality

Step 1: Add 1 to all parts

Step 2: Divide all parts by 3

✅ Solution: x is greater than 1 and less than or equal to 3

Example 2: “Or” compound inequality

Step 1: Solve each inequality

1️⃣ 2x − 5 < −1
Add 5 → 2x < 4
Divide 2 → x < 2

2️⃣ 3x + 2 ≥ 11
Subtract 2 → 3x ≥ 9
Divide 3 → x ≥ 3

Step 2: Combine using “or”

✅ Solution: x < 2 OR x ≥ 3

Graphing compound inequalities

• “And” → shade only the overlap

• “Or” → shade both solution sets

• Use open circle for < or >

• Use closed circle for ≤ or ≥

Common beginner mistakes

1. ❌ Mixing up “and” vs. “or”

2. ❌ Forgetting to solve each part completely

3. ❌ Graphing incorrectly (overlap vs. union)

4. ❌ Not flipping inequality when dividing/multiplying by negative

Why compound inequalities matter

• Represent more complex real-world conditions

• Useful for budget limits, ranges, and constraints

• Foundation for advanced algebra and systems of inequalities