How to Solve Compound Inequalities

 Solving Compound Inequalities

Compound inequalities are two inequalities joined by "and" or "or". They describe a range of values that satisfy both or either condition.

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🔸 Types of Compound Inequalities

1. “And” Compound Inequality

– Both conditions must be true

– The solution is where the two inequalities overlap

 Example: 1 < x ≤ 5

 Means x is greater than 1 and less than or equal to 5

2. “Or” Compound Inequality

– Either condition can be true

– The solution includes both sides

Example: x < –3 or x > 2

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✅ Example 1: Solve 1 < x + 2 ≤ 5

Step 1: Split into two parts:

– 1 < x + 2

– x + 2 ≤ 5

Step 2: Solve each:

– 1 < x + 2 → Subtract 2 → –1 < x

– x + 2 ≤ 5 → Subtract 2 → x ≤ 3

✅ Final Answer: –1 < x ≤ 3

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✅ Example 2: Solve x – 4 < –1 or x + 2 > 7

Step 1: Solve both parts:

– x – 4 < –1 → Add 4 → x < 3

– x + 2 > 7 → Subtract 2 → x > 5

✅ Final Answer: x < 3 or x > 5

Summary of Compound Inequalities 

Type: "And"

Both conditions must be true.

The solution is the overlap (intersection) of the two inequalities.

Example: -1 < x ≤ 3

(This means x is greater than -1 and less than or equal to 3.)

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Type: "Or"

At least one condition must be true.

The solution includes values that satisfy either one of the inequalities.

Example: x < 3 or x > 5

(This means x can be less than 3 or greater than 5.)

📝 Practice Questions 

Solve these compound inequalities:

1. –2 < x + 1 ≤ 4

2. x – 3 > 2 or x + 4 < 0

3. 0 ≤ 2x + 2 < 6

4. x – 5 < –3 or x + 2 ≥ 7

5. –1 ≤ x – 2 < 5

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