1. Introduction
A compound inequality is formed when two simple inequalities are combined into one statement. These inequalities help describe a range of values more accurately.
Compound inequalities use the words AND or OR to join two comparisons.
2. Types of Compound Inequalities
There are two main types of compound inequalities:
2.1. 1. AND Inequality (Intersection)
AND inequalities show values that satisfy both conditions at the same time.
Example: 2 < x < 7
This means x must be greater than 2 and less than 7.
The solution is the overlap of the two inequalities.
2.2. 2. OR Inequality (Union)
OR inequalities show values that satisfy at least one of the conditions.
Example: x < −3 OR x ≥ 5
This gives two separate solution regions on the number line.
3. Solving AND Inequalities
AND inequalities often appear in a combined form like:
a < x < b
To solve:
- Treat the inequality as three parts.
- Perform the operation on all three sides.
- Keep the inequality signs in the same direction.
3.1. Example 1
Solve: 1 < 2x < 9
Divide all three sides by 2:
1/2 < x < 9/2
3.2. Example 2
Solve: −4 ≤ 3x + 2 < 8
Subtract 2:
−6 ≤ 3x < 6
Divide by 3:
−2 ≤ x < 2
4. Solving OR Inequalities
OR inequalities represent values that satisfy at least one condition. They give two separate intervals.
4.1. Example 1
Solve: x − 3 > 2 OR x + 1 ≤ −4
Case 1: x − 3 > 2 → x > 5
Case 2: x + 1 ≤ −4 → x ≤ −5
Final Answer: x > 5 OR x ≤ −5
4.2. Example 2
Solve: 2x ≥ 6 OR −3x > 9
Case 1: 2x ≥ 6 → x ≥ 3
Case 2: −3x > 9 → divide by −3 → x < −3
Final Answer: x ≥ 3 OR x < −3
5. Graphical Representation
AND and OR inequalities look different on the number line:
5.1. Graphing AND Inequalities
AND → overlapping or continuous region.
Example: 1 ≤ x < 4 → closed at 1, open at 4, shaded between.
5.2. Graphing OR Inequalities
OR → two separate shaded regions.
Example: x < −2 OR x ≥ 3 → open at −2, closed at 3, arrows outward.
6. Examples
- Solve: 0 < x + 4 ≤ 10 → −4 < x ≤ 6
- Solve: 2x ≤ −8 OR x − 1 > 4 → x ≤ −4 OR x > 5
- Solve: −3x > 9 → x < −3
7. Common Mistakes
- Forgetting to flip the inequality when multiplying by a negative number.
- Mixing AND with OR conditions.
- Incorrectly combining solution sets.
- Using closed circles for strict inequalities.
- Not applying operations to all three parts in AND inequalities.
8. Quick Practice
Solve the following:
- −2 < 3x ≤ 9
- x + 5 ≥ 12 OR x − 3 < −1
- 4 − 2x > 0
- 2x − 1 < 7 AND x + 3 ≥ 5
9. Summary
- Compound inequalities combine two conditions.
- AND → intersection of solutions.
- OR → union of solutions.
- AND has continuous regions; OR has two separated regions.
- Always check if boundaries are included or not.