Lesson 11: Compound Inequalities
A compound inequality combines two simple inequalities with the words “and†or “orâ€. The solution is the set of all numbers that make the statement true.
- AND → both inequalities must be true at the same time (intersection).
- OR → either inequality can be true (union).
AND Inequalities
Example: Solve \(2 < x \le 5\).
This means \(x\) is greater than 2 and less than or equal to 5. So the solution is all numbers between 2 and 5, not including 2 but including 5.
OR Inequalities
Example: Solve \(x < -1 \; \text{or} \; x \ge 3\).
This means \(x\) is less than -1 or \(x\) is greater than or equal to 3. The solution has two separate parts.
Steps to Solve Compound Inequalities
- Solve each inequality separately.
- For AND: take the overlap (intersection) of the solutions.
- For OR: combine (union) the solutions.
- Graph the solution on a number line.
More Examples
Example 1 (AND):
Solve: \(x - 1 > 2 \; \text{and} \; x + 3 \le 10\).
- First: \(x > 3\)
- Second: \(x \le 7\)
- Combined: \(3 < x \le 7\)
Example 2 (OR):
Solve: \(2x - 5 < -3 \; \text{or} \; x + 4 \ge 10\).
- First: \(2x < 2 \Rightarrow x < 1\)
- Second: \(x \ge 6\)
- Combined: \(x < 1 \; \text{or} \; x \ge 6\)
Practice Problems
- Solve and graph: \(0 < x \le 4\)
- Solve and graph: \(x \le -2 \; \text{or} \; x > 5\)
- Solve and graph: \(2x - 1 > 3 \; \text{and} \; x + 2 \le 9\)
- Solve and graph: \(x - 4 < 0 \; \text{or} \; 3x \ge 12\)
Lesson Summary
Compound inequalities combine two inequalities with AND or OR. AND → solutions overlap (between two values). OR → solutions combine (two separate regions). Always solve each inequality separately, then merge the results based on the connector.