Combining Open Sentences
10-4Solving Combined Inequalities
Overview
AND: both must be true (intersection). OR: at least one must be true (union).
Key ideas
- Compound AND: solutions satisfy both. (e.g., 2 < x < 5)
- Compound OR: solutions satisfy either. (e.g., x < −1 or x > 3)
- Graph on a number line.
Worked examples
Example 1
Problem. Solve 2 ≤ x + 1 < 6.
Solution. Subtract 1: 1 ≤ x < 5.
Example 2
Problem. Solve x < −2 or x > 4.
Solution. Two separate rays on a number line.
Practice
Try each one. Click Show answer when ready.
- 1.
Solve −3 < 2x − 1 < 5.
- 2.
Solve x ≤ 0 or x ≥ 5.
- 3.
Solve x + 2 > 0 AND x − 3 < 0.
Challenge problems
A little tougher — great for test prep. Click Show answer when ready.
- 1.
Solve: −2 < 3x + 1 ≤ 10.
- 2.
Solve: 5 ≤ 2 − x < 8.