Two conditions at once
A compound inequality joins two inequalities.
| Type | Meaning | Example |
|---|---|---|
| And | both must be true | x > 2 and x < 7 |
| Or | at least one must be true | x < −1 or x > 4 |
"And" inequalities
"x > 2 and x < 7" can be written as 2 < x < 7: x is between 2 and 7. The solution is the intersection (overlap) of the two sets.
Number line: open circles at 2 and 7, shaded between them.
Solving
Do the same operation to all three parts.
Example: −3 ≤ 2x + 1 < 9
- Subtract 1: −4 ≤ 2x < 8
- Divide by 2: −2 ≤ x < 4
If you multiply or divide by a negative, reverse both signs.
Example: 1 < −x < 5 → multiply by −1 → −1 > x > −5 → −5 < x < −1.
"Or" inequalities
The solution is the union: everything in either set.
Example: 3x < −6 or x − 2 > 3
- x < −2 or x > 5
Number line: shading to the left of −2 and to the right of 5, with a gap between.
Interval notation
Interval notation uses brackets:
- [ ] means the endpoint is included (≤ or ≥)
- ( ) means it’s not included (< or >)
- ∞ always gets ( )
| Inequality | Interval notation |
|---|---|
| 2 < x < 7 | (2, 7) |
| −2 ≤ x < 4 | [−2, 4) |
| x ≥ 3 | [3, ∞) |
| x < −2 or x > 5 | (−∞, −2) ∪ (5, ∞) |
Real-life examples
- A safe body temperature: 36.1 ≤ T ≤ 37.2 (°C)
- A ride requires height h ≥ 120 cm and h ≤ 200 cm
- A discount applies if age < 12 or age ≥ 65
When there’s no solution
"x < 2 and x > 5" has no solution: no number is both less than 2 and greater than 5.
"x < 5 or x > 2" covers all real numbers.
Worked examples
Example 1: Solve 4 < x + 3 ≤ 10 → 1 < x ≤ 7.
Example 2: Solve 2x ≤ −4 or 3x > 9 → x ≤ −2 or x > 3.
Example 3: Write −1 ≤ x < 6 in interval notation → [−1, 6).
Common mistakes
- Only applying an operation to two of the three parts.
- Shading between the values for an "or" inequality.
- Using the wrong bracket type in interval notation.
Try it yourself
- Solve −5 < x − 2 < 4.
- Solve 6 ≤ 3x ≤ 18.
- Solve x + 1 < 0 or 2x ≥ 10.
- Write x > −3 and x ≤ 8 in interval notation.
- Does "x > 6 and x < 1" have a solution?
Answers: 1) −3 < x < 6 2) 2 ≤ x ≤ 6 3) x < −1 or x ≥ 5 4) (−3, 8] 5) no
Quick recap
- And = overlap (intersection); or = combine (union).
- Apply operations to all parts of a three-part inequality.
- Use interval notation with [ ] for included and ( ) for excluded endpoints.