Interval Notation Calculator
Convert an inequality to interval notation, or interval notation back to an inequality, including unions with or.
Answer
(−∞, 1] ∪ (4, ∞)
- Interval notation
- (−∞, 1] ∪ (4, ∞)
- Inequality
- x ≤ 1 or x > 4
In interval notation: (−∞, 1] ∪ (4, ∞). As an inequality: x ≤ 1 or x > 4.
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| Separate intervals | 2 joined with a union | 2 |
How it works
Choose the direction, then type the inequality or the interval.
Worked example
x ≤ 1 or x > 4:
- x ≤ 1 includes 1 and goes down forever: (−∞, 1]
- x > 4 leaves out 4 and goes up forever: (4, ∞)
- Together: (−∞, 1] ∪ (4, ∞)
Sources
Frequently asked questions
What is interval notation?
Interval notation writes the set of numbers that satisfy an inequality as two end points. A parenthesis leaves the end point out and a bracket includes it. So x > 3 is (3, ∞) and x ≥ 3 is [3, ∞).
Why does infinity always get a parenthesis?
Infinity is not a number that a value can reach, so it is never included. You write (−∞, 1], not [−∞, 1].
How do I write more than one interval?
Join them with a union sign ∪. x ≤ 1 or x > 4 is (−∞, 1] ∪ (4, ∞). In the calculator you can type or, and, or the letter U between intervals.
How do I handle a compound inequality like -2 < x ≤ 5?
Put the lower number first. The result is (−2, 5], with a parenthesis at −2 because it is less than and a bracket at 5 because it is less than or equal.
What if the answer is empty?
If no number satisfies every condition, such as x > 5 and x < 2, the calculator shows ∅, the empty set.