Interval Notation Calculator
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Interval notation writes a set of numbers with brackets: [ ] includes an endpoint, ( ) leaves it out, and ∞ always gets a round bracket. So −2 ≤ x < 5 is [−2, 5), and x < 1 or x ≥ 4 is (−∞, 1) ∪ [4, ∞).
Examples: x > 3, 2x + 3 > 7, x^2 – 4 >= 0, |x – 2| < 3, x < 1 or x >= 4, [2, 5), (-inf, 3] U (5, inf).
Filled dots are included, open dots are excluded, arrows continue to infinity.
| Inequality | \(-2 \le x < 5\) |
| Set-builder notation | \(\left\{x \mid -2 \le x < 5\right\}\) |
| Interval notation | \(\left[-2,\ 5\right)\) |
Show steps
- Solve \(-2 \le x\): critical points \(-2\); test a value in each interval.
- Solve \(x < 5\): critical points \(5\); test a value in each interval.
- A double inequality means both parts hold.
How to use the interval notation calculator
- Type an inequality (it may need solving, like 2x + 3 > 7) or an interval such as [2, 5).
- Join conditions with “or” or “and”; a double inequality like −3 < 2x − 1 ≤ 5 works too.
- Press Convert to get the interval, the inequality, set-builder notation and a number line.
Reading interval notation
| Inequality | Interval | Number line |
|---|---|---|
| a < x < b | (a, b) | open dots at both ends |
| a ≤ x ≤ b | [a, b] | filled dots |
| a ≤ x < b | [a, b) | filled, then open |
| x > a | (a, ∞) | open dot, arrow right |
| x ≤ b | (−∞, b] | filled dot, arrow left |
| all real numbers | (−∞, ∞) | whole line |
Solving before converting
Solve like an equation, but flip the inequality sign when you multiply or divide by a negative number. For polynomial and rational inequalities, find the zeros and the values where the denominator is 0, then test one number in each interval. Example: \(x^{2} – 4 \ge 0\) has critical points ±2; testing gives \((-\infty, -2] \cup [2, \infty)\).
Unions and intersections
“Or” combines sets with a union (∪): every number in either set. “And” (or a double inequality) keeps only numbers in both: an intersection. \(|x – 2| < 3\) means \(-3 < x – 2 < 3\), so \((-1, 5)\).
Interval notation practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you write an inequality in interval notation?
Write the smallest and largest values in order, separated by a comma. Use [ or ] when the endpoint is included (≤ or ≥) and ( or ) when it is not (< or >). Infinity always uses a parenthesis.
What does the ∪ symbol mean?
Union: the set includes every number in either interval. It is used for “or” statements.
Can the calculator solve the inequality first?
Yes. It solves linear, quadratic, polynomial, rational and absolute value inequalities in one variable before converting.
What is set-builder notation?
A description like {x | x > 3}, read “the set of x such that x is greater than 3”.
How do I type infinity?
Type inf or ∞, for example (-inf, 3].
Sources
- OpenStax College Algebra 2e: Linear Inequalities and Absolute Value Inequalities
- OpenStax Intermediate Algebra 2e: Solve Linear Inequalities
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