Interval Notation Calculator
Convert inequalities to interval notation step by step. Enter expressions like "x > 3", "2 < x < 8", or "x < -1 or x > 3".
Try these examples:
How to Use
Supported Formats:
- • Simple:
x > 3 - • Compound:
2 < x < 8 - • With equals:
1 ≤ x ≤ 4 - • Mixed:
-3 ≤ x < 6
Symbols:
- •
>greater than - •
≥greater than or equal - •
<less than - •
≤less than or equal
What is Interval Notation?
Interval notation is a mathematical notation used to describe sets of real numbers between two endpoints. It provides a concise way to represent continuous sets of numbers using brackets and parentheses to indicate whether the endpoints are included or excluded from the set.
Key Concept
Interval notation uses square brackets [ ] to include endpoints and parentheses ( ) to exclude endpoints. This creates a precise mathematical language for describing number ranges.
Types of Intervals
Closed Interval
Both endpoints are included. The interval contains all real numbers x such that a ≤ x ≤ b.
Open Interval
Neither endpoint is included. The interval contains all real numbers x such that a < x < b.
Half-Open [a, b)
Left endpoint included, right endpoint excluded. Contains all x such that a ≤ x < b.
Half-Open (a, b]
Left endpoint excluded, right endpoint included. Contains all x such that a < x ≤ b.
Notation Symbols Guide
Square Brackets [ ]
- ✓Include the endpoint in the interval
- ✓Used for "greater than or equal to" (≥) and "less than or equal to" (≤)
- ✓The endpoint is part of the solution set
Parentheses ( )
- ✗Exclude the endpoint from the interval
- ✗Used for "greater than" (>) and "less than" (<)
- ✗The endpoint is not part of the solution set
Step-by-Step Examples
Example 1: Basic Interval
Problem: Express the set of all real numbers x such that 1 ≤ x ≤ 4 in interval notation.
Solution: Since both endpoints are included, we use square brackets: [1, 4]
Example 2: Open Interval
Problem: Express the set of all real numbers x such that -2 < x < 3 in interval notation.
Solution: Since neither endpoint is included, we use parentheses: (-2, 3)
Example 3: Half-Open Interval
Problem: Express the set of all real numbers x such that 0 ≤ x < 5 in interval notation.
Solution: Left endpoint included, right excluded: [0, 5)
Common Applications
Mathematics
Used in calculus, algebra, and analysis to define domains, ranges, and solution sets for equations and inequalities.
Statistics
Defines confidence intervals, ranges for data analysis, and probability distributions in statistical studies.
Programming
Used in algorithms, data structures, and validation to define ranges for variables and input parameters.
Practice Problems
Test Your Knowledge
Problem 1
Express "all real numbers x such that x > -1 and x ≤ 3" in interval notation.
Problem 2
Express "all real numbers x such that -2 ≤ x < 7" in interval notation.