Interval Notation Calculator - Convert Inequalities
Free calculator to convert between inequality and interval notation. Get instant conversions and learn interval representations
Interval Notation Calculator
Notation Results
Common Examples
What is an Interval Notation Calculator?
An interval notation calculator is a mathematical tool that converts between different notations for representing ranges of real numbers. It translates inequalities like -2 ≤ x < 5 into interval notation [-2, 5), set-builder notation, and other equivalent forms, making it easier to understand and communicate mathematical ranges.
Interval notation appears throughout mathematics including algebra, calculus, statistics, and real analysis. It provides a concise way to describe domains and ranges of functions, solution sets of inequalities, confidence intervals in statistics, and continuous sets of real numbers. Understanding interval notation is fundamental to advanced mathematical communication.
This calculator handles all interval types including finite intervals with included or excluded endpoints, infinite intervals extending to positive or negative infinity, and unions of multiple intervals. It shows multiple equivalent representations simultaneously, helping students understand connections between different mathematical notations for the same concept.
For solving inequalities that produce intervals, try our absolute value equation calculator. The quadratic formula calculator finds solutions that define intervals. For systems, use our system of equations calculator.
How the Interval Notation Calculator Works
The calculator takes endpoint values and endpoint types (open or closed) to construct interval notation. Open endpoints use parentheses ( ) indicating the value is not included, while closed endpoints use brackets [ ] indicating the value is included. The interval (a, b) represents all numbers strictly between a and b.
For inequality notation, the calculator translates bracket types to inequality symbols. Closed left bracket becomes ≤, open left parenthesis becomes <. Similarly, closed right bracket becomes ≤, open right parenthesis becomes <. The result shows the variable x between the endpoints with appropriate inequality symbols.
Set-builder notation expresses intervals as {x | condition}, read "the set of all x such that condition holds." The calculator converts interval conditions to logical statements about x. Infinity endpoints always use open notation since infinity is a concept, not a number that can be included or excluded.
Key Concepts Explained
Open Interval
Uses parentheses (a, b) to exclude endpoints. Represents a < x < b. The interval contains all numbers strictly between a and b without including the endpoints themselves.
Closed Interval
Uses brackets [a, b] to include endpoints. Represents a ≤ x ≤ b. The interval contains all numbers between and including both a and b.
Half-Open Interval
Mixes brackets and parentheses like [a, b) or (a, b]. Includes one endpoint but not the other. Useful for describing asymmetric ranges.
Unbounded Interval
Extends to infinity: (-∞, b] or [a, ∞). Infinity always uses parentheses. Represents all numbers less than or greater than a specific value.
How to Use This Calculator
Enter Endpoints
Input left and right endpoint values. Use actual numbers for finite intervals or check infinity box for unbounded intervals
Select Types
Choose whether each endpoint is open (excluded) or closed (included) based on your inequality
Convert Notation
Click Convert to see interval notation, inequality notation, and set-builder notation instantly
Understand Results
Compare different notation forms to understand how they represent the same mathematical range
Benefits of Using This Calculator
Using this interval notation calculator helps students understand different mathematical notation systems and their equivalences. It eliminates confusion about bracket versus parenthesis usage and inequality symbols.
Learn Equivalences: See how interval, inequality, and set-builder notations represent the same concept
Avoid Notation Errors: Eliminate mistakes in bracket/parenthesis usage and inequality symbol direction
Verify Homework: Check your interval notation conversions for accuracy before submission
Understand Domains: Express function domains and ranges in multiple equivalent notations
Quick Reference: Use as a reference guide for converting between notation systems
Factors That Affect Your Results
The endpoint values and types you select completely determine the interval representation. Understanding these choices helps correctly express mathematical ranges.
Endpoint Inclusion
Open endpoints exclude the value (use < or >). Closed endpoints include the value (use ≤ or ≥). This choice affects whether boundary values are in the set.
Infinity Usage
Infinity symbols always require parentheses, never brackets. You cannot "include" infinity since it's not a real number. Intervals like [5, ∞) mean x ≥ 5.
Interval Order
Left endpoint must be less than right endpoint for valid interval. Equal endpoints create single-point set when both closed: [a, a] = {a}.
Frequently Asked Questions (FAQ)
Q: What is interval notation?
A: Interval notation is a mathematical notation for representing ranges of real numbers. It uses brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints. For example, [2, 5) means all numbers from 2 (included) to 5 (excluded).
Q: What is the difference between brackets and parentheses?
A: Brackets [ ] indicate the endpoint is included in the interval (closed). Parentheses ( ) indicate the endpoint is not included (open). For example, [3, 7] includes both 3 and 7, while (3, 7) excludes both endpoints.
Q: How do you represent infinity in interval notation?
A: Infinity (∞) and negative infinity (-∞) always use parentheses, never brackets, because infinity is not a real number that can be included. For example, (5, ∞) means all numbers greater than 5.
Q: What is the difference between interval notation and inequality notation?
A: Interval notation uses brackets and parentheses to show ranges: [2, 5]. Inequality notation uses symbols: 2 ≤ x < 5. They represent the same mathematical concept but use different formats. Both describe sets of real numbers.
Q: How do you write union of intervals?
A: Use the union symbol ∪ to combine multiple intervals. For example, (-∞, 2) ∪ (5, ∞) represents all numbers less than 2 or greater than 5. This corresponds to the inequality x < 2 or x > 5.