Inequality to Interval Notation Calculator

Convert any mathematical inequality into its corresponding interval notation form.

Enter a valid mathematical inequality involving a variable (like x) to see its interval notation. This tool helps visualize solution sets on the number line.

Use a variable (like x, y, z), numbers, and comparison operators (>, >=, <, <=).

Examples

Click on an example to load it into the calculator.

Simple Greater Than

inequality

A basic inequality showing all numbers greater than a value.

Inequality: x > 3

Compound Inequality (Inclusive/Exclusive)

inequality

An inequality representing a range between two values.

Inequality: -2 <= y < 5

Union of Two Intervals ('or')

inequality

Two separate intervals combined with a logical 'or'.

Inequality: a <= 0 or a >= 10

Simple Less Than or Equal To

inequality

An inequality representing all numbers up to and including a value.

Inequality: b <= -1.5

Other Titles
Understanding Inequality to Interval Notation: A Comprehensive Guide
Learn how to represent solution sets of inequalities using interval notation, a fundamental concept in mathematics.

What is Interval Notation? The Basics Explained

  • A method for writing solution sets of inequalities
  • Uses parentheses and brackets to denote inclusivity or exclusivity
  • Provides a concise way to describe a range of numbers
Interval notation is a way of describing a continuous set of real numbers by the pair of numbers that are at its ends. For example, the set of numbers x satisfying 0 ≤ x ≤ 5 is an interval which contains 0, 5, and all numbers in between. Interval notation is a more concise and standardized method compared to writing out inequalities.
Key Symbols and Their Meanings
  • Parentheses ( ): Used to indicate that an endpoint is not included in the interval. This corresponds to the '<' (less than) and '>' (greater than) symbols.
  • Brackets : Used to indicate that an endpoint is included in the interval. This corresponds to the '≤' (less than or equal to) and '≥' (greater than or equal to) symbols.
  • Infinity (∞): Represents a boundless endpoint. Since infinity is a concept and not a real number, it is always paired with a parenthesis.
  • Union Symbol (U): Used to combine two or more separate intervals.

Basic Interval Notation Examples

  • Inequality: x > 2 → Interval: (2, ∞)
  • Inequality: x ≤ -1 → Interval: (-∞, -1]
  • Inequality: -3 < x ≤ 4 → Interval: (-3, 4]
  • Inequality: x < 0 or x > 5 → Interval: (-∞, 0) U (5, ∞)

Step-by-Step Guide to Using the Inequality to Interval Notation Calculator

  • How to correctly format your inequality input
  • Understanding the different types of inequalities supported
  • Interpreting the converted interval notation result
Our calculator is designed to be intuitive and powerful, handling a wide variety of inequality formats. Follow these steps for a smooth experience.
Inputting Your Inequality
1. Identify the Variable: Use any single letter for your variable, like 'x', 'y', or 'a'.
2. Use Comparison Operators: The calculator recognizes '>', '>=', '<', and '<='.
3. Enter Numbers: You can use integers (5), negative numbers (-10), and decimals (3.14).
4. Compound Inequalities: For ranges, type the inequality as it's written mathematically (e.g., -1 < x <= 5). For unions, use the word 'or' between the two parts (e.g., x < 0 or x > 2).
Getting and Understanding the Result
After clicking 'Convert', the result will be displayed in the standard interval notation format. The output will show parentheses for exclusive bounds, brackets for inclusive bounds, and the union symbol 'U' if your inequality described two separate ranges.

Supported Input Formats

  • Simple: y >= -4
  • Compound: 0 <= x < 10
  • Union: a < -5 or a > 5
  • Decimal: z > 9.5

Real-World Applications of Interval Notation

  • Describing tolerance ranges in engineering and manufacturing
  • Expressing confidence intervals in statistics
  • Defining constraints in optimization problems
Interval notation is not just an abstract mathematical concept; it's a practical tool used across many fields to define ranges and constraints with precision.
Engineering and Manufacturing
A machine part might need to be manufactured with a diameter of 5cm, with a tolerance of ±0.01cm. The acceptable range of diameters can be expressed as the interval [4.99, 5.01].
Statistics and Data Science
When researchers conduct a poll, they often report the result with a margin of error. For example, if 55% of people support a candidate with a 3% margin of error, the confidence interval for the true support is [52%, 58%], or in decimal form, [0.52, 0.58].
Computer Science
In programming, conditions often check if a value falls within a certain range. For example, a color component in RGB might be valid only if its value is in the interval [0, 255].

Practical Application Examples

  • Acceptable temperature range for a chemical reaction: (20, 30) degrees Celsius
  • A passing grade for a test might be in the score range [70, 100]
  • Safe operating voltage for an electronic device: [110, 120] volts

Common Misconceptions and Correct Methods

  • Confusing parentheses ( ) and brackets [ ]
  • Incorrectly handling infinity in notation
  • Mistakes in writing compound and union intervals
While interval notation is efficient, a few common mistakes can lead to incorrect interpretations. Understanding these pitfalls is key to mastering the concept.
Parentheses vs. Brackets
Misconception: They are interchangeable. Correction: The choice is critical. A parenthesis '(' or ')' means the endpoint is not included. A bracket '[' or ']' means the endpoint is included. For example, (2, 5] includes 5 but excludes 2.
The Infinity Symbol (∞)
Misconception: You can have an inclusive bound with infinity, like [∞]. Correction: Infinity is not a number that can be 'reached' or 'included'. Therefore, infinity and negative infinity (-∞) are always paired with a parenthesis.
Writing Compound Inequalities
Misconception: Writing 5 < x < 2. Correction: The numbers in a compound interval must be in ascending order from left to right. The correct way to write this would depend on the intended logic, but as written, it represents an empty set because no number is both greater than 5 and less than 2. The proper form is always smaller_number < x < larger_number.

Examples of Correct Usage

  • Correct: `x < 3` is `(-∞, 3)`
  • Incorrect: `x < 3` is `(-∞, 3]`
  • Correct: `x >= -1` is `[-1, ∞)`
  • Incorrect: `x >= -1` is `[-1, ∞]`

Mathematical Derivation and Logic

  • Translating inequality symbols to interval boundaries
  • The logic behind compound inequalities (intersections)
  • The logic behind 'or' inequalities (unions)
The conversion from an inequality to interval notation is based on a direct logical mapping. Each part of the inequality corresponds to a specific element in the notation.
From Symbols to Boundaries
The core of the conversion lies in mapping the comparison operators to the correct type of boundary. A strict inequality (< or >) creates an 'open' boundary, denoted by a parenthesis. A non-strict inequality (≤ or ≥) creates a 'closed' boundary, denoted by a bracket.
Compound Inequalities as Intersections
A compound inequality like -2 ≤ x < 5 is a shorthand for x ≥ -2 AND x < 5. In set theory, the 'AND' corresponds to an intersection. We are looking for the numbers that are in both the set [-2, ∞) and the set (-∞, 5). The intersection of these two sets is the interval [-2, 5).
Union Inequalities
An inequality like x ≤ 0 or x > 8 involves a logical 'OR', which corresponds to a union in set theory. We are looking for numbers that are in either the set (-∞, 0] or the set (8, ∞). Since these two sets do not overlap, we connect them with the union symbol 'U': (-∞, 0] U (8, ∞).

Logical Derivations

  • `x > 5` → Endpoint is 5, boundary is open → (5, ∞)
  • `-10 <= x <= -2` → `x >= -10` AND `x <= -2` → Intersection of `[-10, ∞)` and `(-∞, -2]` is `[-10, -2]`