Inequality to Interval Notation Calculator

Convert inequalities into standard interval notation

Enter an inequality to see its interval notation equivalent. Use 'x' as the variable.

Examples

  • x >= -1 --> [-1, ∞)
  • -4 < x <= 4 --> (-4, 4]
  • x < 5 or x > 10 --> (-∞, 5) U (10, ∞)
  • x != 3 --> (-∞, 3) U (3, ∞)

Notation Guide

Parentheses ( ) are used for exclusive bounds (< or >), meaning the endpoint is not included. Brackets [ ] are used for inclusive bounds (<= or >=), meaning the endpoint is included. Use 'U' for the union of two separate intervals.

Other Titles
Understanding Inequality to Interval Notation: A Comprehensive Guide
Learn how to represent the solution set of an inequality using interval notation, a standard method in mathematics.

Understanding the Calculator: A Comprehensive Guide

  • Learn the difference between parentheses () and brackets []
  • Understand how to represent infinity (∞)
  • Recognize how 'and' and 'or' inequalities translate to intervals
Interval notation is a way of writing subsets of the real number line. Instead of describing a set of numbers with an inequality, you can use a concise notation that clearly shows the boundaries of the set.
Key Symbols:
  • ( ) Parentheses: Indicate that an endpoint is not included in the interval. Used with greater than (>) or less than (<).
  • Brackets: Indicate that an endpoint is included in the interval. Used with greater than or equal to (>=) or less than or equal to (<=).
  • ∞ Infinity: Represents a boundless endpoint. Always used with a parenthesis.
  • U Union: Used to combine two or more separate intervals.

Basic Conversion Examples

  • x > 5 becomes (5, ∞)
  • x <= -2 becomes (-∞, -2]
  • -1 <= x < 3 becomes [-1, 3)

Step-by-Step Guide to Using the Calculator

  • How to type in different kinds of inequalities
  • Handling compound inequalities with 'and' or 'or'
  • Interpreting the converted interval notation
Our converter handles a wide variety of inequality formats.
Input Guidelines:
  • Simple Inequalities: Type them as you would write them, e.g., x >= 10 or x < -2.5.
  • Compound 'And' Inequalities: You can write them as one statement, e.g., -1 < x < 5.
  • Compound 'Or' Inequalities: Use the word 'or' to separate the parts, e.g., x <= 0 or x > 6.

Usage Examples

  • Input: 0 <= x < 100 -> Output: [0, 100)
  • Input: x < -1 or x > 1 -> Output: (-∞, -1) U (1, ∞)

Real-World Applications of Interval Notation

  • Science: Expressing valid temperature ranges for an experiment
  • Engineering: Defining tolerance levels for a component's size
  • Statistics: Representing confidence intervals
Interval notation is used across many fields to concisely define a range of possible or acceptable values.
Engineering and Manufacturing:
  • A manufactured part might need a diameter 'd' that is 5cm with a tolerance of ±0.01cm. This can be written as the inequality 4.99 <= d <= 5.01, or more concisely in interval notation as [4.99, 5.01].
Statistics:
  • When statisticians estimate a population parameter, they often provide a confidence interval. For example, they might be 95% confident that the true average height of a population is within the interval [170cm, 175cm].

Real-World Examples

  • Safe operating temperature for a device: [0, 40] degrees Celsius.
  • A passing grade 'g' on a test: g >= 65, which is [65, 100] assuming 100 is max.

Common Misconceptions and Correct Methods

  • Mixing up parentheses and brackets
  • Using a bracket with infinity
  • Incorrectly writing compound intervals
The details are crucial for correct interval notation.
Misconception 1: Bracket with Infinity
  • Wrong: Writing [5, ∞] or (-∞, 2].
  • Correct: Infinity is not a number that can be included in a set. Therefore, it must always be paired with a parenthesis: (5, ∞) and (-∞, 2).
Misconception 2: 'And' vs. 'Or'
  • Wrong: Expressing 'x < 2 and x > 5' as (2, 5). There are no numbers that are simultaneously less than 2 AND greater than 5.
  • Correct: This situation represents an empty set. An 'or' statement, like 'x < 2 or x > 5', represents two separate regions and is correctly written with a union: (-∞, 2) U (5, ∞).

Correction Examples

  • x <= 3 is inclusive -> [-∞, 3]
  • x < 3 is exclusive -> (-∞, 3)

Mathematical Derivation and Examples

  • Visualizing intervals on the number line
  • Understanding set-builder notation as a precursor
Interval notation is a shorthand for set-builder notation, which provides a more formal definition.
From Set-Builder to Interval
  • Set-Builder Notation: {x | x is a real number and x > 2}
  • Interval Notation: (2, ∞)
Visualizing on a Number Line
Interval notation corresponds directly to graphs on a number line. An open circle (o) is used for a parenthesis, and a closed circle (●) is used for a bracket. For example, the interval [-1, 3) would be a number line with a closed circle at -1, an open circle at 3, and the line between them shaded.