Interval Notation Calculator

Convert between inequalities and interval notation

Conversion Type

Inequality

Examples

Inequality: x >= 3 => Interval: [3, ∞)
Inequality: -1 < x <= 5 => Interval: (-1, 5]
Interval: (-∞, 2) => Inequality: x < 2
Interval: [-4, 4] => Inequality: -4 <= x <= 4

Important Note

Use 'inf' for positive infinity and '-inf' for negative infinity. Use parentheses () for open intervals (exclusive) and brackets [] for closed intervals (inclusive).

Understanding Interval Notation: A Comprehensive Guide

Interval notation is a method of writing down a set of numbers. It is commonly used to describe the domain and range of functions, and to represent solution sets of inequalities. An interval is a continuous, or unbroken, portion of the number line.

The notation uses parentheses () and/or brackets [] to show whether the endpoints are included or excluded. A parenthesis indicates that the endpoint is not included (exclusive), while a bracket indicates that it is included (inclusive).

Types of Intervals

1. Open Interval: (a, b) represents the set of all real numbers x such that a < x < b. Both endpoints are excluded.

2. Closed Interval: [a, b] represents the set of all real numbers x such that a ≤ x ≤ b. Both endpoints are included.

3. Half-Open Intervals: (a, b] represents a < x ≤ b, and [a, b) represents a ≤ x < b. One endpoint is included, and the other is not.

4. Unbounded Intervals: These extend infinitely in one or both directions, using the infinity symbol (∞). For example, (a, ∞) represents x > a.

Basic Examples

Inequality: x > 2, Interval: (2, ∞)
Inequality: x ≤ -1, Interval: (-∞, -1]
Inequality: -3 < x < 5, Interval: (-3, 5)
Inequality: 0 ≤ x ≤ 10, Interval: [0, 10]

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

Our calculator simplifies the conversion between standard inequality notation and interval notation.

1. Select the Conversion Type

Choose whether you want to convert from 'Inequality to Interval' or 'Interval to Inequality' from the dropdown menu.

2. Enter Your Input

For Inequality to Interval: Type the inequality in the input box. Use standard comparison operators like <, <=, >, >=. You can represent a single inequality (e.g., x < 7) or a compound inequality (e.g., -1 <= x < 4). Use a variable like 'x'.

For Interval to Inequality: Enter the interval using parentheses and brackets. Use 'inf' for positive infinity and '-inf' for negative infinity. For example, [-5, inf).

3. Convert and View the Result

Click the 'Convert' button. The calculator will display the corresponding notation. If there's an error in your input format, a helpful message will appear.

Usage Examples

To convert `x <= 15`: Select 'Inequality to Interval', enter `x <= 15`, and get `(-∞, 15]`.
To convert `(-2, 8)`: Select 'Interval to Inequality', enter `(-2, 8)`, and get `-2 < x < 8`.
For compound inequality `1 < y <= 100`: Enter `1 < y <= 100` and get `(1, 100]`.

Real-World Applications of Interval Notation

Interval notation is not just for algebra class; it's used in many fields to define ranges and constraints.

Science and Engineering

Scientists use interval notation to specify tolerance ranges for measurements. For example, a component's temperature must be within the interval [−10, 50] degrees Celsius to function correctly.

Computer Programming

Programmers use intervals to validate user input. For instance, a user's age might need to be in the interval [18, 120] to register for a service.

Statistics

Confidence intervals in statistics are expressed using interval notation. A 95% confidence interval of [9.2, 9.8] for a product's weight means we are 95% confident the true average weight falls in this range.

Practical Examples

Safe operating temperature for a device: T ∈ [0, 85] °C.
Acceptable pH level for a chemical reaction: pH ∈ (6.5, 7.5).
A movie rated for ages 13 and up: Age ∈ [13, ∞).

Common Misconceptions and Correct Methods in Interval Notation

Misconception 1: Confusing Parentheses and Brackets

Wrong: Using (3, 7] when you mean to include 3. This notation (3, 7] means 3 < x <= 7.

Correct: Use [3, 7] for 3 <= x <= 7. Always use a bracket [ or ] for an endpoint that is included in the set.

Misconception 2: Using Brackets with Infinity

Wrong: Writing [5, ∞] or (-∞, 10].

Correct: Infinity (∞) and negative infinity (-∞) are not actual numbers that can be included in a set. Therefore, always use parentheses with them. The correct forms are [5, ∞) and (-∞, 10).

Misconception 3: Order of Endpoints

Wrong: Writing (8, 2).

Correct: In interval notation, the smaller number must always be written first. The correct form is (2, 8).

Correction Examples

For 'x is greater than 5': Use (5, ∞), not [5, ∞).
For 'x is at most 0': Use (-∞, 0], not (-∞, 0).
For '-10 to 10, inclusive': Use [-10, 10], not (-10, 10).

Mathematical Derivation and Examples

The concept of an interval is fundamental in real analysis and topology. It is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. This property is called connectedness.

Set-Builder Notation vs. Interval Notation

Interval notation is a more concise alternative to set-builder notation for connected sets of real numbers.

Set-Builder: { x ∈ ℝ | a < x < b } --- Interval: (a, b)

Set-Builder: { x ∈ ℝ | x ≥ a } --- Interval: [a, ∞)

Union of Intervals

When a set consists of two or more disjoint intervals, the union symbol (∪) is used to combine them. For example, the solution to the inequality x² > 9 is x < -3 or x > 3. In interval notation, this is written as (-∞, -3) ∪ (3, ∞).

Advanced Examples

Solution for `|x| > 2` is `(-∞, -2) ∪ (2, ∞)`
The domain of `f(x) = 1/(x-1)` is `(-∞, 1) ∪ (1, ∞)`
The range of `y = x²` is `[0, ∞)`