Inequalities in Interval Notation Calculator

This guide explains how to convert inequalities to interval notation and vice versa. Use the calculator on the right to practice converting between these mathematical representations.

How to Use This Calculator

To convert an inequality to interval notation:

Enter your inequality in the input field (e.g., x > 3 or 1 ≤ y < 5)
Click "Calculate" to see the interval notation result
Review the step-by-step conversion process
Use the chart to visualize the interval on a number line

The calculator handles both single-variable inequalities and compound inequalities. For example, "x > 3" converts to (3, ∞) while "1 ≤ y < 5" converts to [1, 5).

Conversion Rules

Basic Conversion Rules

x > a becomes (a, ∞)
x ≥ a becomes [a, ∞)
x < a becomes (-∞, a)
x ≤ a becomes (-∞, a]
a < x < b becomes (a, b)
a ≤ x ≤ b becomes [a, b]
a < x ≤ b becomes (a, b]
a ≤ x < b becomes [a, b)

For compound inequalities, the order of the numbers in the interval notation matches the order in the inequality. The parentheses and brackets indicate whether the endpoints are included:

Parentheses ( ) indicate that the endpoint is not included
Brackets [ ] indicate that the endpoint is included

Examples

Example 1: Simple Inequality

Inequality: x > 4

Interval Notation: (4, ∞)

Explanation: The number 4 is not included in the solution set, so we use a parenthesis.

Example 2: Compound Inequality

Inequality: -2 ≤ y < 5

Interval Notation: [-2, 5)

Explanation: The -2 is included (≤) so we use a bracket, while 5 is not included (<) so we use a parenthesis.

Common Mistakes

Confusing which endpoints should be included: Remember that ≤ and ≥ indicate included endpoints (brackets), while < and > indicate excluded endpoints (parentheses)
Reversing the order of numbers: The smaller number always comes first in interval notation
Forgetting to include infinity symbols: Use ∞ for positive infinity and -∞ for negative infinity
Using the wrong symbol for infinity: ∞ is correct, infinity is not

FAQ

What is interval notation?

Interval notation is a way to represent sets of real numbers using parentheses and brackets. It's commonly used in calculus and algebra to describe ranges of values.

How do I know if to use a parenthesis or bracket?

Use a bracket [ or ] when the endpoint is included in the solution set (≤ or ≥). Use a parenthesis ( or ) when the endpoint is not included (< or >).

What does ∞ mean in interval notation?

∞ represents positive infinity, indicating that the interval extends without bound in the positive direction. Similarly, -∞ represents negative infinity.