Inequality Using Interval Notation Calculator

What is Interval Notation?

Interval notation is a way to represent sets of real numbers using parentheses and brackets. It's commonly used in mathematics to describe the solution set of inequalities.

There are four main types of intervals:

• (a, b) - Open interval, does not include a and b
• [a, b] - Closed interval, includes both a and b
• (a, b] - Half-open interval, includes b but not a
• [a, b) - Half-open interval, includes a but not b

Infinite intervals can also be represented:

• (a, ∞) - All numbers greater than a
• (-∞, b) - All numbers less than b
• (-∞, ∞) - All real numbers

How to Convert Inequalities to Interval Notation

To convert an inequality to interval notation, follow these steps:

1. Identify the inequality symbol: <, >, ≤, or ≥
2. Determine if the endpoints are included (use brackets) or excluded (use parentheses)
3. Write the interval in the correct order from smallest to largest

For compound inequalities, combine the intervals:

• x < a or x > b → (-∞, a) ∪ (b, ∞)
• a ≤ x ≤ b or x ≥ c → [a, b] ∪ [c, ∞)

Examples

Example 1: Simple Inequality

Convert x > 3 to interval notation.

Solution: Since x must be greater than 3 but not equal to 3, we use a parenthesis for 3.

Answer: (3, ∞)

Example 2: Compound Inequality

Convert -2 ≤ x < 5 to interval notation.

Solution: x must be greater than or equal to -2 and less than 5. We use a bracket for -2 and a parenthesis for 5.

Answer: [-2, 5)

Example 3: Union of Intervals

Convert x < -1 or x > 4 to interval notation.

Solution: This represents all numbers less than -1 or greater than 4. We use parentheses for both endpoints.

Answer: (-∞, -1) ∪ (4, ∞)