Interval Notation Calculator Symbolab

Convert Inequality to Interval Notation:

\[ x > a \Rightarrow (a, \infty) \] \[ x \geq a \Rightarrow [a, \infty) \] \[ x < a \Rightarrow (-\infty, a) \] \[ x \leq a \Rightarrow (-\infty, a] \]

Interval Notation:

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1. What is Interval Notation?

Interval notation is a mathematical notation used to represent sets of real numbers with inequalities. It uses parentheses and brackets to describe intervals on the real number line.

2. How to Convert Inequalities

Basic conversion rules:

\[ x > a \Rightarrow (a, \infty) \] \[ x \geq a \Rightarrow [a, \infty) \] \[ x < a \Rightarrow (-\infty, a) \] \[ x \leq a \Rightarrow (-\infty, a] \]

Explanation:

Parentheses ( ) indicate the endpoint is not included
Brackets [ ] indicate the endpoint is included
∞ always uses parenthesis since infinity is not a finite number

3. Examples of Conversion

Examples:

\( x > 5 \) → (5, ∞)
\( x \geq -2 \) → [-2, ∞)
\( x < 3 \) → (-∞, 3)
\( x \leq 0 \) → (-∞, 0]

4. Using the Calculator

Instructions: Select the inequality type and enter the boundary value. The calculator will automatically convert it to proper interval notation.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between ( ) and [ ]?
A: Parentheses exclude the endpoint, brackets include it. For example, (2,5] means greater than 2 and less than or equal to 5.

Q2: How do you represent all real numbers?
A: The interval (-∞, ∞) represents all real numbers.

Q3: What about compound inequalities?
A: For a < x ≤ b, the interval would be (a, b]. The calculator handles single inequalities only.

Q4: Can I represent empty sets?
A: Empty sets are represented by ∅ or {}. This occurs with contradictions like x > 2 and x < 1.

Q5: How are intervals used in calculus?
A: They're essential for defining domains, ranges, and intervals of increase/decrease in functions.