Writing Inequalities in Interval Notation Calculator

Interval notation is a concise way to represent sets of real numbers. This calculator helps you convert inequalities to interval notation and vice versa. Whether you're studying calculus, algebra, or any other math subject, understanding interval notation is essential.

How to Use This Calculator

To use the interval notation calculator on the right sidebar:

Enter your inequality in the input field (e.g., x > 3 or 1 ≤ x < 5).
Select whether the inequality includes or excludes endpoints.
Click "Calculate" to see the interval notation.
Review the result and the step-by-step explanation.

The calculator will display the interval notation and provide a visual representation when possible.

Rules for Writing Interval Notation

Interval notation uses square brackets [ ] and parentheses ( ) to represent closed and open intervals, respectively. Here are the basic rules:

Square brackets [ ] indicate that an endpoint is included in the interval.
Parentheses ( ) indicate that an endpoint is not included in the interval.
Infinity symbol ∞ is used to represent unbounded intervals.
Intervals are written from the lower bound to the upper bound, separated by a comma.

For example, the interval [3, 5] includes all numbers from 3 to 5, including 3 and 5. The interval (3, 5) includes all numbers from 3 to 5, excluding 3 and 5.

Examples of Interval Notation

Here are some common examples of inequalities and their interval notation equivalents:

Inequality	Interval Notation	Description
x > 2	(2, ∞)	All numbers greater than 2
x ≤ 5	(-∞, 5]	All numbers less than or equal to 5
1 < x < 5	(1, 5)	All numbers between 1 and 5, excluding 1 and 5
2 ≤ x ≤ 8	[2, 8]	All numbers between 2 and 8, including 2 and 8

Example Calculation

Convert the inequality -3 ≤ x < 7 to interval notation.

Solution: The inequality includes -3 but excludes 7. Therefore, the interval notation is [-3, 7).

Common Mistakes to Avoid

When writing interval notation, it's easy to make a few common mistakes. Here are some to watch out for:

Incorrect brackets: Using square brackets when you should use parentheses, or vice versa.
Order of bounds: Writing the lower bound after the upper bound.
Infinity symbol: Using the wrong symbol for infinity or omitting it entirely.
Missing endpoints: Forgetting to include or exclude endpoints based on the inequality.

Always double-check your interval notation to ensure that the brackets and order are correct. A small mistake can lead to a completely different interval.

Frequently Asked Questions

What is interval notation?: Interval notation is a way to represent sets of real numbers using intervals and brackets. It's commonly used in calculus and algebra to describe ranges of values.
How do I know when to use square brackets vs. parentheses?: Use square brackets [ ] when the endpoint is included in the interval and parentheses ( ) when the endpoint is excluded. For example, [3, 5] includes 3 and 5, while (3, 5) excludes them.
Can interval notation represent all real numbers?: Yes, the interval (-∞, ∞) represents all real numbers. This is often used to describe the domain of a function that is defined for all real numbers.
What does the infinity symbol ∞ mean in interval notation?: The infinity symbol ∞ is used to represent unbounded intervals. For example, (2, ∞) represents all numbers greater than 2, and (-∞, 5] represents all numbers less than or equal to 5.
How do I convert an inequality to interval notation?: To convert an inequality to interval notation, identify the lower and upper bounds, determine whether each endpoint is included or excluded, and then write the interval using the appropriate brackets.