Free Interval Notation Calculator

Introducing the Interval Notation Converter

This all‑in‑one interval notation calculator serves as both an inequality to interval notation converter and an interval notation solver, allowing you to switch seamlessly between inequality and interval representations. Whether you need to convert to interval notation from an inequality or translate an interval back into an inequality, this tool provides immediate, accurate results. It also handles compound inequalities, making it a practical companion for homework, exam preparation, or quick verification.

Understanding Interval Notation

Interval notation is a compact method for describing connected subsets of the real number line. Each interval is defined by two endpoints, and the symbols surrounding them indicate whether those endpoints belong to the set.

• Closed interval $[a, b]$ – includes both endpoints: $a \le x \le b$
• Open interval $(a, b)$ – excludes both endpoints: $a < x < b$
• Half‑open intervals $[a, b)$ or $(a, b]$ – include exactly one endpoint: $a \le x < b$ or $a < x \le b$

This notation appears throughout mathematics and the sciences. For instance, confidence intervals in statistics are frequently expressed in interval form, illustrating the range in which a population parameter is expected to lie.

How to Use the Calculator

Putting the interval notation converter to work is simple:

1. Choose the conversion direction – decide whether you want to go from an inequality to interval notation or from an interval back to an inequality.
2. Enter your input – type the inequality (e.g., $x > 3$) or the interval endpoints (e.g., $(-2, 5]$). The converter can handle compound inequalities such as $-\infty < x \le 2$ or $3 \le x < 7$.
3. Obtain the result – the tool instantly displays the equivalent expression, simplifying it when possible.

The calculator deals with both basic and more complex cases, offering instant feedback that helps you confirm your manual conversions.

Manual Method: Writing an Interval by Hand

If you need to write in interval notation manually, follow these steps:

• Identify the smallest value (infimum) and the largest value (supremum) that bound the set.
• For the lower bound: use a square bracket $[$ if the bound belongs to the set; use a parenthesis $($ if it does not.
• Write the lower marker, a comma, and then the upper bound.
• For the upper bound: use a square bracket $]$ if the bound belongs to the set; use a parenthesis $)$ if it does not.

For example, the set containing all numbers from $-3$ up to but not including $5$ is written as $[-3, 5)$.

Special Case: All Real Numbers

The set of all real numbers is expressed in interval notation as $(-\infty, \infty)$. Because $\infty$ (infinity) is not a concrete number but a concept that represents unboundedness, parentheses are always used around $\infty$ and $-\infty$. This notation tells us the set extends indefinitely in both directions with no finite boundaries.

Common Conversions at a Glance

Inequality	Interval Notation
$x > 3$	$(3, \infty)$
$x \le -1$	$(-\infty, -1]$
$-2 < x \le 4$	$(-2, 4]$
$0 \le x < 5$	$[0, 5)$
$-\infty < x < \infty$	$(-\infty, \infty)$

Why Use an Interval Notation Tool?

Mastering interval notation is essential for solving inequalities, describing domain and range, and working with functions. An inequality to interval notation converter speeds up assignments while reinforcing the relationship between the two forms. By delivering immediate feedback, it helps learners build confidence and accuracy in translating mathematical statements.