Inequality to Interval Notation Calculator

Understanding Inequality and Interval Notation

An inequality describes a mathematical relationship comparing values (such as $x > 3$ or $-2 \le x \le 5$). Interval notation is a compact, standardized method used in algebra and calculus to write subsets of the real number line $\mathbb{R}$ using parentheses $(,)$ and brackets $[,]$.

Bracket vs Parenthesis Rules

The fundamental rule of interval notation determines whether an endpoint value is included in the solution set:

Square Bracket $[$ or $]$: Used for inclusive (non-strict) inequalities ($\le$ or $\ge$). The endpoint is part of the interval, visualized as a solid dot (●) on a number line.
Round Parenthesis $($ or $)$: Used for strict inequalities ($<$ or $>$). The endpoint is excluded from the interval, visualized as a hollow circle (○) on a number line.
Infinities ($-\infty$ and $\infty$): Always enclosed by parentheses because infinity is an unbounded concept, not a real number that can be reached or included.

Interval and Inequality Conversion Reference Table

Inequality Form	Interval Notation	Set-Builder Notation	Classification
$a < x < b$	$(a, b)$	$\{x \in \mathbb{R} \mid a < x < b\}$	Open Bounded
$a \le x \le b$	$[a, b]$	$\{x \in \mathbb{R} \mid a \le x \le b\}$	Closed Bounded
$a \le x < b$	$[a, b)$	$\{x \in \mathbb{R} \mid a \le x < b\}$	Half-Open (Left-Closed)
$a < x \le b$	$(a, b]$	$\{x \in \mathbb{R} \mid a < x \le b\}$	Half-Open (Right-Closed)
$x > a$	$(a, \infty)$	$\{x \in \mathbb{R} \mid x > a\}$	Open Right Ray
$x \ge a$	$[a, \infty)$	$\{x \in \mathbb{R} \mid x \ge a\}$	Closed Right Ray
$x < b$	$(-\infty, b)$	$\{x \in \mathbb{R} \mid x < b\}$	Open Left Ray
$x \le b$	$(-\infty, b]$	$\{x \in \mathbb{R} \mid x \le b\}$	Closed Left Ray
$x < a \text{ or } x > b$	$(-\infty, a) \cup (b, \infty)$	$\{x \in \mathbb{R} \mid x < a \lor x > b\}$	Union of Disjoint Rays
All Real Numbers	$(-\infty, \infty)$	$\{x \in \mathbb{R}\}$	Complete Real Line

How to Convert Compound Inequalities

Compound inequalities join two conditions using "AND" or "OR":

AND Statements (Intersection $\cap$): Both conditions must hold simultaneously, yielding a single bounded interval such as $x \ge -2 \text{ and } x < 5 \implies [-2, 5)$.
OR Statements (Union $\cup$): At least one condition must hold, creating a disjoint union of intervals such as $x \le -3 \text{ or } x \ge 4 \implies (-\infty, -3] \cup [4, \infty)$.

For related algebraic tools, explore our Inequality Solver, Absolute Value Inequality Solver, and Confidence Interval Calculator.

Frequently Asked Questions

How do you turn an inequality into interval notation?

Identify the lower bound and upper bound, separate them with a comma (e.g. $a, b$), and use square brackets $[,]$ if the bound is included ($\le, \ge$) or round parentheses $(,)$ if excluded ($<, >$). Unbounded sides use $-\infty$ or $\infty$ with parentheses.

What is the interval notation for -1 ≤ x ≤ 1?

The interval notation is $[-1, 1]$ because both endpoints $-1$ and $1$ are included with $\le$ inequalities (closed interval).

Why does infinity always have a parenthesis instead of a bracket?

Infinity ($\infty$ or $-\infty$) is an abstract mathematical concept indicating unbounded growth, not a discrete reachable number. Because you can never include infinity itself, it is always written with a parenthesis.

What does the union symbol (∪) mean in interval notation?

The union symbol $\cup$ combines two or more separate (disjoint) intervals into a single solution set, representing values that satisfy either condition (an "OR" inequality).

What is the difference between [2, 5) and (2, 5]?

$[2, 5)$ includes $2$ but excludes $5$ ($2 \le x < 5$). In contrast, $(2, 5]$ excludes $2$ but includes $5$ ($2 < x \le 5$).

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