Inequality to Interval Notation Converter
Convert single rays, compound bounded intervals, and disjoint "OR" inequalities into standard interval notation [a, b) and set-builder format with interactive number line visualization.
Input Method
Type freeform inequality or use guided interval builder
| Inequality Form | Interval Notation | Type Name | Bracket Rule |
|---|---|---|---|
| a < x < b | (a, b) | Open Interval | Both excluded: parentheses |
| a ≤ x ≤ b | [a, b] | Closed Interval | Both included: square brackets |
| a < x ≤ b | (a, b] | Half-Open (Left-Open) | Left excluded, right included |
| a ≤ x < b | [a, b) | Half-Open (Right-Open) | Left included, right excluded |
| x > a | (a, ∞) | Right-Ray (Open) | ∞ always uses parenthesis |
| x ≥ a | [a, ∞) | Right-Ray (Closed) | Left bracket, right parenthesis |
| x < b | (−∞, b) | Left-Ray (Open) | −∞ always uses parenthesis |
| x ≤ b | (−∞, b] | Left-Ray (Closed) | Left parenthesis, right bracket |
How to Convert Inequalities to Interval Notation
To convert an inequality to interval notation, write the lower boundary followed by the upper boundary separated by a comma. Use round parentheses ( ) when an endpoint is excluded (<, >, ±∞). Use square brackets [ ] when an endpoint is included (≤, ≥). For compound inequalities like a < x ≤ b, write (a, b].
The Master Rules of Interval Notation
In algebra, calculus, and set theory, interval notation provides a concise, standardized notation for writing continuous subsets of the real number line ℝ. Rather than writing inequality symbols like <, ≤, >, or ≥, mathematicians enclose the boundary values inside brackets and parentheses:
Indicates an excluded boundary (open endpoint). Applied whenever the inequality uses strict symbols (< or >) and always on infinite limits (−∞ or +∞).
Indicates an included boundary (closed endpoint). Applied whenever the inequality uses non-strict "or equal to" symbols (≤ or ≥).
The 4 Core Interval Topologies
Every continuous interval belongs to one of four fundamental geometric topologies based on boundary closure:
| Classification | Inequality Condition | Interval Notation | Number Line Appearance |
|---|---|---|---|
| Open Interval | a < x < b | (a, b) | Two hollow circles ○ — ○ |
| Closed Interval | a ≤ x ≤ b | [a, b] | Two solid filled dots • — • |
| Half-Open (Left) | a < x ≤ b | (a, b] | Hollow left ○, solid right • |
| Half-Open (Right) | a ≤ x < b | [a, b) | Solid left •, hollow right ○ |
| Infinite Left-Ray | x ≤ b | (−∞, b] | Arrow pointing left ← • |
| Infinite Right-Ray | x > a | (a, ∞) | ○ → Arrow pointing right |
Disjoint Unions & Compound "OR" Inequalities
When solving absolute value inequalities like |x| > 3 or quadratic inequalities whose graph lies above a horizontal axis, the solution set consists of two disconnected branches. In logic, this is written with the keyword "or":
Because a single interval cannot jump across a gap, interval notation joins the two independent rays using the set union operator ∪:
Visualizing on the Real Number Line
Graphing inequalities on a real number line translates abstract algebraic constraints into visual segments. The graphical conventions map 1-to-1 to interval brackets:
Step-by-Step Worked Conversion Examples
Convert -3 < x ≤ 7 to Interval Notation
Convert x ≥ 0 to Interval Notation
Convert x < -2 or x ≥ 4 to Interval Notation
Common Pitfalls and Notation Mistakes
Putting Square Brackets on Infinity
Writing [-∞, 5] is universally incorrect. Infinity is an open boundless direction, never a discrete member of the set. Always write (-∞, 5].
Writing Intervals in Descending Order
Writing (7, -3] instead of (-3, 7]. The lower value must strictly appear first on the left.
Confusing Intersection (∩) with Union (∪)
"OR" conditions require union ∪. Using intersection ∩ would produce an empty set ∅ because no single number x can simultaneously be < -2 AND > 4.
Swapping Strict and Inclusive Brackets
Remember: < and > map to round ( ), whereas ≤ and ≥ map to square [ ].
Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.
Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.