Interval Notation to Inequality Converter
Convert bounded intervals, infinite rays, and disjoint unions into standard algebraic inequalities and set-builder notation a < x ≤ b with interactive auto-scaling number lines and step-by-step mathematical proofs.
Input Method
Type interval notation directly or construct using guided bracket selectors
| Interval Notation | Inequality Form | Classification | Bracket Rule |
|---|---|---|---|
| (a, b) | a < x < b | Open Interval | Both parentheses ⇒ strict < |
| [a, b] | a ≤ x ≤ b | Closed Interval | Both brackets ⇒ inclusive ≤ |
| (a, b] | a < x ≤ b | Half-Open (Left-Open) | Left <, right ≤ |
| [a, b) | a ≤ x < b | Half-Open (Right-Open) | Left ≤, right < |
| (a, ∞) | x > a | Right-Ray (Open) | Strict lower bound, no upper cap |
| [a, ∞) | x ≥ a | Right-Ray (Closed) | Inclusive lower bound, no upper cap |
| (−∞, b) | x < b | Left-Ray (Open) | No lower limit, strict upper bound |
| (−∞, b] | x ≤ b | Left-Ray (Closed) | No lower limit, inclusive upper bound |
How to Convert Interval Notation to Inequalities
To convert interval notation into an inequality, replace round parentheses ( or ) with strict inequalities (< or >) and square brackets [ or ] with non-strict inequalities (≤ or ≥). Bounded intervals like (a, b] become compound sandwich inequalities a < x ≤ b, while unbounded rays like (-∞, b] become single inequalities x ≤ b.
The Translation Code: From Brackets to Inequalities
Translating mathematical intervals into inequalities requires decoding two fundamental pieces of information from each expression: boundary values (the numbers listed inside) and enclosure symbols (parentheses versus square brackets).
Represents an open, excluded boundary point. When converting to inequality form, use strict operators: < for upper bounds or lower sandwich bounds, and > for unbounded lower bounds.
Represents a closed, included boundary point. When converting to inequality form, use inclusive "or equal to" operators: ≤ or ≥.
The 8 Canonical Conversion Patterns
Every finite or infinite single interval maps to one of eight standard mathematical inequality templates:
| Interval Notation | Inequality Equivalent | Set-Builder Form | Classification |
|---|---|---|---|
| (a, b) | a < x < b | { x ∈ ℝ | a < x < b } | Open Interval |
| [a, b] | a ≤ x ≤ b | { x ∈ ℝ | a ≤ x ≤ b } | Closed Interval |
| (a, b] | a < x ≤ b | { x ∈ ℝ | a < x ≤ b } | Half-Open (Left-Open) |
| [a, b) | a ≤ x < b | { x ∈ ℝ | a ≤ x < b } | Half-Open (Right-Open) |
| (a, ∞) | x > a | { x ∈ ℝ | x > a } | Infinite Right-Ray (Open) |
| [a, ∞) | x ≥ a | { x ∈ ℝ | x ≥ a } | Infinite Right-Ray (Closed) |
| (−∞, b) | x < b | { x ∈ ℝ | x < b } | Infinite Left-Ray (Open) |
| (−∞, b] | x ≤ b | { x ∈ ℝ | x ≤ b } | Infinite Left-Ray (Closed) |
Translating Disjoint Unions (∪) to "OR"
In set theory, the union symbol ∪ joins sets of elements together. In logic and algebra, membership in A ∪ B means an element belongs to set A or set B.
Never combine disjoint intervals with "and". Writing x < -2 and x ≥ 3 is mathematically impossible because no single real number can be simultaneously less than -2 and greater than 3.
Entire Real Line and Punctured Intervals
Two special cases frequently occur in calculus domain problems:
All Real Numbers: (-∞, ∞)
The interval extends without limit across the entire continuum. In inequality notation, this represents the condition that holds for every number: −∞ < x < ∞ or simply x ∈ ℝ.
Punctured Real Line: (-∞, c) ∪ (c, ∞)
Represents all real numbers except a single excluded point c. This translates into x < c or x > c, commonly written as the exclusion constraint x ≠ c.
Step-by-Step Worked Conversion Examples
Convert (-3, 7] to Inequality Notation
Convert (-∞, 4] to Inequality Notation
Convert (-∞, -2) ∪ [3, ∞) to Inequality Notation
Common Pitfalls and Reversal Errors
Flipping Inequality Signs on Upper Bounds
Misinterpreting (-∞, 5] as x ≥ 5. Numbers to the left of 5 are less than 5, so the correct inequality is x ≤ 5.
Writing "and" Instead of "or" for Unions
Writing x < -1 and x > 4. This represents an impossible intersection ∅. Always use "or" when translating disjoint intervals with ∪.
Inverting Sandwich Inequalities
Writing 7 ≤ x < -3. Inequalities must progress from lowest to highest from left to right: -3 < x ≤ 7.
Confusing Strict (<) with Non-Strict (≤)
Double-check the bracket shape: a smooth curve ( indicates strict inequality; a square angle [ indicates "or equal to".
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Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.