Inequality to Interval Notation Calculator
Convert inequalities to interval notation
Category: Mathematics
Inequality to Interval Notation Calculator Inputs
Inequality to Interval Notation Calculator Formula
Equation
x > a arrow (a,∞), \quad x ≤ b arrow (-∞,b], \quad a < x < b arrow (a,b)
Excel Formula
=x>a(a,),xb(-,b],a<x<b(a,b)
Variables
- Inequality — Enter inequality (e.g., "x > 3", "2 ≤ x < 5", "x < -2 or x > 4")
How the Inequality to Interval Notation Calculator Works
Interval notation is a mathematical notation used to represent subsets of real numbers on the number line. It provides a concise way to express ranges of values using brackets and parentheses. This notation is fundamental in calculus, algebra, and real analysis for describing domains, ranges, and solution sets of inequalities. Understanding interval notation is essential for working with continuous functions, limits, and optimization problems.
The core relationship is x > a \rightarrow (a,\infty), \quad x \leq b \rightarrow (-\infty,b], \quad a < x < b \rightarrow (a,b). Typical inputs include Inequality.
Enter your values in the inequality to interval notation calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Inequality to Interval Notation Calculator Theory & Explanation
Bracket Types and Meanings
Interval notation uses two types of brackets with specific meanings: Square brackets [ ] indicate closed intervals where endpoints are included (≤ or ≥). Parentheses ( ) indicate open intervals where endpoints are excluded (< or >). These can be mixed to create half-open (or half-closed) intervals. The choice of bracket directly corresponds to the inequality symbol used.
[a,b] = \x \in \mathbbR : a ≤ x ≤ b\, \quad (a,b) = \x \in \mathbbR : a < x < b\
Closed Intervals
A closed interval [a,b] includes both endpoints a and b. It represents all real numbers x such that a ≤ x ≤ b. In set-builder notation: [a,b] = x ∈ ℝ : a ≤ x ≤ b. Closed intervals are compact in topology and are used when boundary values are part of the solution. Example: [0,1] includes 0, 1, and all numbers between them.
[a,b] = \x : a ≤ x ≤ b\ \quad \text(both endpoints included)
Open Intervals
An open interval (a,b) excludes both endpoints a and b. It represents all real numbers x such that a < x < b. In set-builder notation: (a,b) = x ∈ ℝ : a < x < b. Open intervals are important in analysis for neighborhoods and continuity. Example: (0,1) includes numbers like 0.5, 0.9999, but not 0 or 1 themselves.
(a,b) = \x : a < x < b\ \quad \text(both endpoints excluded)
Half-Open Intervals
Half-open (or half-closed) intervals include one endpoint but not the other. [a,b) includes a but excludes b: a ≤ x < b. (a,b] excludes a but includes b: a < x ≤ b. These are common in discrete mathematics and computer science (array indices). They're also used when one boundary condition differs from the other.
[a,b) = \x : a ≤ x < b\, \quad (a,b] = \x : a < x ≤ b\
Infinite Intervals
When intervals extend to infinity, we always use parentheses with ∞ since infinity is not a real number and cannot be "included." (a,∞) means x > a. [a,∞) means x ≥ a. (-∞,b) means x < b. (-∞,b] means x ≤ b. (-∞,∞) represents all real numbers ℝ. Infinity symbols always get parentheses, never square brackets.
(a,∞) = \x : x > a\, \quad [a,∞) = \x : x ≥ a\, \quad (-∞,∞) = \mathbbR
Union of Intervals
The union symbol ∪ combines non-overlapping intervals to represent "or" conditions. For x < a or x > b, we write (-∞,a) ∪ (b,∞). This represents values in either interval. Graphically, this shows two separate regions on the number line. Union is used when the solution set is disconnected.
(-∞,a) \cup (b,∞) = \x : x < a \text or x > b\
Intersection of Intervals
The intersection symbol ∩ represents "and" conditions—values that satisfy both inequalities simultaneously. For a < x and x < b, we get (a,b). For overlapping intervals [a,c] ∩ [b,d] where a < b < c < d, the result is [b,c]. If intervals don't overlap, the intersection is empty: ∅. Intersection gives the common region.
[a,c] \cap [b,d] = [b,c] \text (if a < b < c < d\text), \quad [0,1] \cap [2,3] = \emptyset
Converting Inequalities
To convert inequalities to intervals: (1) Identify the variable and isolate it. (2) Determine the inequality type (<, >, ≤, ≥). (3) Choose bracket type: < or > → parentheses ( ); ≤ or ≥ → square brackets [ ]. (4) For compound inequalities, use ∪ (or) or ∩ (and). (5) Always use parentheses with ∞. Practice: x ≥ -2 becomes [-2,∞); 3 < x ≤ 7 becomes (3,7].
x > a \Leftrightarrow (a,∞), \quad x ≤ b \Leftrightarrow (-∞,b], \quad a < x ≤ b \Leftrightarrow (a,b]
Number Line Representation
Intervals correspond to regions on the number line. Open endpoints are shown with open circles (○), closed endpoints with filled circles (●). [2,5] shows ● at 2 and 5 with shaded region between. (2,5) shows ○ at 2 and 5 with shaded region between. For unions like (-∞,0) ∪ (2,∞), show two separate shaded regions with gaps.
\text●———● (closed), \quad \text○———○ (open), \quad \text●———○ (half-open)
Set Operations and Properties
Intervals obey set algebra laws. Union is commutative: A ∪ B = B ∪ A. Intersection is commutative: A ∩ B = B ∩ A. Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Empty set: [a,b] where a > b equals ∅. Subset: [a,b] ⊆ [c,d] if c ≤ a and b ≤ d. These properties are essential for solving complex inequality systems.
A \cup B = B \cup A, \quad A \cap (B \cup C) = (A \cap B) \cup (A \cap C)
Inequality to Interval Notation Calculator Worked Examples
Worked Example
Inputs
- inequality: x > 3
Result: (3,∞)
Explanation
x > 3 means all numbers greater than 3, excluding 3 itself. Since we use strict inequality (>), we use parenthesis. The interval extends to positive infinity.
Second Scenario
Inputs
- inequality: x > 3
Result: (3,∞)
Explanation
This scenario uses different inputs (inequality = x > 3) to show how changing one variable affects the inequality to interval notation result. Run the calculator above with these values to get the exact updated output with step-by-step work.
Common Inequality to Interval Notation Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Inequality to Interval Notation homework and study
- Inequality to Interval Notation design and analysis
Inequality to Interval Notation Calculator FAQs
What is the difference between parentheses ( ) and square brackets [ ] in interval notation?
Parentheses ( ) indicate open endpoints that are NOT included in the interval (corresponding to < or >). Square brackets [ ] indicate closed endpoints that ARE included (corresponding to ≤ or ≥). For example: (2,5) means 2 < x < 5 (excluding 2 and 5), while [2,5] means 2 ≤ x ≤ 5 (including 2 and 5). You can mix them: [2,5) means 2 ≤ x < 5 (including 2, excluding 5).
Why do we always use parentheses with infinity (∞)?
Infinity (∞) is not a real number—it's a concept representing unboundedness. Since we can never "reach" or "include" infinity, we always use parentheses, never square brackets. For example: (5,∞) is correct for x > 5, but [5,∞] would be wrong at the infinity end. The correct notation is [5,∞) for x ≥ 5, where the bracket is at 5 (a real number we can include) and the parenthesis is at ∞.
How do I convert compound inequalities like "x < -2 or x > 4"?
For "or" compound inequalities, use the union symbol ∪ to combine separate intervals. "x < -2 or x > 4" becomes (-∞,-2) ∪ (4,∞). This represents two disconnected regions on the number line. For "and" compound inequalities, you typically get a single interval or use intersection ∩. For example, "x > 2 and x < 5" simplifies to (2,5), representing the overlap of both conditions.
What does the union symbol ∪ mean in interval notation?
The union symbol ∪ means "or" and combines multiple intervals that don't overlap. It represents values that belong to at least one of the intervals. For example: (-∞,0) ∪ (2,∞) includes all numbers less than 0 OR greater than 2, but NOT numbers between 0 and 2. On a number line, unions show as separate shaded regions with gaps between them.
What does the intersection symbol ∩ mean?
The intersection symbol ∩ means "and" and represents values that belong to ALL of the intervals simultaneously. For example: [0,5] ∩ [3,8] = [3,5] because only numbers from 3 to 5 are in both intervals. If intervals don't overlap, like [0,2] ∩ [5,7], the intersection is the empty set ∅ (no numbers satisfy both). Intersection finds the common region where conditions overlap.
How do I write "all real numbers" in interval notation?
All real numbers is written as (-∞,∞), pronounced "negative infinity to positive infinity." This represents the entire number line with no restrictions. It's equivalent to the mathematical symbol ℝ. Note that both infinities use parentheses since infinity isn't a number we can include. This interval has no boundaries—it extends infinitely in both directions.
Can an interval be empty?
Yes! An interval can be empty (∅). This happens when there are no real numbers that satisfy the condition. Examples: [5,2] is empty because there's no number that's both ≥5 and ≤2 (the condition is impossible). Similarly, x > 5 and x < 3 has empty intersection: (5,∞) ∩ (-∞,3) = ∅. Empty intervals indicate contradictory or impossible conditions.
How do I handle double inequalities like "2 < x ≤ 7"?
Double inequalities represent a single interval with specific endpoint behavior at each end. "2 < x ≤ 7" means x is between 2 and 7, excluding 2 but including 7. This converts directly to (2,7]—parenthesis at 2 (because 2 is excluded by <) and square bracket at 7 (because 7 is included by ≤). The bracket type at each end matches the inequality symbol pointing to that endpoint.
What's the difference between [a,b] and {a,b}?
[a,b] is interval notation representing ALL real numbers from a to b (infinitely many numbers). a,b is set notation representing exactly TWO numbers: a and b, with nothing in between. For example: [2,5] includes 2, 5, 3.7, 4.999, etc. (all reals from 2 to 5), while 2,5 contains only the two numbers 2 and 5. Interval notation describes ranges; set braces list specific elements.
How do number lines relate to interval notation?
Number lines provide visual representation of intervals. Open endpoints (parentheses) are shown with open circles ○; closed endpoints (brackets) are shown with filled circles ●. The interval region is shaded or bolded. For example, [2,5) shows ● at 2, ○ at 5, with shading between them. Unions appear as multiple separated shaded regions. Drawing number lines helps verify your interval notation is correct before finalizing answers.