Interval Notation Explained
Interval notation is a compact way to describe a set of numbers on the number line using brackets [ ], parentheses ( ), and the infinity symbol ∞. This complete guide covers all 9 types of intervals, how to convert between inequalities and interval notation, union and intersection of intervals, domain and range applications, and 40+ worked examples.
Core rules at a glance
Parentheses ( )
Open endpoints. Use them for strict inequalities and for ±∞.
Brackets [ ]
Closed endpoints. Use them only when the boundary value is included.
Union and intersection
Union means OR. Intersection means AND. Overlap decides the final interval shape.
Quick Interval Notation Converter
Convert a solved inequality into interval notation, or read interval notation back into inequality form without leaving the article.
Result
The inequality converts to interval notation by reading the included and excluded endpoints from the final solution set.
Inequality
Interval Notation
Set-Builder Notation
Number Line
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Zone 2
What Is Interval Notation?
Interval notation is a standardized way to describe a continuous set of real numbers between two endpoints. It tells you the left endpoint, the right endpoint, and whether each endpoint is included or excluded.
The format always runs from smaller value to larger value. That is why a left endpoint appears first, a right endpoint appears second, and the symbols around them do real work instead of acting like decoration.
If the answer is bounded, both endpoints are finite. If it extends forever, use -∞ or +∞. If the answer splits into separate pieces, join them with a union symbol.
Interval Notation Symbols: Brackets vs Parentheses
Parentheses ( )
Parentheses mean an endpoint is open and excluded. Use them for strict inequalities such as x > 3, a < x < b, and every appearance of ±∞.
Square Brackets [ ]
Brackets mean an endpoint is closed and included. Use them only with real boundary values that belong to the set, such as x ≥ 3 or -2 ≤ x ≤ 5.
| Symbol | Meaning | Example | Rule |
|---|---|---|---|
| ( | Open left endpoint | (3, 7) means x > 3 | Use with excluded endpoints. |
| ) | Open right endpoint | (3, 7) means x < 7 | Use with excluded endpoints. |
| [ | Closed left endpoint | [3, 7] means x ≥ 3 | Use with included endpoints. |
| ] | Closed right endpoint | [3, 7] means x ≤ 7 | Use with included endpoints. |
| +∞ | Extends to positive infinity | (3, +∞) means x > 3 | Infinity always uses parentheses. |
| -∞ | Extends to negative infinity | (-∞, 5] means x ≤ 5 | Infinity always uses parentheses. |
| ∪ | Union, meaning OR | (-∞, 2) ∪ (5, +∞) | Keep values from either interval. |
| ∩ | Intersection, meaning AND | [1, 5] ∩ [3, 8] = [3, 5] | Keep only the overlap. |
| ∅ | Empty set | [5, 3] = ∅ | No real number satisfies the condition. |
Critical rule
±∞ always uses parentheses, never brackets, because infinity is not a reachable real endpoint.
The 9 Types of Intervals (Complete List)
These nine patterns cover the interval forms used across algebra, precalculus, calculus, and function domain-range work.
| Type | Interval | Inequality | Number line | Notes |
|---|---|---|---|---|
| Open interval | (a, b) | a < x < b | ○══════○ | Neither endpoint is included. |
| Closed interval | [a, b] | a ≤ x ≤ b | ●══════● | Both endpoints are included. |
| Half-open (left closed) | [a, b) | a ≤ x < b | ●══════○ | Include a, exclude b. |
| Half-open (right closed) | (a, b] | a < x ≤ b | ○══════● | Exclude a, include b. |
| Open ray (right) | (a, +∞) | x > a | ○════════→ | Strict right ray. |
| Closed ray (right) | [a, +∞) | x ≥ a | ●════════→ | Inclusive right ray. |
| Open ray (left) | (-∞, b) | x < b | ←════════○ | Strict left ray. |
| Closed ray (left) | (-∞, b] | x ≤ b | ←════════● | Inclusive left ray. |
| All real numbers | (-∞, +∞) | x ∈ ℝ | ←════════→ | The entire number line. |
Open Intervals
Open intervals use parentheses at both ends and exclude both boundary values. They match double strict inequalities such as a < x < b.
(2, 5)
2 < x < 5
2 and 5 are both excluded.
(-3, 0)
-3 < x < 0
Only the interior values count.
(0, π)
0 < x < π
Use parentheses at both ends.
Closed Intervals
Closed intervals use brackets at both ends and include both endpoints. They match double inclusive inequalities such as a ≤ x ≤ b.
[2, 5]
2 ≤ x ≤ 5
Both endpoints are included.
[-3, 0]
-3 ≤ x ≤ 0
Closed intervals include the boundary values.
[0, 1]
0 ≤ x ≤ 1
This is the unit interval.
Half-Open (Half-Closed) Intervals
Mixed intervals include one endpoint and exclude the other. They appear naturally when one inequality is strict and the other is inclusive.
[0, 5)
0 ≤ x < 5
Closed at 0, open at 5.
(-2, 3]
-2 < x ≤ 3
Open at -2, closed at 3.
[1, 10)
1 ≤ x < 10
Mixed endpoints are common in domain work.
Unbounded Intervals (with Infinity)
Unbounded intervals are rays or whole-line answers. The finite endpoint can be open or closed, but the infinity side always uses a parenthesis.
(a, +∞)
x > a
Open right ray.
[a, +∞)
x ≥ a
Closed right ray.
(-∞, b)
x < b
Open left ray.
(-∞, b]
x ≤ b
Closed left ray.
(-∞, +∞)
all real numbers
Both infinities still use parentheses.
How to Write Interval Notation Step by Step
Step 1: Identify the interval shape
Decide whether the answer is a bounded interval, a left or right ray, all real numbers, or two separate intervals joined by a union.
Step 2: Find the endpoints
The boundary values in the inequality become the left and right endpoints. If the graph continues forever, use -∞ or +∞.
Step 3: Match each symbol to a bracket choice
Strict signs < and > use parentheses. Inclusive signs ≤ and ≥ use brackets. Infinity always keeps parentheses.
Step 4: Write from smaller to larger
Always write the left endpoint first and the right endpoint second, even when the original inequality is written with x on the left.
Step 5: Check against the number line
Open circles must match parentheses, closed circles must match brackets, and split solutions must show a union.
Interval Notation to Inequality (Conversion)
Read the left symbol first, then the right symbol. Parentheses turn into strict signs, brackets turn into inclusive signs, and infinity removes one side of the bound.
Example
(2, 9)
A parenthesis at 2 means x > 2.
A parenthesis at 9 means x < 9.
Because the interval is bounded, combine them into one chain.
Answer
2 < x < 9
Example
[-5, 3]
A bracket at -5 means x ≥ -5.
A bracket at 3 means x ≤ 3.
Together that gives one closed interval.
Answer
-5 ≤ x ≤ 3
Example
(-∞, 6)
There is no lower bound because the interval starts at -∞.
The parenthesis at 6 means x < 6.
Answer
x < 6
Example
(-∞, -2] ∪ [5, +∞)
Read each interval separately.
The union symbol means OR.
Combine the two branches into one statement.
Answer
x ≤ -2 or x ≥ 5
Inequality to Interval Notation (Conversion)
Start with the boundary values, decide whether each boundary is open or closed, and then write the answer from left to right. OR answers become unions.
Example
x > 5
Boundary point is 5.
The inequality is strict, so 5 is open.
The set extends right forever.
Answer
(5, +∞)
Example
x ≤ -3
The solution extends left forever.
-3 is included because the sign is ≤.
Use a bracket at -3 and a parenthesis at -∞.
Answer
(-∞, -3]
Example
-2 < x ≤ 7
Left endpoint is -2 and right endpoint is 7.
-2 is open because the sign is <.
7 is closed because the sign is ≤.
Answer
(-2, 7]
Example
x < -1 or x > 4
This is an OR statement, so the answer splits into two intervals.
The left ray is open at -1.
The right ray is open at 4.
Answer
(-∞, -1) ∪ (4, +∞)
Interval Notation on the Number Line
The number line is the fastest way to check whether your interval notation really matches the underlying inequality. Open circles correspond to parentheses, closed circles correspond to brackets, and unions show separate highlighted regions.
| Inequality | Interval | Number line |
|---|---|---|
| x > a | (a, +∞) | ○════════→ |
| x ≥ a | [a, +∞) | ●════════→ |
| x < b | (-∞, b) | ←════════○ |
| x ≤ b | (-∞, b] | ←════════● |
| a < x < b | (a, b) | ○══════○ |
| a ≤ x ≤ b | [a, b] | ●══════● |
| a ≤ x < b | [a, b) | ●══════○ |
| a < x ≤ b | (a, b] | ○══════● |
| x < a or x > b | (-∞, a) ∪ (b, +∞) | ←══○ ○══→ |
| x ∈ ℝ | (-∞, +∞) | ←════════→ |
| No solution | ∅ | ──────── |
Union of Intervals (∪)
Union means OR. Keep every value that belongs to either interval, and merge the intervals when they overlap or touch through an included endpoint.
x < -2 or x > 5
(-∞, -2) ∪ (5, +∞)
OR keeps both outer regions, so the answer stays split.
x ≤ 0 or x ≥ 3
(-∞, 0] ∪ [3, +∞)
Because 0 and 3 are included, both endpoints use brackets.
(-2, 5) ∪ (3, 8)
(-2, 8)
The intervals overlap, so the union merges into one larger interval.
(-2, 5) ∪ [5, 8)
(-2, 8)
The sets touch through the included point 5, so the union still becomes one interval.
Intersection of Intervals (∩)
Intersection means AND. Keep only the overlap by taking the larger left endpoint and the smaller right endpoint, then choose the more restrictive bracket at each side.
(-1, 7) ∩ (3, 10)
(3, 7)
Take the larger left endpoint and the smaller right endpoint.
[-2, 5] ∩ [1, 8]
[1, 5]
Both bounds are closed in the overlap.
(-3, 6) ∩ [2, 9]
[2, 6)
Use the more restrictive bracket at each endpoint.
(-5, 1) ∩ (3, 8)
∅
There is no shared region, so the result is the empty set.
Interval Notation for Domain and Range
Domain tells you which x-values are allowed. Range tells you which y-values appear. Interval notation is the standard way to summarize both sets once the restrictions or extrema are known.
Domain examples
f(x) = 2x + 3
No restrictions
(-∞, +∞)
f(x) = √(x - 4)
x - 4 ≥ 0, so x ≥ 4
[4, +∞)
f(x) = √(9 - x²)
9 - x² ≥ 0, so -3 ≤ x ≤ 3
[-3, 3]
f(x) = 1 / (x - 2)
x ≠ 2
(-∞, 2) ∪ (2, +∞)
f(x) = (x + 1) / ((x - 3)(x + 5))
x ≠ 3 and x ≠ -5
(-∞, -5) ∪ (-5, 3) ∪ (3, +∞)
f(x) = ln(x + 1)
x + 1 > 0, so x > -1
(-1, +∞)
f(x) = ln(x² - 4)
x² - 4 > 0, so x < -2 or x > 2
(-∞, -2) ∪ (2, +∞)
f(x) = √(x + 3) / (x - 1)
x ≥ -3 and x ≠ 1
[-3, 1) ∪ (1, +∞)
Range examples
f(x) = x²
Minimum value is 0 and there is no maximum.
[0, +∞)
f(x) = -(x - 2)² + 5
Maximum value is 5 and values extend downward forever.
(-∞, 5]
f(x) = √x
Square roots are never negative.
[0, +∞)
f(x) = |x| - 3
The minimum value is -3.
[-3, +∞)
f(x) = 1 / x
Every real value except 0 appears.
(-∞, 0) ∪ (0, +∞)
f(x) = sin(x)
Sine stays between -1 and 1 inclusive.
[-1, 1]
Set-Builder Notation vs Interval Notation
Set-builder notation keeps the condition explicit, while interval notation compresses the same condition into the shape of the number line. Strong students should be able to move between both forms comfortably.
| Set-builder notation | Interval notation | Description |
|---|---|---|
| { x | x > 3 } | (3, +∞) | x greater than 3 |
| { x | x ≥ 3 } | [3, +∞) | x at least 3 |
| { x | x < 5 } | (-∞, 5) | x less than 5 |
| { x | x ≤ 5 } | (-∞, 5] | x at most 5 |
| { x | -2 < x < 7 } | (-2, 7) | strictly between -2 and 7 |
| { x | -2 ≤ x ≤ 7 } | [-2, 7] | inclusive at both ends |
| { x | -2 ≤ x < 7 } | [-2, 7) | left closed, right open |
| { x | x < -1 or x > 4 } | (-∞, -1) ∪ (4, +∞) | two separate regions |
| { x | x ∈ ℝ } | (-∞, +∞) | all real numbers |
| { x | x ≠ 2 } | (-∞, 2) ∪ (2, +∞) | all real numbers except 2 |
Interval Notation Examples (30+ Worked Problems)
These worked problems move from direct conversions to algebra, absolute value, rational inequalities, and domain restrictions. The goal is to make the notation feel routine instead of mysterious.
x > 7
Boundary point is 7.
Strict sign means an open endpoint.
Answer in interval notation
(7, +∞)
x < -3
The interval extends left forever.
The endpoint at -3 is open.
Answer in interval notation
(-∞, -3)
x ≥ 0
0 is included, so use a bracket.
The ray extends right forever.
Answer in interval notation
[0, +∞)
x ≤ 4
4 is included.
The interval extends left forever.
Answer in interval notation
(-∞, 4]
2 < x < 8
Both endpoints are strict.
Use parentheses on both sides.
Answer in interval notation
(2, 8)
-5 ≤ x ≤ 5
Both bounds are inclusive.
Use brackets on both ends.
Answer in interval notation
[-5, 5]
0 ≤ x < 10
0 is closed and 10 is open.
This is a half-open interval.
Answer in interval notation
[0, 10)
-1 < x ≤ 6
Left side is open.
Right side is closed.
Answer in interval notation
(-1, 6]
(2, 9)
Parentheses become strict signs.
The result is a double inequality.
Answer in interval notation
2 < x < 9
[-4, +∞)
The bracket at -4 means include it.
There is no upper bound.
Answer in interval notation
x ≥ -4
Solve 2x - 3 > 7
Add 3, then divide by 2.
The solved inequality is x > 5.
Answer in interval notation
(5, +∞)
Solve -4x ≥ 12
Divide by -4 and flip the sign.
The result is x ≤ -3.
Answer in interval notation
(-∞, -3]
Solve -2 ≤ 3x + 1 < 10
Subtract 1, then divide all parts by 3.
The result is -1 ≤ x < 3.
Answer in interval notation
[-1, 3)
Solve |x - 3| < 5
Rewrite as -5 < x - 3 < 5.
Add 3 to all parts.
Answer in interval notation
(-2, 8)
Solve |2x + 1| ≥ 7
Split into two branches.
Solve each branch and join with OR.
Answer in interval notation
(-∞, -4] ∪ [3, +∞)
Domain of f(x) = √(x + 5)
Require x + 5 ≥ 0.
That gives x ≥ -5.
Answer in interval notation
[-5, +∞)
Domain of f(x) = 1 / (x + 2)
Exclude x = -2.
Everything else is allowed.
Answer in interval notation
(-∞, -2) ∪ (-2, +∞)
(-∞, 4) ∪ (1, +∞)
The intervals overlap.
The union covers the entire real line.
Answer in interval notation
(-∞, +∞)
[-2, 5] ∩ [1, 8]
Take the larger left endpoint and smaller right endpoint.
Both remain closed.
Answer in interval notation
[1, 5]
x ≠ 2
Exclude one point from the real line.
That creates two open rays.
Answer in interval notation
(-∞, 2) ∪ (2, +∞)
Solve x² - 4x - 5 < 0
Factor to (x - 5)(x + 1).
The quadratic is negative between the roots.
Answer in interval notation
(-1, 5)
Solve x² - 9 ≥ 0
Factor to (x - 3)(x + 3).
The quadratic is nonnegative outside the roots.
Answer in interval notation
(-∞, -3] ∪ [3, +∞)
Solve (x + 2) / (x - 1) > 0
Build a sign chart with critical points -2 and 1.
Keep intervals where the quotient is positive.
Answer in interval notation
(-∞, -2) ∪ (1, +∞)
Solve (x - 3) / (x + 4) ≤ 0
Critical points are -4 and 3.
The quotient is nonpositive between them, excluding -4.
Answer in interval notation
(-4, 3]
Domain of f(x) = √(x² - 9)
Require x² - 9 ≥ 0.
That means x ≤ -3 or x ≥ 3.
Answer in interval notation
(-∞, -3] ∪ [3, +∞)
Domain of f(x) = ln(x² - 5x + 6)
Require x² - 5x + 6 > 0.
The logarithm only accepts positive input.
Answer in interval notation
(-∞, 2) ∪ (3, +∞)
(-3, 8) ∩ [2, 12)
The overlap starts at 2 and ends at 8.
Use the more restrictive bracket at each side.
Answer in interval notation
[2, 8)
(-∞, 5] ∩ [3, +∞)
The shared region runs from 3 to 5.
Both endpoints stay closed.
Answer in interval notation
[3, 5]
Solve 2 / (x - 3) < 1
Move everything to one side.
Use a sign chart on the simplified rational expression.
Answer in interval notation
(-∞, 3) ∪ (5, +∞)
Solve x² - 2x - 8 ≤ 0
Factor to (x - 4)(x + 2).
Keep the interval between the roots and include both endpoints.
Answer in interval notation
[-2, 4]
Practice Problems
Try the prompt first, then expand the answer to compare your thinking with the interval notation method.
Practice Problem 1Convert x ≥ -4 to interval notation.
The value -4 is included, so use a bracket.
The set extends right forever, so the infinity side uses a parenthesis.
Answer
[-4, +∞)
Practice Problem 2Convert (-3, 7] to an inequality.
The parenthesis at -3 means x > -3.
The bracket at 7 means x ≤ 7.
Answer
-3 < x ≤ 7
Practice Problem 3Convert -5 < x ≤ 2 to interval notation.
-5 is open because the sign is strict.
2 is closed because the sign is inclusive.
Answer
(-5, 2]
Practice Problem 4Solve 3x + 2 > 11 and write the answer in interval notation.
Subtract 2 to get 3x > 9.
Divide by 3 to get x > 3.
Answer
(3, +∞)
Practice Problem 5Solve -2x - 1 ≤ 5 and write the answer in interval notation.
Add 1 to get -2x ≤ 6.
Divide by -2 and flip the sign to get x ≥ -3.
Answer
[-3, +∞)
Practice Problem 6Solve |x + 3| < 6 and write the answer in interval notation.
Rewrite as -6 < x + 3 < 6.
Subtract 3 from all parts.
Answer
(-9, 3)
Practice Problem 7Find (-1, 6) ∪ (4, 9).
The two intervals overlap.
Merge them into a single interval.
Answer
(-1, 9)
Practice Problem 8Find [-2, 5] ∩ (1, 8).
The overlap starts just after 1 and ends at 5.
1 stays open and 5 stays closed.
Answer
(1, 5]
Practice Problem 9Find the domain of f(x) = √(2x - 6).
Require 2x - 6 ≥ 0.
Solve to get x ≥ 3.
Answer
[3, +∞)
Practice Problem 10Solve x² - 2x - 8 ≤ 0 and write the answer in interval notation.
Factor to (x - 4)(x + 2).
The quadratic is nonpositive between the roots, including both roots.
Answer
[-2, 4]
Frequently Asked Questions
What is interval notation in math?
Interval notation is a compact way to describe a continuous set of real numbers using endpoints, brackets, parentheses, infinity symbols, unions, and the empty set symbol.
What is the difference between ( ) and [ ] in interval notation?
Parentheses mean the endpoint is excluded, while brackets mean the endpoint is included.
When do you use a parenthesis vs a bracket in interval notation?
Use parentheses for strict inequalities such as < and >, and brackets for inclusive inequalities such as ≤ and ≥.
Why does infinity always use a parenthesis in interval notation?
Infinity is not a real number that can be reached or included, so it always uses parentheses.
What does ∪ mean in interval notation?
The union symbol means OR. A value belongs to the result if it is in either interval.
What does ∩ mean in interval notation?
The intersection symbol means AND. A value belongs to the result only if it is in both intervals.
How do you convert an inequality to interval notation?
Find the boundary values, decide whether each endpoint is open or closed, and write the interval from left to right.
How do you convert interval notation to an inequality?
Read the bracket type at each endpoint, turn parentheses into strict signs, brackets into inclusive signs, and keep infinity as an unbounded side.
How do you write x > 5 in interval notation?
x > 5 becomes (5, +∞).
How do you write x ≤ -3 in interval notation?
x ≤ -3 becomes (-∞, -3].
How do you write -2 < x ≤ 7 in interval notation?
-2 < x ≤ 7 becomes (-2, 7].
How do you write (-3, 7] as an inequality?
(-3, 7] means -3 < x ≤ 7.
What is an open interval?
An open interval uses parentheses at both ends and excludes both endpoints, such as (2, 5).
What is a closed interval?
A closed interval uses brackets at both ends and includes both endpoints, such as [2, 5].
What is a half-open interval?
A half-open interval includes one endpoint and excludes the other, such as [0, 5) or (-2, 3].
How do you graph interval notation on a number line?
Mark the endpoints, use open circles for parentheses, closed circles for brackets, and shade the interval or ray that belongs to the solution set.
How do you find the union of two intervals?
Keep every value that belongs to either interval, and merge them when they overlap or touch through an included endpoint.
How do you find the intersection of two intervals?
Keep only the overlap by taking the larger left endpoint and the smaller right endpoint.
How do you write all real numbers in interval notation?
All real numbers are written as (-∞, +∞).
How do you write no solution in interval notation?
No solution is written as ∅, the empty set.
How is interval notation used for domain and range?
After finding the valid x-values or possible y-values of a function, interval notation summarizes those sets compactly and clearly.
What is the difference between set-builder notation and interval notation?
Set-builder notation states the condition directly, while interval notation emphasizes the number-line structure of the same set.