Interval Notation Calculator: Convert Inequalities to Interval Form
Table of Contents - Interval Notation
- How to Use This Calculator
- Understanding Interval Notation
- How to Write Interval Notation Manually
- Real-World Applications
- Common Mistakes and How to Avoid Them
- Related Topics
- How This Calculator Works
- FAQs
How to Use This Calculator - Interval Notation
Enter an inequality or set description. For example, x is greater than or equal to 3 and less than 10, or -5 is less than x which is less than or equal to 2.
Click "Convert" to see the interval notation. The calculator shows bracket types, explains whether endpoints are included, and displays the number line representation.
The results show the interval notation like [3, 10) or (-5, 2], explain what the brackets mean, and verify the conversion with examples.
Understanding Interval Notation
Interval notation is a compact way to describe a range of numbers. Instead of writing long inequality statements, you use brackets and parentheses to show which numbers are included in a set.
The basic syntax:
Use square brackets [ ] when the endpoint is included (closed), and parentheses ( ) when the endpoint is not included (open). The two numbers inside represent the lower and upper bounds.
Reading intervals:
[3, 7] means all numbers from 3 to 7, including both 3 and 7. (3, 7) means all numbers between 3 and 7, not including either endpoint. [3, 7) means 3 is included but 7 is not. (3, 7] means 3 is not included but 7 is.
Infinity symbols:
When an interval extends forever in one direction, use the infinity symbol ∞ or negative infinity -∞. Always use parentheses with infinity because you can never "reach" infinity to include it.
Why use interval notation:
It's much more concise than writing inequalities. Instead of "x is greater than or equal to 2 and less than 5," you write [2, 5). It's the standard mathematical notation for continuous sets.
Closed versus open:
A closed interval includes its endpoints (uses brackets). An open interval excludes its endpoints (uses parentheses). Half-open (or half-closed) intervals include one endpoint but not the other.
Connection to inequalities:
Every interval corresponds to an inequality: [a, b] corresponds to a ≤ x ≤ b (a, b) corresponds to a less than x less than b [a, b) corresponds to a ≤ x less than b (a, b] corresponds to a less than x ≤ b
How to Write Interval Notation Manually
Let me show you how to convert different situations to interval notation.
Example 1: Both endpoints included
Inequality: 3 ≤ x ≤ 8
Step 1: Identify the bounds Lower bound: 3 Upper bound: 8
Step 2: Determine if endpoints are included 3 is included (≤ means less than or equal) 8 is included (≤ means less than or equal)
Step 3: Write interval notation [3, 8]
Use square brackets because both endpoints are included.
Example 2: Both endpoints excluded
Inequality: -5 less than x less than 2
Step 1: Identify bounds Lower: -5 Upper: 2
Step 2: Check inclusion -5 is not included (strict inequality) 2 is not included (strict inequality)
Step 3: Write interval (-5, 2)
Use parentheses because both endpoints are excluded.
Example 3: Mixed endpoints
Inequality: 0 ≤ x less than 10
Step 1: Bounds Lower: 0 Upper: 10
Step 2: Inclusion 0 is included (≤) 10 is not included (less than)
Step 3: Interval notation [0, 10)
Square bracket at 0, parenthesis at 10.
Example 4: Unbounded above
Inequality: x ≥ 7
Step 1: Identify what we know Lower bound: 7 (included) Upper bound: infinity (no upper limit)
Step 2: Write interval [7, ∞)
Always use ( with infinity, never ].
Example 5: Unbounded below
Inequality: x less than -3
Step 1: Bounds Lower: negative infinity Upper: -3 (not included)
Step 2: Interval (-∞, -3)
Always use ) with negative infinity.
Example 6: All real numbers
Inequality: x can be any real number
Interval: (-∞, ∞)
This represents the entire number line.
Example 7: Union of intervals
Inequality: x less than 2 or x ≥ 5
Step 1: Identify two separate intervals First part: x less than 2 → (-∞, 2) Second part: x ≥ 5 → [5, ∞)
Step 2: Use union symbol (-∞, 2) ∪ [5, ∞)
The ∪ means "or" - x is in one interval OR the other.
Example 8: Single point
If x = 3 exactly (no range), write [3, 3]
Both endpoints are the same number, both included.
Example 9: Converting back to inequality
Interval: [-2, 5)
Step 1: Read the brackets [ means -2 is included ) means 5 is not included
Step 2: Write inequality -2 ≤ x less than 5
Example 10: Complex union
x is less than or equal to -1 or between 2 and 6 (excluding 2, including 6) or greater than or equal to 10.
Step 1: Break into parts x ≤ -1 → (-∞, -1] 2 less than x ≤ 6 → (2, 6] x ≥ 10 → [10, ∞)
Step 2: Combine with unions (-∞, -1] ∪ (2, 6] ∪ [10, ∞)
Real-World Applications
Domain and range in mathematics:
When defining the domain (valid inputs) or range (possible outputs) of a function, interval notation expresses these sets concisely and clearly.
Engineering specifications:
Tolerance ranges for manufacturing specify acceptable measurements. An interval like [9.95, 10.05] mm might define acceptable diameters for a part.
Temperature ranges:
Weather forecasts or scientific experiments use intervals. "Temperature between 20°C and 25°C inclusive" becomes [20, 25].
Financial constraints:
Investment guidelines might specify "invest between $1000 and $5000" as an interval [1000, 5000] for allowed investment amounts.
Age restrictions:
Legal restrictions like "must be at least 18 years old" translate to [18, ∞) for describing the allowed age range.
Data analysis:
When binning continuous data into categories, intervals define each bin's boundaries. Income brackets, test score ranges, and measurement classes all use interval notation.
Calculus and analysis:
Convergence criteria, continuity definitions, and limit specifications all use interval notation extensively in advanced mathematics.
Common Mistakes and How to Avoid Them
Mistake 1: Using brackets with infinity
Wrong: [3, ∞]
Right: [3, ∞)
Why it happens: Thinking of infinity as a number. It's not, so it can't be "included." Always use parentheses with ∞ or -∞.
Mistake 2: Reversed order
Wrong: [7, 3] for numbers from 3 to 7
Right: [3, 7]
Why it happens: Not putting the smaller number first. Always write lower bound first, then upper bound.
Mistake 3: Wrong bracket type
Wrong: (5, 10] for 5 ≤ x ≤ 10
Right: [5, 10]
Why it happens: Not matching brackets to the inequality symbols. ≤ means use [, less than means use (.
Mistake 4: Forgetting the comma
Wrong: [3 8]
Right: [3, 8]
Why it happens: Rushing. Always include the comma between the two values.
Mistake 5: Using "or" without union symbol
Wrong: Writing two intervals side by side for "or" conditions
Right: Use ∪ to show union. Example: (-∞, 2) ∪ [5, ∞)
Why it happens: Not knowing the notation for combining intervals. ∪ is standard for "or."
Mistake 6: Empty interval confusion
Wrong: [5, 3] thinking this represents something
Right: This is impossible (5 is not less than 3). It represents the empty set.
Why it happens: Not checking that lower bound is actually less than upper bound.
Mistake 7: Single value notation
Wrong: Writing just [5] for x = 5
Right: [5, 5]
Why it happens: Not understanding that interval notation always has two values. For a single point, both are the same.
Related Topics
- Union and Intersection Calculator - Combine intervals
How This Calculator Works
Step 1: Parse input
Identify type: inequality, description, or existing interval
Extract comparison operators (less than, ≤, greater than, ≥, =)
Identify bound values
Determine unbounded cases
Step 2: Determine endpoint inclusion
For ≤ or ≥: endpoint is included (use bracket [)
For strict less than or greater than: endpoint excluded (use parenthesis ()
For infinity: always use parenthesis
Step 3: Identify interval type
Bounded: has both lower and upper finite bounds
Unbounded above: extends to ∞
Unbounded below: extends to -∞
Unbounded both: (-∞, ∞)
Step 4: Format interval
Place lower bound first
Add comma
Place upper bound second
Choose correct bracket/parenthesis for each end
Step 5: Handle special cases
If "or" condition: create union with ∪
If "and" condition: find intersection
If single value: write [a, a]
If impossible: note empty set
Step 6: Verify
Check lower bound less than upper bound
Confirm bracket types match inequality
Ensure infinity uses parentheses
Step 7: Display
Show interval notation
Explain bracket meanings
Provide number line visualization
Give equivalent inequality
FAQs
What is interval notation?
A mathematical notation using brackets and parentheses to describe a continuous set of numbers between two bounds.
What's the difference between [ and (?
Square bracket [ means the endpoint is included. Parenthesis ( means the endpoint is excluded.
Why use parentheses with infinity?
Infinity isn't a number you can reach or include. It represents unboundedness, so we always use ( or ) with ∞ and -∞.
How do I read [3, 7)?
All numbers from 3 to 7, including 3 but not including 7. In inequality form: 3 ≤ x less than 7.
What does the ∪ symbol mean?
Union. It means "or" - the number is in one interval OR the other. (-∞, 2) ∪ [5, ∞) means x less than 2 or x ≥ 5.
Can I have [5, 5]?
Yes, this represents a single point: x = 5 exactly. Both bounds are the same and both are included.
What's the difference between (a, b) in interval notation and (a, b) as coordinates?
Context tells you. In interval notation, it's a range of numbers. As coordinates, it's a point. Usually the context makes it clear.
How do I write "all real numbers"?
(-∞, ∞). This represents the entire number line.
What if there's no solution?
Write ∅ (empty set symbol) or for the empty set. Or note "no solution."
Can intervals include negative numbers?
Yes, intervals can include any real numbers. [-5, -2] is perfectly valid.
How do I combine intervals with "and"?
Find the intersection (overlap). If you want x ≥ 3 AND x less than 10, that's [3, 10).
What's a half-open interval?
An interval that includes one endpoint but not the other, like [2, 5) or (2, 5].
Is 0 included in (0, 5)?
No, the parenthesis means 0 is not included. Numbers just barely greater than 0 are included, but not 0 itself.
How do I show "x is not equal to 3"?
That's not a single interval. Write (-∞, 3) ∪ (3, ∞) to show all numbers except 3.
What does [−∞, ∞] mean?
This is incorrect notation. Should be (-∞, ∞). Never use brackets with infinity.
Can I write intervals for integers only?
Interval notation traditionally describes continuous real numbers. For discrete sets like integers, use set notation: 12345.
What's the difference between closed and open intervals?
Closed intervals include their endpoints (use brackets). Open intervals exclude their endpoints (use parentheses).
How do I convert interval notation back to inequality?
[ means ≤, ( means less than. So [3, 7) becomes 3 ≤ x less than 7.
Can fractions be endpoints?
Yes. [1/2, 3/4] is valid interval notation.
What if lower bound equals upper bound?
[a, a] represents a single point. (a, a) would be empty (no numbers between a and itself).