Inequality to Interval Notation Calculator: Convert Between Forms
Table of Contents - Inequality to Interval Notation
- How to Use This Calculator
- Understanding the Conversion
- How to Convert Manually
- Real-World Applications
- Common Mistakes and How to Avoid Them
- Related Topics
- How This Calculator Works
- FAQs
How to Use This Calculator - Inequality to Interval Notation
Enter an inequality like x is greater than 5, -3 is less than or equal to x is less than 7, or x is less than -2 OR x is greater than or equal to 4.
Click "Convert" to see the interval notation equivalent. The calculator shows brackets versus parentheses, union symbols for "or" conditions, and the complete interval representation.
The results display the interval notation, explain what the symbols mean, and show the number line representation for visualization.
Understanding the Conversion
Inequalities and interval notation are two ways to represent the same set of numbers. Converting between them requires understanding how inequality symbols map to brackets and parentheses.
The basic mapping:
"Less than or equal" and "greater than or equal" use square brackets [ or ] because the endpoint is included. Strict inequalities "less than" and "greater than" use parentheses ( or ) because the endpoint is excluded.
Reading direction:
For x is greater than 3, you write (3, ∞) - the interval extends from 3 (excluded) to infinity. For x is less than or equal to 5, you write (-∞, 5] - from negative infinity to 5 (included).
Compound inequalities:
"And" inequalities like 2 is less than x is less than 7 become a single interval (2, 7). "Or" inequalities like x is less than -1 OR x is greater than 3 use union: (-∞, -1) ∪ (3, ∞).
Infinity symbols:
Always use parentheses with infinity: (∞ or -∞). You can never "reach" infinity to include it, so brackets would be wrong.
Why convert:
Interval notation is more compact and standard in higher mathematics. It's the preferred way to express solution sets in calculus, analysis, and many applications.
Reversing the process:
You can also go from interval notation to inequality. [2, 5) means 2 is less than or equal to x is less than 5. The bracket types tell you which inequality symbols to use.
Special cases:
A single point [a, a] represents x = a exactly. The empty set ∅ represents no solution. All real numbers (-∞, ∞) represents any x.
How to Convert Manually
Let me show you how to convert different types of inequalities to interval notation.
Example 1: Simple greater than
Convert: x is greater than 4
Step 1: Identify the inequality type Greater than is strict (no equal sign)
Step 2: Determine bounds Lower bound: 4 (excluded) Upper bound: infinity
Step 3: Choose bracket types Excluded endpoint: parenthesis ( Infinity: always parenthesis )
Step 4: Write interval (4, ∞)
Example 2: Less than or equal
Convert: x is less than or equal to -2
Step 1: Inequality type Less than or equal (includes endpoint)
Step 2: Bounds Lower bound: negative infinity Upper bound: -2 (included)
Step 3: Brackets Infinity: ( Included endpoint: ]
Step 4: Interval (-∞, -2]
Example 3: Compound "and" inequality
Convert: 3 is less than or equal to x is less than 10
Step 1: Identify both conditions x is greater than or equal to 3 (3 included) x is less than 10 (10 excluded)
Step 2: Bounds Lower: 3 (included) Upper: 10 (excluded)
Step 3: Brackets 3 included: [ 10 excluded: )
Step 4: Interval [3, 10)
Example 4: Compound "or" inequality
Convert: x is less than -1 OR x is greater than or equal to 5
Step 1: Break into two parts First: x is less than -1 → (-∞, -1) Second: x is greater than or equal to 5 → [5, ∞)
Step 2: Combine with union (-∞, -1) ∪ [5, ∞)
The ∪ symbol means "or" - x is in one interval OR the other.
Example 5: Greater than or equal
Convert: x is greater than or equal to 0
Step 1: Type Greater than or equal (includes 0)
Step 2: Bounds Lower: 0 (included) Upper: infinity
Step 3: Interval [0, ∞)
Example 6: Between values (both excluded)
Convert: -5 is less than x is less than 2
Step 1: Both inequalities strict x is greater than -5 (excluded) x is less than 2 (excluded)
Step 2: Bounds Lower: -5 (excluded) Upper: 2 (excluded)
Step 3: Interval (-5, 2)
Both parentheses because both excluded.
Example 7: Between values (both included)
Convert: 1 is less than or equal to x is less than or equal to 8
Step 1: Both include endpoints x is greater than or equal to 1 x is less than or equal to 8
Step 2: Interval [1, 8]
Both brackets because both included.
Example 8: All real numbers
Convert: x is less than 100 OR x is greater than or equal to -50
Step 1: Analyze coverage Every number is either less than 100 or greater than or equal to -50 (in fact, every number satisfies at least one condition)
Step 2: Simplify This covers all real numbers
Step 3: Interval (-∞, ∞)
Example 9: No solution
Convert: x is greater than 5 AND x is less than 2
Step 1: Check if possible No number is simultaneously greater than 5 and less than 2
Step 2: Conclusion
Empty set: ∅ or { }
Example 10: Reverse direction - interval to inequality
Convert [−3, 7) to inequality
Step 1: Read the brackets [ at -3 means -3 is included (less than or equal or greater than or equal) ) at 7 means 7 is excluded (strict inequality)
Step 2: Determine direction x is between -3 and 7
Step 3: Write inequality -3 is less than or equal to x is less than 7
Or separately: x is greater than or equal to -3 AND x is less than 7
Real-World Applications
Domain specification:
Functions have domains expressed in interval notation. f(x) = sqrt(x) has domain [0, ∞) meaning x is greater than or equal to 0.
Range specification:
Similarly, the range (possible outputs) of functions uses interval notation. The range of x² is [0, ∞).
Continuous intervals:
Manufacturing tolerances like "diameter between 9.9mm and 10.1mm inclusive" become [9.9, 10.1] in interval notation.
Temperature ranges:
Safe storage "below 40°F" is (-∞, 40), while "between 2°C and 8°C" is [2, 8].
Time windows:
"Available after 9am" is [9, ∞), "before 5pm" is (-∞, 5), and "9am to 5pm" is [9, 5] (in 24-hour: [9, 17]).
Scientific measurements:
pH levels, concentrations, or other measured quantities with valid ranges are expressed in interval notation for precision.
Solution sets:
Solutions to equations and inequalities in mathematics courses are standard ly written in interval notation.
Common Mistakes and How to Avoid Them
Mistake 1: Brackets with infinity
Wrong: [5, ∞] or (-∞, 3]
Right: [5, ∞) and (-∞, 3]
Why it happens: Not remembering infinity can't be included. Always ( or ) with ∞ or -∞.
Mistake 2: Reversed bracket type
Wrong: For x is greater than or equal to 2, writing (2, ∞)
Right: [2, ∞) - the bracket [ shows 2 is included.
Why it happens: Confusing which bracket means included. [ and ] include, ( and ) exclude.
Mistake 3: Wrong order
Wrong: Writing [7, 2] for numbers between 2 and 7
Right: [2, 7] - smaller number first, larger number second.
Why it happens: Not following convention. Always write lower bound, then upper bound.
Mistake 4: Forgetting union symbol
Wrong: Writing two intervals side by side for "or"
Right: Use ∪ between intervals. (-∞, 2) ∪ [5, ∞)
Why it happens: Not knowing the symbol. ∪ is standard for union ("or").
Mistake 5: Using "and" incorrectly
Wrong: For x is less than 2 OR x is greater than 5, writing one interval
Right: This is union of two separate intervals: (-∞, 2) ∪ (5, ∞)
Why it happens: Confusing AND (single interval between) with OR (separate intervals).
Mistake 6: Single point notation
Wrong: Writing [5] for x = 5
Right: [5, 5] - both bounds are the same value.
Why it happens: Not knowing the convention. Interval notation always has two values.
Mistake 7: Missing comma
Wrong: [2 7]
Right: [2, 7] - must have comma separating the two bounds.
Why it happens: Carelessness. Always include the comma.
Related Topics
- Interval Notation Calculator - General interval operations
How This Calculator Works
Step 1: Parse inequality input
Identify inequality symbol(s): less than, greater than, ≤, ≥
Extract boundary value(s)
Determine if compound (AND/OR)
Note which endpoints are included/excluded
Step 2: Determine interval bounds
For x is greater than a: bounds are (a, ∞)
For x is less than a: bounds are (-∞, a)
For a is less than x is less than b: bounds are (a, b)
Step 3: Select bracket types
If endpoint included (≤ or ≥): use [ or ]
If endpoint excluded (strict inequality): use ( or )
Infinity: always use ( or )
Step 4: Handle compound inequalities
If AND (between): single interval
If OR: separate intervals with ∪
Step 5: Format output
Write lower bound
Add comma
Write upper bound
Enclose in appropriate brackets
Add ∪ if union needed
Step 6: Verify
Check order (lower less than upper)
Verify bracket types match inequality symbols
Confirm special cases (empty set, all reals)
Step 7: Provide explanation
Show conversion steps
Explain bracket choices
Display number line if helpful
Give reverse conversion
FAQs
What is interval notation?
A compact way to represent a range of numbers using brackets and parentheses: (a, b), [a, b], (a, b], or [a, b).
What's the difference between ( and [?
Parenthesis ( means the endpoint is excluded. Bracket [ means included. Same for ) vs ].
How do I convert x is greater than 5?
(5, ∞) - parenthesis because 5 is excluded, extends to infinity.
What about x is less than or equal to 3?
(-∞, 3] - from negative infinity to 3, bracket because 3 is included.
How do I show "or" conditions?
Use the union symbol ∪. Example: x is less than 2 OR x is greater than 5 becomes (-∞, 2) ∪ (5, ∞).
Can I use brackets with infinity?
No, always use parentheses with ∞ or -∞. You can't include infinity.
What does [2, 2] mean?
A single point: x = 2 exactly. Both bounds equal.
How do I write all real numbers?
(-∞, ∞)
What about no solution?
Empty set: ∅ or { }
Is (3, 5] valid?
Yes, this means 3 is less than x is less than or equal to 5. Mixed brackets are fine.
How do I convert back to inequality?
Read the brackets: [ or ] means "or equal," ( or ) means strict. [3, 7) is 3 is less than or equal to x is less than 7.
What if both endpoints are included?
Use brackets on both ends: [a, b] means a is less than or equal to x is less than or equal to b.
Do I need a comma?
Yes, always separate the two bounds with a comma: (a, b) not (a b).
Can negative numbers be bounds?
Yes, [-5, -2] is perfectly valid interval notation.
What about fractions?
Works the same: [1/2, 3/4] represents one-half is less than or equal to x is less than or equal to three-fourths.
How do I represent x is not equal to 5?
Technically (-∞, 5) ∪ (5, ∞), though this is rarely written this way.
What's the difference between AND and OR?
AND creates one interval (between two values). OR creates separate intervals needing ∪.
Can I have three intervals?
Yes, use multiple ∪ symbols: (-∞, -2) ∪ (1, 3) ∪ [5, ∞)
Does order matter?
Yes, write smaller bound first: [2, 7] not [7, 2].
What if I write [5, 3]?
This represents the empty set since 5 is not less than 3. It's technically invalid.