Interval Notation Calculator
Free Interval notation Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Interval Notation Calculator
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Formula: {x | a <= x < b} = [a, b)
Worked example — Interval: [-2, 7) | Length: 9 | Midpoint: 2.5 | Type: Half-Open
Formula
{x | a <= x < b} = [a, b)Where square brackets [ ] indicate the endpoint is included (closed), parentheses ( ) indicate the endpoint is excluded (open), a is the lower bound, and b is the upper bound. The interval represents all real numbers x satisfying the given inequality conditions.
Worked Examples
Example 1: Converting Inequality to Interval Notation
Problem:Express the inequality -2 <= x < 7 in interval notation, set-builder notation, and determine the interval type.
Solution:The inequality -2 <= x < 7 means x is greater than or equal to -2 AND less than 7. Left endpoint: -2 is included (<=) so we use a square bracket [ Right endpoint: 7 is excluded (<) so we use a parenthesis ) Interval notation: [-2, 7) Set-builder notation: {x | -2 <= x < 7} Length: 7 - (-2) = 9 Midpoint: (-2 + 7) / 2 = 2.5 Type: Half-open (one endpoint included, one excluded)
Result:Interval: [-2, 7) | Length: 9 | Midpoint: 2.5 | Type: Half-Open
Example 2: Testing Membership in an Interval
Problem:Given the interval (1, 10], determine whether x = 1, x = 5.5, and x = 10 are members of the set.
Solution:The interval (1, 10] means 1 < x <= 10. Test x = 1: Is 1 > 1? No (not strictly greater). x = 1 is NOT in the interval. Test x = 5.5: Is 1 < 5.5 <= 10? Yes. x = 5.5 IS in the interval. Test x = 10: Is 1 < 10 <= 10? Yes. x = 10 IS in the interval. Complement: (-infinity, 1] U (10, infinity)
Result:x=1: Not in interval | x=5.5: In interval | x=10: In interval
Frequently Asked Questions
What is interval notation and why is it used in mathematics?
Interval notation is a compact mathematical shorthand used to describe a continuous range of real numbers between two endpoints. It uses brackets and parentheses to indicate whether endpoints are included or excluded from the set. Square brackets like [a, b] mean the endpoint is included (closed), while parentheses like (a, b) mean the endpoint is excluded (open). This notation is far more efficient than writing out full inequality statements, especially in calculus and analysis where intervals appear frequently. Mathematicians prefer interval notation because it clearly communicates both the range and boundary conditions in a single expression.
What is the difference between open and closed intervals?
An open interval (a, b) excludes both endpoints, meaning the set contains all numbers strictly between a and b but not a or b themselves. A closed interval [a, b] includes both endpoints, so a and b are part of the set along with every number between them. There are also half-open intervals like [a, b) or (a, b] that include one endpoint but not the other. The distinction is crucial in calculus because many theorems require specific interval types. For example, the Extreme Value Theorem guarantees a continuous function attains its maximum and minimum on a closed interval, but this guarantee does not hold for open intervals.
How do you convert between interval notation and inequality notation?
Converting between these notations is straightforward once you understand the bracket conventions. The interval [2, 7) translates to the inequality 2 <= x < 7, where the square bracket becomes a less-than-or-equal sign and the parenthesis becomes a strict less-than sign. Going the other way, if you have -3 < x <= 5, the strict inequality on the left means an open parenthesis and the inclusive inequality on the right means a closed bracket, giving you (-3, 5]. For unbounded intervals, use infinity symbols with always-open parentheses since infinity is not a number that can be reached or included.
What is set-builder notation and how does it relate to interval notation?
Set-builder notation describes a set by stating the properties its members must satisfy, typically written as {x | condition}. The vertical bar means such that, so {x | 2 <= x < 5} reads as the set of all x such that x is greater than or equal to 2 and less than 5. This is equivalent to the interval notation [2, 5). Set-builder notation is more flexible than interval notation because it can describe sets that are not simple intervals, such as {x | x is an even integer} or {x | x squared < 9}. However, for simple continuous ranges, interval notation is preferred because it is more concise and immediately conveys the boundary information.
How do you represent unions and intersections of intervals?
The union of two intervals combines all elements from both sets and is written with the symbol U. For example, (-3, 1) U (4, 8] represents all numbers that are either in (-3, 1) or in (4, 8] or both. The intersection uses the symbol and represents only the elements common to both intervals. For instance, [0, 5] intersected with [3, 7] gives [3, 5] because only numbers between 3 and 5 belong to both intervals. Unions are commonly used to express domains of functions that have discontinuities or holes, while intersections appear when combining multiple constraints on a variable.
Why do we always use parentheses with infinity in interval notation?
Infinity is not a real number but rather a concept representing unboundedness, so it can never be included as an endpoint in a set of real numbers. Since square brackets indicate inclusion, using them with infinity would incorrectly suggest that infinity is a member of the set. This is why we always write (negative infinity, a] or [b, infinity) with parentheses on the infinity side. This convention is universal across mathematics and prevents logical errors. The same rule applies to negative infinity. Even in extended real number systems where infinity is formally treated, standard interval notation conventions maintain parentheses for clarity and consistency.
How is interval notation used in calculus and real analysis?
Interval notation is fundamental in calculus for expressing domains, ranges, and regions of interest. When finding where a function is increasing, you might state f is increasing on (2, 7). The domain of the square root function is [0, infinity), and the range of the natural logarithm is (-infinity, infinity). In integration, definite integrals are computed over specific intervals. The Mean Value Theorem states that for a function continuous on [a, b] and differentiable on (a, b), there exists a point c in (a, b) where the instantaneous rate equals the average rate. Notice how the theorem carefully specifies closed versus open intervals for different conditions.
What is the complement of an interval and how do you find it?
The complement of an interval is the set of all real numbers that are NOT in the interval. For the interval [2, 5), the complement consists of all numbers less than 2 or greater than or equal to 5, written as (-infinity, 2) U [5, infinity). Notice that the bracket type reverses at each endpoint: the closed bracket at 2 in the original interval becomes an open parenthesis in the complement because 2 is included in the original and therefore excluded from the complement. Finding complements is important when solving inequalities by negation, computing probabilities of complementary events, and working with set theory problems in real analysis.
How do you handle degenerate intervals and single-point sets?
A degenerate interval occurs when the lower and upper bounds are equal, such as [3, 3], which contains only the single point 3. The open interval (3, 3) is actually the empty set because no number is strictly between 3 and 3. Similarly, half-open intervals like [3, 3) or (3, 3] are also empty. Single-point sets are more commonly written as {3} rather than [3, 3] for clarity. These edge cases are important in topology and measure theory, where single points have measure zero and degenerate intervals arise as limiting cases. Understanding these distinctions helps avoid errors when working with boundary conditions in mathematical proofs.
What common mistakes should students avoid with interval notation?
The most frequent error is reversing bracket meanings, using parentheses for inclusion or brackets for exclusion. Another common mistake is writing intervals with the larger number first, like [5, 2], which is not valid since the lower bound must be listed first. Students also sometimes use brackets with infinity, writing [3, infinity] instead of the correct [3, infinity). Confusing union and intersection is another pitfall, particularly when solving compound inequalities. Writing x < 3 or x > 7 as (3, 7) instead of (-infinity, 3) U (7, infinity) is a widespread error. Always verify your interval by testing a few points to confirm they belong to the correct set.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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