Inequality to Interval Notation Calculator
Free Inequality interval notation Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs.
Calculator
Adjust values & calculateAll real numbers strictly less than 5
Open circle at 5, shade left toward negative infinity
Quick Reference
Formula
Parentheses ( ) exclude endpoints (strict inequalities < >). Brackets [ ] include endpoints (non-strict inequalities <= >=). Infinity always gets parentheses. The union symbol U combines separate intervals for OR inequalities.
Last reviewed: December 2025
Worked Examples
Example 1: Convert x <= 7 to Interval Notation
Example 2: Convert -3 < x <= 8 to Interval Notation
Background & Theory
The Inequality to Interval Notation Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Inequality to Interval Notation Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
Formula
x < a becomes (-Infinity, a); x >= a becomes [a, Infinity)
Parentheses ( ) exclude endpoints (strict inequalities < >). Brackets [ ] include endpoints (non-strict inequalities <= >=). Infinity always gets parentheses. The union symbol U combines separate intervals for OR inequalities.
Worked Examples
Example 1: Convert x <= 7 to Interval Notation
Problem: Express the inequality x <= 7 in interval notation and set-builder notation.
Solution: The inequality x <= 7 includes all numbers from negative infinity up to and including 7.\nSince 7 is included (<=), use a bracket at 7.\nSince negative infinity is never reached, use a parenthesis.\nInterval notation: (-Infinity, 7]\nSet-builder: {x | x <= 7}\nNumber line: Closed circle at 7, shade to the left.
Result: (-Infinity, 7]
Example 2: Convert -3 < x <= 8 to Interval Notation
Problem: Express the compound inequality -3 < x <= 8 in interval notation.
Solution: This is a compound AND inequality: x is between -3 and 8.\n-3 is excluded (strict <), so use a parenthesis.\n8 is included (<=), so use a bracket.\nInterval notation: (-3, 8]\nSet-builder: {x | -3 < x <= 8}\nNumber line: Open circle at -3, closed circle at 8, shade between.
Result: (-3, 8]
Frequently Asked Questions
What is interval notation and why is it used?
Interval notation is a mathematical shorthand for describing continuous sets of numbers on the real number line using brackets, parentheses, and the infinity symbol. It replaces wordy descriptions with compact, precise expressions. A square bracket [ or ] indicates that the endpoint is included in the set (closed endpoint), while a parenthesis ( or ) indicates the endpoint is excluded (open endpoint). Infinity and negative infinity always receive parentheses since they are not actual numbers. Interval notation is widely used in calculus, analysis, and precalculus because it clearly communicates domains, ranges, and solution sets. It is more compact than set-builder notation and less ambiguous than verbal descriptions.
How do absolute value inequalities convert to interval notation?
Absolute value inequalities translate directly into compound inequalities and then to interval notation. For |x - c| < r (less-than type), the solution is all x within distance r of center c: c - r < x < c + r, which becomes (c - r, c + r) in interval notation. For |x - c| > r (greater-than type), the solution is all x further than distance r from c: x < c - r or x > c + r, which becomes (-Infinity, c - r) U (c + r, Infinity). The less-than case always produces a single bounded interval (AND type), while the greater-than case always produces two unbounded intervals (OR type). This pattern holds for all absolute value inequalities: less-than means between, greater-than means outside.
What is the difference between parentheses and brackets in interval notation?
Parentheses and brackets serve as the critical distinguishing marks in interval notation, conveying whether boundary values are included or excluded from the set. Parentheses ( and ) indicate open endpoints, meaning the boundary value is NOT part of the set. This corresponds to strict inequalities < and > and is visualized with an open (hollow) circle on the number line. Brackets [ and ] indicate closed endpoints, meaning the boundary value IS part of the set. This corresponds to non-strict inequalities <= and >= and is visualized with a closed (filled) circle. Infinity and negative infinity always use parentheses because they are concepts, not actual numbers that can be reached. A single point can be represented as [a, a], though set notation {a} is more common.
How do you convert interval notation back to an inequality?
Converting from interval notation back to inequality form reverses the process. For a single interval like (a, b], read it as a < x <= b: the parenthesis at a means strict inequality (a is excluded), and the bracket at b means non-strict inequality (b is included). For (-Infinity, c), the result is x < c. For [d, Infinity), the result is x >= d. For union intervals like (-Infinity, a) U (b, Infinity), convert each interval separately and join with OR: x < a or x > b. For single bounded intervals, express as a compound AND: (2, 7] becomes 2 < x <= 7. The key is consistently matching parentheses to strict inequalities and brackets to non-strict inequalities while remembering that infinity always corresponds to unbounded directions.
What is set-builder notation and how does it compare to interval notation?
Set-builder notation describes sets by specifying the properties that elements must satisfy, written as {variable | condition}. For example, {x | -3 < x <= 5} describes all x between -3 (exclusive) and 5 (inclusive). The equivalent interval notation is (-3, 5]. Both notations describe the same set but differ in form and flexibility. Interval notation is more compact for continuous intervals on the real line and is preferred in calculus. Set-builder notation is more versatile because it can describe sets that are not continuous intervals, such as {x | x is an integer and x > 0} or {x | x^2 < 9}. In practice, mathematicians use both notations interchangeably for simple intervals, choosing whichever is more convenient for the context at hand.
Can interval notation represent a single point or an empty set?
Yes, interval notation can handle these special cases, though alternative notations are sometimes preferred. A single point a can be written as [a, a], which is a degenerate interval containing only one number, though {a} using set notation is more standard. The empty set (no solutions) can be written as the empty set symbol or simply stated as no solution, since there is no meaningful interval to write. An empty set arises when conditions are contradictory, like x > 5 AND x < 3. Some textbooks use the notation (a, a) = empty set to emphasize that an open interval with equal endpoints contains no numbers. Understanding these edge cases is important for complete mastery of interval notation and for correctly expressing solution sets of inequalities that have unusual solutions.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy