Inequality to Interval Notation Calculator
Free Inequality interval notation Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy