In algebra, the solution to an inequality is often a range of numbers, not just a single value. While writing this solution out (e.g., "x is greater than 5") is descriptive, mathematicians prefer a more concise and universal language called interval notation. Our Inequality to Interval Notation Calculator is a simple tool designed to help students master this conversion. It translates common algebraic inequalities into their proper interval notation format, providing a quick way to check homework and build fluency in this essential mathematical language.
How to Use the Inequality to Interval Notation Calculator
Converting your inequality is an easy, one-step process:
- Enter Your Inequality: Type a simple or compound linear inequality into the input field. The calculator understands common formats.
- Convert: Click the "Convert" button.
- View the Interval Notation: The calculator will instantly display the solution set in standard interval notation.
The Rules of Interval Notation
Interval notation is a way of describing a set of numbers by stating its endpoints. It uses specific symbols—parentheses and brackets—to indicate whether those endpoints are included in the set. The rules are simple and consistent.
Parentheses vs. Brackets: The Endpoints
- Parentheses ( ): Use parentheses when an endpoint is not included in the solution set. This corresponds to the strict inequality symbols "less than" (<) and "greater than" (>). An open parenthesis indicates an "open" interval.
- Brackets [ ]: Use brackets when an endpoint is included in the solution set. This corresponds to the inequality symbols "less than or equal to" (≤) and "greater than or equal to" (≥). A square bracket indicates a "closed" interval.
For example, the interval (3, 7] includes all real numbers between 3 and 7, including 7 but not including 3.
Unbounded Intervals and the Infinity Symbol (∞)
When an inequality's solution set continues forever in one direction, we use the infinity symbol (∞) or negative infinity symbol (-∞) to represent this. Because infinity is a concept and not a real number that can be reached, it is always used with a parenthesis.
- The inequality
x ≥ 5is written in interval notation as[5, ∞). - The inequality
x < -2is written in interval notation as(-∞, -2).
Compound Inequalities and the Union Symbol (U)
Some inequalities result in a solution set that consists of two separate, non-overlapping intervals. In these cases, we use the union symbol (U) to connect the two sets.
- An "or" statement like
x < 1 or x ≥ 4results in two separate intervals. In interval notation, this is written as(-∞, 1) U [4, ∞). - An "and" statement like
-3 ≤ x < 8represents a single, bounded interval and is written as[-3, 8).
Frequently Asked Questions
How does this relate to graphing on a number line?
Interval notation and number line graphs are two ways of showing the same information. An open circle (o) on a number line is equivalent to a parenthesis ( ). A closed circle (•) is equivalent to a bracket [ ]. Shading to the right corresponds to an interval ending in ∞), and shading to the left corresponds to an interval beginning with (-∞.
What is the interval notation for "all real numbers"?
The set of all real numbers is an unbounded interval that stretches from negative infinity to positive infinity. It is written as (-∞, ∞).
What is the difference between interval notation and set-builder notation?
They are two different ways to formally describe a set of numbers. Interval notation is more concise and visual, describing the set by its endpoints. Set-builder notation is more descriptive and formal, defining the set by the properties its members must satisfy. For example, the interval [5, ∞) would be written in set-builder notation as {x | x ≥ 5}, which is read as "the set of all x such that x is greater than or equal to 5."