Interval Notation

Definition

Interval notation is a standardized mathematical shorthand used to represent a continuous range of real numbers bounded by two values, using parentheses to denote exclusivity and brackets to denote inclusivity.

Why It Matters

Interval notation compactly defines subsets of the real number line. It is universally used in calculus to define domains, ranges, intervals of continuity, and regions where functions are increasing or decreasing.

Core Concepts

• Open Interval $(a, b)$: All numbers between $a$ and $b$, excluding endpoints.
  • How to read: “The open interval from a to b.”
  • Meaning: $a < x < b$; endpoints not included (parentheses).
• Closed Interval $[a, b]$: All numbers between $a$ and $b$, including endpoints.
  • How to read: “The closed interval a, b.”
  • Meaning: $a \le x \le b$; endpoints included (brackets).
• Half-Open Interval $[a, b)$ or $(a, b]$: Includes one endpoint, excludes the other.
• **Infinite Interval $Linear Inequalities

• Polynomial Inequalities
• Rational Inequalities
• Absolute Value Inequalities