Interval Notation

Definition

Interval notation is a mathematical notation used to represent a range of values along the number line. It uses parentheses and brackets to indicate whether endpoints are included or excluded, making it a concise way to express intervals. Understanding interval notation is essential for working with inequalities, functions, and sets of numbers in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. In interval notation, an interval is written as (a, b) for an open interval or [a, b] for a closed interval, clearly indicating whether endpoints are part of the interval.
2. When using parentheses, it means the endpoints are excluded; with brackets, the endpoints are included.
3. The notation extends to infinite intervals such as (-∞, a) or (b, ∞), which represent all numbers less than a or greater than b, respectively.
4. Intervals can be combined using union notation. For example, (1, 3) ∪ (4, 5) represents values in either interval.
5. Understanding how to read and write intervals in notation is crucial for solving inequalities and graphing on the number line.

Review Questions

• How do you differentiate between open and closed intervals in interval notation?
  • Open intervals in interval notation use parentheses to indicate that the endpoints are not included in the range of values. For instance, (a, b) means that neither a nor b is part of the interval. In contrast, closed intervals use brackets to show that the endpoints are included. For example, [a, b] indicates that both a and b are part of the interval. Recognizing these differences is important for accurately interpreting ranges in mathematical problems.
• What role does interval notation play in solving inequalities?
  • Interval notation is crucial for expressing the solution set of inequalities succinctly. When solving an inequality like x < 5 or x ≥ 2, we can use interval notation to clearly represent the solution set. For x < 5, the solution can be written as (-∞, 5), indicating all values less than 5 but not including it. Meanwhile, x ≥ 2 can be represented as [2, ∞), showing that 2 is included while all greater values are also part of the solution. This concise representation makes it easier to visualize and communicate results.
• Evaluate how combining intervals in union notation enhances understanding of multiple ranges of values.
  • Combining intervals using union notation provides clarity when expressing multiple ranges that do not overlap. For example, if one needs to represent both x < 3 and x > 5 together, it can be expressed as (-∞, 3) ∪ (5, ∞). This shows two separate intervals on the number line where the variable x can exist without confusion. Understanding this concept helps in analyzing complex sets and inequalities by making it clear which values are included without redundancy or ambiguity.

Related terms

Closed Interval: A closed interval includes its endpoints and is denoted with square brackets, e.g., [a, b].

Open Interval: An open interval does not include its endpoints and is denoted with parentheses, e.g., (a, b).

Half-Open Interval: A half-open interval includes one endpoint but not the other, represented as [a, b) or (a, b].