Graph each interval on a number line. See Section 2.5. (-∞,...

Key Concepts

Union of Intervals

The union of intervals represents the combination of two or more intervals into one set, meaning that any element belonging to any of the intervals is included in the union. Graphically, this involves marking each distinct interval on a number line and indicating that together they form one complete set.

Infinite Intervals

Infinite intervals involve endpoints that extend indefinitely in one direction, using the symbols -? or ?. These intervals, such as (-?, a] or [b, ?), indicate that the set continues without bound on one side, and are always depicted with an open end at the infinity since infinity is not a specific numerical value.

Graphing on a Number Line

Graphing on a number line involves marking the relevant points and shading the regions corresponding to the interval. Special attention is given to endpoints by using either solid dots, which indicate inclusion, or open circles, which indicate exclusion. This visual representation aids in understanding the spread and limits of the intervals described in interval notation.

Interval Notation

Interval notation is a method for describing a set of numbers between two endpoints. It uses brackets and parentheses to indicate whether endpoints are included (closed interval with square brackets) or excluded (open interval with parentheses). This notation is a fundamental tool in describing continuous subsets of the real numbers.

Open and Closed Intervals

Open and closed intervals denote whether the endpoints of an interval are included in the set. A closed interval, represented by square brackets [a, b], includes its endpoints, whereas an open interval, represented by parentheses (a, b), excludes them. This distinction is critical when visualizing and graphing intervals as it affects how the boundaries are represented.

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