In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. (- ∞, 3)

Key Concepts

Set-Builder Notation

Set-builder notation is a mathematical shorthand used to describe a set by stating the properties that its members must satisfy. For example, the interval (-∞, 3) can be expressed in set-builder notation as {x | x < 3}, meaning 'the set of all x such that x is less than 3'. This notation is particularly useful for defining intervals and sets in a concise manner.

Intervals

An interval is a range of numbers between two endpoints. Intervals can be open, closed, or half-open, depending on whether the endpoints are included. The interval (-∞, 3) is an open interval that includes all real numbers less than 3 but does not include 3 itself. Understanding the types of intervals is crucial for accurately expressing and interpreting them in mathematical contexts.

Graphing on a Number Line

Graphing an interval on a number line visually represents the set of numbers included in that interval. For the interval (-∞, 3), you would draw a number line and shade all the points to the left of 3, using an open circle at 3 to indicate that it is not included in the interval. This graphical representation helps in understanding the range of values that satisfy the given conditions.