In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. (- ∞, 5.5)

Verified step by step guidance

1. Identify the type of interval given: \((-\infty, 5.5)\) is an open interval.
2. In set-builder notation, an open interval \((-\infty, 5.5)\) can be expressed as \(\{ x \mid x < 5.5 \}\).
3. To graph this interval on a number line, draw a line and place an open circle at 5.5 to indicate that 5.5 is not included in the interval.
4. Shade the number line to the left of 5.5 to represent all numbers less than 5.5.
5. Ensure the shading extends indefinitely to the left, indicating that the interval includes all numbers less than 5.5, approaching negative infinity.

Key Concepts

Set-Builder Notation

Set-builder notation is a mathematical shorthand used to describe a set by specifying a property that its members must satisfy. For example, the interval (-∞, 5.5) can be expressed in set-builder notation as {x | x < 5.5}, meaning 'the set of all x such that x is less than 5.5'. This notation is particularly useful for defining intervals that extend infinitely in one direction.

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 (-∞, 5.5) is an open interval that includes all real numbers less than 5.5 but does not include 5.5 itself. Understanding the types of intervals is crucial for accurately expressing them in set-builder notation.

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 (-∞, 5.5), you would draw a number line, place an open circle at 5.5 to indicate that it is not included, and shade the line to the left towards negative infinity. This graphical representation helps in understanding the extent and limits of the interval.