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 interval given: \((-\infty, 5.5)\), which includes all real numbers less than 5.5 but does not include 5.5 itself.
2. Recall that in set-builder notation, we describe the set of all elements \(x\) that satisfy a certain condition. Here, the condition is that \(x\) is less than 5.5.
3. Write the set-builder notation as: \(\{ x \mid x < 5.5 \}\), which reads as "the set of all \(x\) such that \(x\) is less than 5.5."
4. To graph this interval on a number line, draw a line and mark the point 5.5. Use an open circle at 5.5 to indicate that 5.5 is not included in the interval.
5. Shade the number line to the left of 5.5, extending towards negative infinity, to represent all numbers less than 5.5.

Key Concepts

Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints. For example, (-∞, 5.5) includes all real numbers less than 5.5 but does not include 5.5 itself, indicated by the parenthesis. Understanding this notation helps in translating intervals into other forms.

Set-Builder Notation

Set-builder notation describes a set by specifying a property that its members must satisfy. For the interval (-∞, 5.5), it can be written as {x | x < 5.5}, meaning the set of all x such that x is less than 5.5. This form is useful for expressing intervals in a more formal mathematical language.

Graphing Intervals on a Number Line

Graphing intervals involves shading the portion of the number line that represents all numbers in the interval. For (-∞, 5.5), the line is shaded to the left of 5.5, with an open circle at 5.5 to show it is not included. This visual helps in understanding the range of values covered by the interval.