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

Verified step by step guidance

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

Key Concepts

Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints. For example, (-∞, 3) includes all real numbers less than 3 but does not include 3 itself, indicated by the parenthesis. It is a concise way to describe continuous ranges on the number line.

Set-Builder Notation

Set-builder notation describes a set by specifying a property that its members satisfy. For the interval (-∞, 3), it can be written as {x | x < 3}, meaning the set of all x such that x is less than 3. This notation emphasizes the condition defining the set.

Graphing Intervals on a Number Line

Graphing intervals involves shading the portion of the number line that represents the set. For (-∞, 3), the line is shaded to the left of 3, with an open circle at 3 to show it is not included. This visual helps understand the range of values in the interval.