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

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Identify the interval given: $(-\infty, 3)$, which includes all real numbers less than 3 but does not include 3 itself.

Recall that in set-builder notation, we describe the set of all elements $x$ that satisfy a certain condition.

Since the interval includes all numbers less than 3, write the condition as $x < 3$.

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."

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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