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Chapter 0: Problem 14

Express each interval in set-builder notation and graph the interval on a number line. $$(-\infty, 3.5]$$

Short Answer

Expert verified
The interval \(-\infty, 3.5] \) can be expressed in set-builder notation as \( \{x \in \mathbb{R} | x \leq 3.5\}\), and its representation on a number line is a line that starts from \(3.5\) (which is included, marked by a solid dot) and extends to the left to represent all numbers less than \(3.5\).

Step by step solution

01

Understand and express in set-builder notation

The set-builder notation is a method to describe a set by stating the properties that its members must satisfy. In our case, the interval \(-\infty, 3.5] \) should be seen as all real numbers that are less than or equal to \(3.5\). The open parenthesis before \(-\infty\) signals that \(-\infty\) is not a part of the interval, while the closed bracket after \(3.5\) indicates that \(3.5\) is included in the interval. This can be written in set-builder notation as: \( \{x \in \mathbb{R} | x \leq 3.5\}\), meaning the set of all real numbers \(x\) such that \(x\) is less than or equal to \(3.5\).
02

Graphing the interval on a number line

To graph the interval on a number line, first draw a straight horizontal line and mark it with regular intervals. Because our interval includes the number \(3.5\), you should make sure this value is marked on your line. Then, place a closed circle at \(3.5\) to indicate that this value is included in the interval. Finally, shade in or draw an arrow pointing to the left from \(3.5\), going all the way to the end of the line. The arrow indicates that all values less than \(3.5\) (down to negative infinity) are included in the interval.

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