Problem 5 - College Algebra Essentials

Question

Express each interval in set-builder notation and graph the interval on a number line. $$[-3,1]$$

Step-by-Step Solution

Answer

The set-builder notation for the interval \([-3,1]\) is \(\{x \mid -3 \leq x \leq 1\}\). The graph on the number line includes a line segment extending from a filled circle at -3 to another filled circle at 1.

Step 1: Set-Builder Notation

To convert the interval \([-3,1]\) to set-builder notation, consider it as a set of all real numbers \(x\) such that \(x\) is greater than or equal to -3 and \(x\) is less than or equal to 1. This can be written as: \[ \{x \mid -3 \leq x \leq 1\} \]

Step 2: Graphing the Interval on the Number Line

To graph this interval on a number line, a line is drawn and labelled with numbers. -3 and 1 are marked distinctly on the line because they are the boundaries of the interval. A closed circle is drawn at both these points indicating that they are included in the interval, and a line is drawn connecting these two points.