Problem 12 - College Algebra Essentials

Question

Express each interval in set-builder notation and graph the interval on a number line. $$(-\infty, 2)$$

Step-by-Step Solution

Answer

Set-builder notation for the interval \(-\infty, 2\) is \(\{x \in \mathbb{R} \mid x < 2\}\). The graph of the interval on a number line contains all numbers less than 2 with an open circle at 2.

Step 1: Express the Interval in Set-Builder Notation

Set-builder notation is a short way to describe or define a set without listing all of its elements. The set is described by properties its elements have in common. \n\nThe interval \(-\infty, 2\) is the set of all real numbers less than 2. The set-builder notation for the given interval is \(\{x \in \mathbb{R} \mid x < 2\}\). The notation \(\{x \in \mathbb{R} \mid x < 2\}\) is read as 'The set of all x in the real numbers such that x is less than 2.'

Step 2: Sketch the Interval on a Number Line

To graph the interval \(-\infty, 2\) on a number line, mark the point 2 on the number line. As it is an open interval (indicated by the parenthesis on the 2), draw an open circle at 2 to show that 2 is not included in the interval. Then, draw a line to the left of 2 extending indefinitely, indicating all numbers less than 2 are part of the interval.