Problem 13 - College Algebra Essentials

Question

Express each interval in set-builder notation and graph the interval on a number line. $$(-\infty, 5.5)$$

Step-by-Step Solution

Answer

The set-builder notation for the interval \(-\infty, 5.5)\) is \(x: x < 5.5\). On a number line, this is represented by an arrow pointing to the left (indicating all numbers less than 5.5), starting from an open circle at 5.5.

Step 1: Understanding the Interval

The provided interval is \(-\infty, 5.5)\). In this interval, it implies that x is greater than \(-\infty\) and less than 5.5, so x can be any number less than 5.5.

Step 2: Converting to Set-Builder Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. Based on our initial understanding, this interval can be expressed as \(x: x < 5.5\), which means the set of all x such that x is less than 5.5.

Step 3: Drawing the Number Line

A number line represents numbers as points on a line. The interval \(-\infty, 5.5)\) on a number line can be represented by drawing a line, marking a point at 5.5, and then drawing an arrow to the left to indicate all numbers less than 5.5. Remember to draw an open circle at 5.5 to show that 5.5 is not included in the interval.