Problem 11 - College Algebra Essentials

Question

Express each interval in set-builder notation and graph the interval on a number line. $$(-\infty, 3)$$

Step-by-Step Solution

Answer

In set-builder notation, the interval \(-\infty, 3)\ is \(x: x < 3\). The graph would be a hollow circle at 3, with a line extending to the left towards \(-\infty\).

Step 1: Interpret the Open Intervals

Open intervals are given by \((a, b)\), where \(a\) and \(b\) are the endpoints. If \(a\) or \(b\) is replaced by \(-\infty\), that means all numbers going towards negative infinity are included. For this exercise, \(-\infty\) corresponds to the lower limit, while 3 corresponds to the upper limit. It is open at 3, means all the numbers less than 3 are considered here. None of the end points are included, it is denoted by hollow circles.

Step 2: Convert to Set-Builder Notation

Set-builder notation is another way to denote intervals. For \(-\infty, 3)\, all numbers less than 3 are included. This can expressed in set-builder notation as: \(x: x < 3\), where 'x' is the element of the set.

Step 3: Graph the Interval on a Number Line

Draw a number line and mark the point corresponding to 3. Since the interval is open at 3, indicate this with a hollow circle at 3. Draw a line starting from 3 and extending to the left towards \(-\infty\). This represents all numbers less than 3.