Problem 24 - College Algebra Essentials

Question

Use graphs to find each set. $$[2, \infty) \cap(4, \infty)$$

Step-by-Step Solution

Answer

The intersection of [2, \infty) and (4, \infty) is (4, \infty).

Step 1: Understand the intervals

Let's start by understanding our two sets: [2, \infty) and (4, \infty). The first interval, [2, \infty), is a closed interval that begins from 2 and goes up to \infty. The second interval, (4, \infty), is an open interval that begins just after 4 and goes up to \infty.

Step 2: Draw the graphs

Now, graph the first set on a number line. It starts from 2 and continues towards \infty. 2 is included in the set which is represented by a closed dot on 2. For the second set, draw a separate line. This graph will look identical to the first one, but this time the dot for 4 will be open because 4 is not included in the set.

Step 3: Find the intersection

The intersection of two sets is the set of elements that are common to both sets. Upon comparing the two graphs, it is apparent that the intersection of these two sets is (4, \infty) because this interval contains all the numbers that are in both the initial sets.