Problem 23 - College Algebra Essentials

Question

Use graphs to find each set. $$[3, \infty) \cap(6, \infty)$$

Step-by-Step Solution

Answer

The intersection of sets [3, ∞) and (6, ∞) is (6, ∞).

Step 1: Analyzing each set

The set [3, ∞) includes every number greater than or equal to 3. This is marked on the number line with a filled point at 3, indicating 3 is part of the set, extending to the right to show all larger numbers are included. On the other hand, the set (6, ∞) includes every number strictly greater than 6, and this is illustrated on the number line with an empty point at 6, representing that 6 itself is not part of the set, and extending to the right to represent all larger numbers.

Step 2: Finding the intersection

The intersection of the sets is the section of the number line where the two sets overlap. In this case, since (6, ∞) includes all the numbers greater than 6, but does not include 6 itself, and [3, ∞) includes all the numbers greater than or equal to 3, the numbers that are common to both sets are the numbers strictly greater than 6. So, the intersection is (6, ∞).

Step 3: Graphical representation

Graphing this on the number line would be as follows: a line drawn extending to the right from the point 6, with an empty point at 6 to indicate that 6 itself is not part of the set. The resulting figure shows that the intersection of [3, ∞) and (6, ∞) is (6, ∞).