In Exercises 15–26, use graphs to find each set. [3, ∞) ∩ (6, ∞)

Verified step by step guidance

1. Step 1: Understand the notation. The interval [3, ∞) represents all numbers greater than or equal to 3, and the interval (6, ∞) represents all numbers greater than 6.
2. Step 2: Visualize the intervals on a number line. Draw a number line and mark the point 3 with a closed circle (indicating that 3 is included) and shade to the right. Then, mark the point 6 with an open circle (indicating that 6 is not included) and shade to the right.
3. Step 3: Identify the intersection of the two intervals. The intersection is the set of numbers that are in both intervals.
4. Step 4: Notice that the interval [3, ∞) includes all numbers from 3 onwards, while (6, ∞) includes numbers greater than 6. The overlap starts just after 6.
5. Step 5: Conclude that the intersection of [3, ∞) and (6, ∞) is the interval (6, ∞), as this is the set of numbers that satisfy both conditions.

Key Concepts

Intervals

Intervals are a way to describe a range of numbers on the real number line. They can be open, closed, or half-open, depending on whether the endpoints are included. For example, the interval [3, ∞) includes all numbers starting from 3 and going to infinity, while (6, ∞) includes all numbers greater than 6 but not 6 itself.

Intersection of Sets

The intersection of two sets is the set of elements that are common to both sets. In the context of intervals, this means finding the values that belong to both intervals simultaneously. For instance, to find the intersection of [3, ∞) and (6, ∞), we look for numbers that are greater than or equal to 3 and also greater than 6.

Graphing Intervals

Graphing intervals involves representing the ranges of numbers visually on a number line. This helps in understanding the relationships between different intervals. By plotting [3, ∞) and (6, ∞) on a number line, one can easily see where the two intervals overlap, which is essential for determining their intersection.