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 solid line starting at 3 and extending to the right for [3, ∞), and a dashed line starting just after 6 and extending to the right for (6, ∞).
3. Step 3: Identify the union of the two intervals. The union, denoted by ⋃, includes all numbers that are in either of the intervals.
4. Step 4: Determine the combined interval. Since [3, ∞) includes all numbers from 3 onwards and (6, ∞) includes numbers greater than 6, the union will cover all numbers starting from 3.
5. Step 5: Conclude the union. The union of [3, ∞) and (6, ∞) is [3, ∞) because it encompasses all numbers from 3 onwards, including those greater than 6.

Key Concepts

Set Notation

Set notation is a mathematical way to describe a collection of numbers or elements. In this context, the notation [3, ∞) represents all numbers starting from 3 and extending to infinity, including 3 itself. The notation (6, ∞) indicates all numbers greater than 6, not including 6. Understanding these notations is crucial for interpreting the sets involved.

Union of Sets

The union of sets is a fundamental operation in set theory that combines all unique elements from two or more sets. The symbol '∪' denotes this operation. For example, the union of [3, ∞) and (6, ∞) includes all numbers from 3 to infinity, effectively merging the two sets while eliminating any duplicates. This concept is essential for solving the given problem.

Graphing Inequalities

Graphing inequalities involves representing the solutions of inequalities on a number line or coordinate plane. For the sets [3, ∞) and (6, ∞), one would shade the region starting from 3 to the right, including 3, and another region starting just after 6 to infinity. Visualizing these sets helps in understanding their union and the overall solution to the problem.