In Exercises 15–26, use graphs to find each set. (- 3, 0) ⋃ [- 1, 2]

Verified step by step guidance

1. Step 1: Understand the notation. The problem involves two intervals: (-3, 0) and [-1, 2].
2. Step 2: Interpret the intervals. The interval (-3, 0) is an open interval, meaning it includes all numbers between -3 and 0, but not -3 and 0 themselves. The interval [-1, 2] is a closed interval, meaning it includes all numbers between -1 and 2, including -1 and 2.
3. Step 3: Visualize the intervals on a number line. Draw a number line and mark the points -3, 0, -1, and 2. Use open circles for -3 and 0, and closed circles for -1 and 2.
4. Step 4: Shade the regions corresponding to each interval. For (-3, 0), shade the region between -3 and 0 without including the endpoints. For [-1, 2], shade the region between -1 and 2, including the endpoints.
5. Step 5: Combine the shaded regions to find the union. The union of the two intervals is the set of all points that are in either interval. Shade the combined region on the number line to represent the union.

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. An open interval, like (-3, 0), does not include its endpoints, while a closed interval, like [-1, 2], includes its endpoints. Understanding how to interpret these intervals is crucial for graphing and finding unions of sets.

Union of Sets

The union of sets combines all elements from the involved sets without duplication. For example, the union of the intervals (-3, 0) and [-1, 2] includes all numbers from -3 to 2, covering both intervals. This concept is essential for determining the complete set of values represented by the combined intervals.

Graphing

Graphing involves visually representing mathematical concepts on a coordinate plane. For intervals, this means plotting the endpoints and shading the appropriate regions. Understanding how to accurately graph intervals and their unions helps in visualizing the solution and confirming the correctness of the combined sets.