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

Verified step by step guidance

1. Understand the notation: \((-3, 0)\) is an open interval, meaning it includes all numbers between -3 and 0, but not -3 and 0 themselves. \([-1, 2]\) is a closed interval, meaning it includes all numbers between -1 and 2, including -1 and 2.
2. Visualize the intervals on a number line. The interval \((-3, 0)\) is represented by a line starting just after -3 and ending just before 0. The interval \([-1, 2]\) is represented by a line starting at -1 and ending at 2.
3. Identify the overlap (intersection) of the two intervals. The intersection is the set of numbers that are common to both intervals.
4. The intersection of \((-3, 0)\) and \([-1, 2]\) is the set of numbers that are in both intervals. This will be the part of the number line where both intervals overlap.
5. Write the intersection as an interval. Since the overlap starts at -1 and ends just before 0, the intersection is \([-1, 0)\).

Key Concepts

Intervals

An interval is a set of real numbers that fall between two endpoints. Intervals can be open, closed, or half-open. For example, the interval (-3, 0) is open, meaning it includes all numbers greater than -3 and less than 0, but not -3 and 0 themselves. The interval [-1, 2] is closed, including both endpoints -1 and 2.

Intersection of Sets

The intersection of two sets is the set of elements that are common to both sets. In this case, we are looking for the numbers that are present in both the interval (-3, 0) and the interval [-1, 2]. The intersection is found by identifying the overlapping values between the two intervals.

Graphing Intervals

Graphing intervals involves representing the sets of numbers visually on a number line. Open intervals are shown with parentheses, while closed intervals are represented with brackets. This visual representation helps in easily identifying overlaps and intersections between different sets, making it easier to determine the solution to the problem.