In Exercises 107–110, use graphs to find each set. (-2,1] ∩ [-1,3)

Verified step by step guidance

1. Identify the intervals given in the problem. The first interval is (-2,1], which means it includes all numbers between -2 and 1, including 1 but not -2. The second interval is [-1,3), which includes all numbers from -1 to 3, including -1 but not 3.
2. Visualize or sketch the intervals on a number line. Place a circle at -2 and a filled dot at 1 for the first interval (-2,1]. For the second interval [-1,3), place a filled dot at -1 and a circle at 3.
3. Look for the overlap of the two intervals on the number line. The overlap is the region where both intervals cover the same numbers.
4. Determine the endpoints of the overlapping interval. The overlap starts where the first interval becomes inclusive, which is at -1, and ends where the first interval stops being inclusive, which is at 1.
5. Write the interval notation for the overlapping section. Since -1 is included in the overlap and 1 is also included, the interval notation for the intersection is [-1,1].

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included or excluded. For example, in the interval (-2, 1], -2 is excluded (parenthesis) while 1 is included (bracket). Understanding this notation is crucial for accurately interpreting and combining intervals.

Graphing Intervals

Graphing intervals involves visually representing the range of values on a number line. Each interval is depicted as a line segment, with open circles indicating excluded endpoints and closed circles for included endpoints. This visual representation helps in understanding the overlap between different intervals, which is essential for solving problems involving intersections.

Set Intersection

Set intersection refers to the common elements shared between two or more sets. In the context of intervals, the intersection of two intervals is the set of values that lie within both ranges. For example, to find the intersection of (-2, 1] and [-1, 3), one must identify the overlapping values, which is a key step in solving the given problem.