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

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the intersection of two intervals: (-2,1] and [-1,3). The intersection represents the set of numbers that are common to both intervals.
2. Step 2: Analyze the first interval (-2,1]. This interval includes all numbers greater than -2 and up to and including 1. In mathematical terms, it is expressed as: -2 < x ≤ 1.
3. Step 3: Analyze the second interval [-1,3). This interval includes all numbers greater than or equal to -1 and less than 3. In mathematical terms, it is expressed as: -1 ≤ x < 3.
4. Step 4: Graph both intervals on a number line. For (-2,1], draw an open circle at -2 (indicating -2 is not included) and a closed circle at 1 (indicating 1 is included). For [-1,3), draw a closed circle at -1 (indicating -1 is included) and an open circle at 3 (indicating 3 is not included).
5. Step 5: Identify the overlap (intersection) of the two intervals on the number line. The intersection is the set of numbers that are in both intervals. This will be the range of x-values that satisfy both conditions simultaneously.

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, the interval (-2, 1] includes all numbers greater than -2 and up to and including 1, while [-1, 3) includes -1 and all numbers less than 3 but not including 3.

Set Intersection

Set intersection refers to the operation of finding common elements between two sets. In the context of intervals, the intersection of two intervals A and B, denoted as A ∩ B, includes all values that are present in both intervals. This concept is crucial for determining the overlapping range of values when working with multiple intervals.

Graphing Intervals

Graphing intervals involves visually representing the range of values on a number line. Each interval is depicted with a line segment, where closed endpoints are marked with solid dots and open endpoints with open circles. This visual representation helps in easily identifying the intersection of intervals by observing where the segments overlap on the number line.