Intersection Of Intervals (-3, 5] And [-1, 2) Explained With Diagram

Introduction to Interval Intersection

In the realm of mathematics, specifically within set theory and real analysis, understanding interval intersection is a foundational concept. An interval represents a continuous set of real numbers bounded by two endpoints. The intersection of two or more intervals is the set of numbers that are common to all the intervals. This article delves into the process of finding the intersection of two specific intervals: (-3, 5] and [-1, 2). We will not only determine the resultant interval but also illustrate this intersection using a diagram for better comprehension. Grasping the concept of interval intersection is crucial for various mathematical applications, including solving inequalities, determining the domain and range of functions, and analyzing solutions in optimization problems. Before we dive into our specific example, it's important to understand the notation and terminology associated with intervals. Intervals can be open, closed, or half-open (or half-closed), each having a distinct meaning. An open interval, denoted by parentheses, does not include its endpoints. For instance, (a, b) represents all real numbers between a and b, excluding a and b themselves. A closed interval, denoted by square brackets, includes its endpoints. Thus, [a, b] represents all real numbers between a and b, including a and b. Half-open intervals use a combination of parentheses and square brackets to indicate that one endpoint is included while the other is not. For example, (a, b] includes b but excludes a, while [a, b) includes a but excludes b. Understanding this notation is paramount to accurately finding the intersection of intervals. Now, let's consider the task at hand: finding the intersection of the intervals (-3, 5] and [-1, 2). This means we need to identify the set of real numbers that belong to both intervals simultaneously. To do this effectively, we can visualize the intervals on a number line, which will provide a clear picture of the overlapping region. This visual representation will not only help us determine the intersection but also solidify our understanding of interval operations. In the subsequent sections, we will walk through the step-by-step process of identifying the intersection and then represent it graphically. This comprehensive approach will provide a strong foundation for tackling more complex interval problems in the future.

Step-by-Step Solution

To find the intersection of the two intervals (-3, 5] and [-1, 2), we need to identify the range of real numbers that are common to both intervals. Let's break down each interval separately first. The interval (-3, 5] includes all real numbers greater than -3 and less than or equal to 5. This can be written in inequality form as -3 < x ≤ 5. The parenthesis '(' indicates that -3 is not included in the interval, while the square bracket ']' indicates that 5 is included. On the other hand, the interval [-1, 2) includes all real numbers greater than or equal to -1 and strictly less than 2. In inequality form, this is represented as -1 ≤ x < 2. The square bracket '[' signifies that -1 is part of the interval, and the parenthesis ')' indicates that 2 is not included. Now, to find the intersection, we look for the overlapping region between these two intervals. This means we need to find the values of x that satisfy both -3 < x ≤ 5 and -1 ≤ x < 2. To do this, we can consider the lower and upper bounds separately. For the lower bound, we have -3 < x and -1 ≤ x. Since -1 is greater than -3, the lower bound of the intersection will be -1, and since -1 is included in the second interval, it will also be included in the intersection. For the upper bound, we have x ≤ 5 and x < 2. The upper bound of the intersection will be 2, as it is smaller than 5. However, since 2 is not included in the second interval (due to the parenthesis), it will not be included in the intersection. Therefore, the intersection of the two intervals (-3, 5] and [-1, 2) is the interval [-1, 2). This interval includes all real numbers greater than or equal to -1 and strictly less than 2. In other words, any number within this range belongs to both of the original intervals. This step-by-step approach helps clarify how the intersection is derived by considering the individual intervals and their respective boundaries. In the next section, we will visualize this intersection using a diagram, which will further solidify our understanding of this concept. This visual representation is a powerful tool for grasping interval operations and will be particularly helpful for more complex problems involving multiple intervals or inequalities.

Diagrammatic Representation

Visualizing mathematical concepts often enhances understanding, and this holds true for interval intersection. To represent the intersection of the intervals (-3, 5] and [-1, 2), we can use a number line. First, we draw a number line and mark the key points of interest: -3, -1, 2, and 5. These points are the endpoints of our intervals and will define the regions we need to consider. Now, let's represent the interval (-3, 5] on the number line. We draw an open circle at -3 to indicate that it is not included in the interval, and a closed circle (or a bracket) at 5 to show that it is included. We then shade the region between -3 and 5, representing all the real numbers within this range. Next, we represent the interval [-1, 2) on the same number line. We draw a closed circle (or a bracket) at -1 to indicate its inclusion and an open circle at 2 to indicate its exclusion. We shade the region between -1 and 2, representing all the real numbers within this interval. The intersection of these two intervals is the region where the shaded portions overlap. Looking at the diagram, we can clearly see that the overlap occurs between -1 and 2. The overlapping region starts at -1, which is included in both intervals (as indicated by the closed circle or bracket). It extends up to 2, but 2 is not included in the interval [-1, 2) (as indicated by the open circle). Therefore, the intersection is the interval [-1, 2). The diagram provides a visual confirmation of our earlier algebraic calculation. It allows us to see at a glance which numbers belong to both intervals and which do not. This visual approach is particularly useful when dealing with more complex interval operations, such as unions and differences, or when working with multiple intervals. By drawing a number line, we can easily identify the overlapping or non-overlapping regions and determine the resulting interval. In summary, the diagrammatic representation not only reinforces our understanding of interval intersection but also provides a powerful tool for solving related problems. It complements the algebraic method and offers a more intuitive way to grasp the concept. In the next section, we will provide a concise conclusion summarizing our findings and highlighting the key takeaways from this article. This will ensure that the main points are clearly understood and can be readily applied to future problems.

Conclusion

In this article, we have successfully found the intersection of the two intervals (-3, 5] and [-1, 2). By employing both algebraic and diagrammatic methods, we have demonstrated a comprehensive approach to solving this type of problem. We began by defining what an interval is and distinguishing between open, closed, and half-open intervals. Understanding these distinctions is crucial for accurately determining the intersection of intervals. We then proceeded with the algebraic method, carefully considering the endpoints and their inclusion or exclusion in each interval. By analyzing the inequalities representing the intervals (-3, 5] and [-1, 2), we identified the overlapping region as the interval [-1, 2). This means that all real numbers greater than or equal to -1 and strictly less than 2 belong to both intervals. To further solidify our understanding, we utilized a diagrammatic representation. By drawing the intervals on a number line, we visually confirmed the intersection. The diagram clearly showed the overlapping region between -1 (inclusive) and 2 (exclusive), reinforcing our algebraic solution. This visual approach is particularly beneficial for complex problems involving multiple intervals or inequalities, as it provides an intuitive way to grasp the relationships between different intervals. The key takeaway from this exercise is the importance of considering both the endpoints and the type of interval (open, closed, or half-open) when finding the intersection. The use of a number line as a visual aid can greatly simplify the process and reduce the likelihood of errors. Furthermore, this exercise highlights the fundamental role of interval intersection in various mathematical applications. Understanding how to find the intersection of intervals is essential for solving inequalities, determining domains and ranges of functions, and tackling optimization problems. By mastering this concept, you will be well-equipped to handle a wide range of mathematical challenges. In conclusion, finding the intersection of intervals involves careful consideration of endpoints and interval types, and the use of both algebraic and diagrammatic methods can enhance understanding and accuracy. The intersection of the intervals (-3, 5] and [-1, 2) is [-1, 2), a result we have thoroughly demonstrated through both calculation and visualization.