Sets A And B: Unions, Intersections, And Differences

Let's dive into a classic set theory problem! We've got two sets, A and B, defined by inequalities involving absolute values. Our mission, should we choose to accept it (and we totally do!), is to find their union (A ∪ B), intersection (A ∩ B), and the set differences (A \ B and B \ A). This sounds a bit daunting, but don't worry, we'll break it down step by step. So, grab your pencils, and let's get started!

Understanding the Sets A and B

First, let's understand the sets A and B. Set A is defined as all real numbers x such that |2x + 7| ≥ 15. This inequality means that the distance between 2x + 7 and 0 is greater than or equal to 15. Similarly, set B is defined as all real numbers x such that |2x - 5| > 9. This means the distance between 2x - 5 and 0 is greater than 9. To work with these sets, we need to translate these absolute value inequalities into simpler, more manageable inequalities. We'll start by tackling set A.

To decipher set A, we need to consider two scenarios arising from the absolute value: either 2x + 7 is greater than or equal to 15, or 2x + 7 is less than or equal to -15. Let's break down each case. First, consider 2x + 7 ≥ 15. Subtracting 7 from both sides gives us 2x ≥ 8. Dividing both sides by 2, we get x ≥ 4. So, one part of set A consists of all real numbers greater than or equal to 4. Now, let's consider the second case: 2x + 7 ≤ -15. Subtracting 7 from both sides gives us 2x ≤ -22. Dividing both sides by 2, we get x ≤ -11. Therefore, the other part of set A consists of all real numbers less than or equal to -11. Combining these two results, we can express set A as the union of two intervals: A = (-∞, -11] ∪ [4, ∞). This means A includes all real numbers from negative infinity up to -11 (inclusive), and all real numbers from 4 (inclusive) to positive infinity. Guys, this is like finding the edges of our first puzzle piece!

Now, let's tackle set B, which is defined by the inequality |2x - 5| > 9. Just like with set A, we have two cases to consider. Either 2x - 5 is greater than 9, or 2x - 5 is less than -9. Let's look at the first case: 2x - 5 > 9. Adding 5 to both sides, we get 2x > 14. Dividing both sides by 2, we find x > 7. So, one part of set B includes all real numbers greater than 7. Next, let's consider the second case: 2x - 5 < -9. Adding 5 to both sides gives us 2x < -4. Dividing both sides by 2, we get x < -2. Thus, the other part of set B consists of all real numbers less than -2. Combining these, we can express set B as the union of two intervals: B = (-∞, -2) ∪ (7, ∞). This means B includes all real numbers from negative infinity up to -2 (exclusive), and all real numbers from 7 (exclusive) to positive infinity. See? We're already halfway there! Understanding these individual sets is crucial before we start combining them. Now that we know what A and B look like, we can find their union, intersection, and differences.

Finding the Union: A ∪ B

Okay, let's find the union of A and B, which is written as A ∪ B. Remember, the union of two sets is a new set that contains all the elements from both sets. Think of it like merging two groups of friends – the union is the entire group of everyone! In our case, A = (-∞, -11] ∪ [4, ∞) and B = (-∞, -2) ∪ (7, ∞). To find A ∪ B, we need to combine these intervals. A ∪ B will include all real numbers that are in A, in B, or in both.

To visualize this, it's super helpful to draw a number line. Let's mark the key points: -11, -2, 4, and 7. Set A covers everything from -∞ to -11 (including -11) and from 4 to ∞ (including 4). Set B covers everything from -∞ to -2 (excluding -2) and from 7 to ∞ (excluding 7). When we combine these, we essentially want to shade all the regions covered by either A or B. We see that the interval (-∞, -11] from A is included in the union. Then, B covers the interval (-∞, -2), which extends beyond -11, so we have (-∞, -2). The interval [4, ∞) from A is also included. Finally, B contributes (7, ∞), which is already covered by A's [4, ∞). Now, what about the space in between? We see that A covers [4, ∞) and B covers (-∞, -2). So, between -2 and 4, there's a gap. However, beyond 4, A covers everything, and beyond -2, B covers everything up to negative infinity. Thus, the union will cover everything up to -2 and from 4 onwards. Putting it all together, we see that A ∪ B covers almost the entire number line except for the small gap between -2 (inclusive) and 4 (exclusive). Therefore, A ∪ B = (-∞, -2) ∪ [4, ∞). Awesome! We've found our first combined set. Now, let's move on to finding the intersection.

Finding the Intersection: A ∩ B

Next up, let's find the intersection of A and B, denoted as A ∩ B. The intersection of two sets is a new set containing only the elements that are in both sets. Think of it as finding the common ground between two groups – the intersection is the group of friends who are in both groups! Again, we have A = (-∞, -11] ∪ [4, ∞) and B = (-∞, -2) ∪ (7, ∞). To find A ∩ B, we need to identify the intervals where A and B overlap. This means we're looking for the regions that are shaded in both sets on our number line.

Looking back at our number line visualization, remember that A covers (-∞, -11] and [4, ∞), while B covers (-∞, -2) and (7, ∞). The intersection will be where these intervals overlap. Let's start with the negative infinity end. A covers (-∞, -11], and B covers (-∞, -2). The overlap here is (-∞, -11] since this interval is contained in both sets. Now, let's look at the positive infinity end. A covers [4, ∞), and B covers (7, ∞). The overlap here is (7, ∞), as this is the region where both sets have elements. There's no overlap between the interval [4, ∞) of A and the interval (-∞, -2) of B, and similarly, there's no overlap between the interval (-∞, -11] of A and the interval (7, ∞) of B. So, we only have two regions of overlap. Therefore, the intersection A ∩ B consists of the intervals (-∞, -11] and (7, ∞). We can write this as A ∩ B = (-∞, -11] ∪ (7, ∞). Great job! We've nailed the intersection. Now, let's tackle the set differences, which are a slightly different beast.

Finding the Set Difference: A \ B

Now, let's find the set difference A ** B. The set difference A \ B (sometimes written as A – B) represents all the elements that are in set A but not in set B. Think of it as taking set A and removing anything that's also in set B. It's like if you have a group of friends (set A) and some of them are also in another group (set B), A \ B is the group of friends who are only in your group, not the other one. We still have A = (-∞, -11] ∪ [4, ∞) and B = (-∞, -2) ∪ (7, ∞).

To find A \ B, we need to consider what parts of A are not also in B. Visualizing this on the number line is crucial. A covers (-∞, -11] and [4, ∞), while B covers (-∞, -2) and (7, ∞). Let's consider the interval (-∞, -11] from A. The interval (-∞, -2) from B overlaps with this. So, we need to remove the overlap from A. The overlap is (-∞, -11], as this entire interval is part of A. However, B only covers up to -2, so we need to find the part of A that isn't covered by B. This means we need to remove the interval (-∞, -11] from A. This leaves us with the interval (-11, -2]. Now, let's consider the interval [4, ∞) from A. The interval (7, ∞) from B overlaps with this. So, we need to remove the part of A that overlaps with B. The overlap is (7, ∞). So, we subtract this from [4, ∞), which leaves us with the interval [4, 7]. Combining these two results, we get A \ B = (-11, -2] ∪ [4, 7]. Fantastic! We're getting the hang of this. Only one more set operation to go!

Finding the Set Difference: B \ A

Finally, let's find the set difference B ** A. This time, we want to find all the elements that are in set B but not in set A. It's the reverse of what we just did! Think of it as taking set B and removing anything that's also in set A. So, if we switch our perspective from the previous example, B \ A represents the group of friends who are only in the second group, not the first one. We have A = (-∞, -11] ∪ [4, ∞) and B = (-∞, -2) ∪ (7, ∞).

To find B \ A, we need to consider what parts of B are not also in A. We'll continue to use the number line visualization. A covers (-∞, -11] and [4, ∞), while B covers (-∞, -2) and (7, ∞). Let's consider the interval (-∞, -2) from B. The interval (-∞, -11] from A overlaps with this. The overlap is the interval (-∞, -11]. So, we need to subtract this from B. Since B covers (-∞, -2), removing the part that's also in A means removing the overlap (-∞, -11]. This leaves us with the interval (-11, -2). Now, let's consider the interval (7, ∞) from B. The interval [4, ∞) from A overlaps with this. So, we need to remove the overlap from B. The overlap is (7, ∞). This means we need to remove the interval (7, ∞) from (7, ∞), which leaves us with nothing (the empty set). Combining these two results, we get B \ A = (-11, -2). Woohoo! We've found all the set operations.

Conclusion

Wow, we did it! We successfully determined A ∪ B, A ∩ B, A \ B, and B \ A for the given sets A and B. To recap, we found:

• A ∪ B = (-∞, -2) ∪ [4, ∞)
• A ∩ B = (-∞, -11] ∪ (7, ∞)
• A \ B = (-11, -2] ∪ [4, 7]
• B \ A = (-11, -2)

By breaking down the absolute value inequalities, visualizing the intervals on a number line, and carefully considering the definitions of union, intersection, and set difference, we were able to solve this problem step by step. Remember, practice makes perfect, guys! The more you work with sets and set operations, the more comfortable you'll become. Keep exploring, keep learning, and keep having fun with math! We navigated through the world of set theory, and I hope you found this explanation clear and helpful. Keep up the awesome work, and remember, math is not just about answers; it's about understanding the journey to get there!