Solving Set Operations: Union And Intersection In Math

Hey guys! Let's dive into a fun math problem today! We're going to tackle a set theory question that involves absolute values, inequalities, and a bit of set operations. Sounds complicated? Don't worry, we'll break it down step by step. The core concepts we'll be exploring are the union and intersection of sets, along with solving inequalities. This is a classic type of problem that you might encounter in high school math or even in some introductory college courses. So, grab your pencils and let's get started! We'll look at the sets A and B, define them using inequalities, and then find the union of A and B, focusing only on the integer elements within that union. Are you ready to become a set theory whiz?

Understanding the Problem and Key Concepts

Alright, let's get straight to the point. The problem gives us two sets, A and B, defined using mathematical conditions. Set A includes all real numbers x such that the absolute value of x - 4 is less than or equal to 3. In simpler terms, we are looking for all the numbers whose distance from 4 is at most 3 units. Set B, on the other hand, includes all real numbers x that satisfy the inequality (3x + 8) / 2 < 13. This means we want to find all the numbers that, when we perform some operations, give us a result less than 13. The ultimate goal? To determine the union of sets A and B, and then find all the integer values in this combined set. Remember, the union of two sets means combining all the elements from both sets into one. The intersection of two sets, which isn't the focus of this particular problem, would mean finding the elements common to both sets. In this context, we are especially interested in the elements that belong to the integer set, Z. So, basically, we are trying to find what integers are in A or B or both. The key concepts involved here are: absolute value, inequalities, the set of real numbers (ℝ), the set of integers (ℤ), and set operations (union). We'll be doing some basic algebra to find the solution. This kind of problem really tests your understanding of the basics. It's all about applying what you already know to solve a real-world math problem. It is useful in a lot of programming applications and is really used in everyday life.

To approach this, we need to solve the inequalities and understand what these sets actually represent. Once we have a clear picture of the elements in set A and set B, we can easily perform the union operation. The final step will be identifying which of those elements are integers. Let's start by cracking the code of set A. This set is defined using an absolute value inequality, which means we're going to have to remember how absolute values work. Remember the definition of absolute value, it is the distance from zero. This will require a bit of algebraic manipulation. With these, we should be able to visualize the solutions in an easy way. Also, always remember the properties of inequalities, like how to solve them, or what happens when you multiply or divide both sides by a negative number. Being confident with these basics will help you breeze through problems like these. These are essential building blocks for more complex topics. So, let's get to work.

Solving for Set A: Absolute Value Inequality

Let's take a closer look at set A. It's defined by the absolute value inequality: |x - 4| ≤ 3. When you see an absolute value inequality like this, think about it as representing a distance on a number line. In this case, we're saying that the distance between x and 4 is less than or equal to 3. To solve this, we can rewrite the absolute value inequality as a compound inequality. We know that if the absolute value of x - 4 is less than or equal to 3, then x - 4 must be between -3 and 3, inclusive. So, we can write this as: -3 ≤ x - 4 ≤ 3. Now, we have a straightforward compound inequality to solve. To isolate x, we need to add 4 to all parts of the inequality. Adding 4 to each part, we get: -3 + 4 ≤ x - 4 + 4 ≤ 3 + 4, which simplifies to 1 ≤ x ≤ 7. This means set A includes all real numbers from 1 to 7, including 1 and 7. In interval notation, we can write this as A = [1, 7]. So, we've successfully decoded set A! We know precisely which real numbers belong to it. What does this mean? This means that the interval starts from 1 and goes up to 7. In other words, any real number between 1 and 7, including 1 and 7, is a member of the set A. You can also visualize this on a number line. Imagine a line representing all real numbers. Then, mark the points 1 and 7. Everything in between these points, and including these points, constitutes set A. This is an important skill because it helps you picture the problem and makes it easier to solve. We'll be using this skill later. Don't underestimate the importance of visualization when it comes to math problems like these. Okay, next, let's find out what set B is all about.

Solving for Set B: Linear Inequality

Now, let's turn our attention to set B. It's defined by the linear inequality: (3x + 8) / 2 < 13. Our goal is to isolate x and find the range of real numbers that satisfy this inequality. First, to get rid of the fraction, we can multiply both sides of the inequality by 2. This gives us: 3x + 8 < 26. Next, we need to get rid of the 8 on the left side. We can subtract 8 from both sides: 3x < 26 - 8, which simplifies to 3x < 18. Finally, to solve for x, we divide both sides by 3: x < 6. This means set B includes all real numbers less than 6. In interval notation, this can be written as B = (-∞, 6). Notice the parenthesis next to 6, which means it does not include 6. So, set B represents all real numbers less than 6. On a number line, you can imagine a ray starting from 6 and going all the way to negative infinity. The open circle on the 6 indicates that 6 itself is not included. This inequality is much easier to deal with compared to the absolute value inequality. The algebraic steps are pretty basic, and it helps when you have a solid understanding of the rules of inequalities. Always remember that when multiplying or dividing by a negative number, you must flip the inequality sign. Now, that we have solved for both sets A and B, next we should do the union operation.

Finding the Union: A ∪ B

Now that we have solved the inequalities for sets A and B, we can determine their union, which is denoted as A ∪ B. Remember, the union of two sets includes all the elements that are in either set A or set B or in both. Set A consists of all real numbers between 1 and 7, inclusive (i.e., [1, 7]). Set B includes all real numbers less than 6 (i.e., (-∞, 6)). To find the union, think about combining the ranges. Set A has numbers from 1 up to 7, and set B has numbers from negative infinity up to 6. Since set B includes all numbers less than 6 and set A includes numbers from 1 to 7, the union will essentially include all the numbers from negative infinity up to 7, inclusive. In interval notation, this can be expressed as (-∞, 7]. Visually, we are merging the number lines of A and B. The union will cover everything from the leftmost point in B (which goes to negative infinity) up to the rightmost point in A (which is 7). Therefore, A ∪ B = (-∞, 7]. So, all real numbers less than or equal to 7 are in the union. This means all values that are in A, B, or both. Now, we have performed the union operation, and the result is a set that includes all real numbers up to 7. Congratulations, you are almost there! But, we are not done yet. Remember, the goal is to find the integer members of A ∪ B. Let's move to the next step.

Finding the Integer Elements: (A ∪ B) ∩ ℤ

Our last step is to find the intersection of (A ∪ B) with the set of integers (ℤ). Remember that we are looking for the integers within the range of our union, which we previously found to be (-∞, 7]. This means we need to identify all the integers that are less than or equal to 7. Integers are whole numbers, both positive and negative, including zero. So, the integers in (A ∪ B) are: ..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. We start from negative infinity and go up to 7, including 7. We are looking for the integers in the interval (-∞, 7]. This is just a matter of listing the integers, and the result will be every whole number up to and including 7. The intersection of a set with the set of integers essentially asks us to select the integer numbers that belong to that set. Therefore, (A ∪ B) ∩ ℤ = {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7}. This set represents all integers less than or equal to 7. The answer is an infinite set containing all whole numbers from negative infinity up to 7. And there you have it! We've successfully solved the problem. We have found the integer values that are included in the union of sets A and B. You have shown your ability to handle absolute value inequalities, linear inequalities, set operations, and the set of integers. Good job!

Conclusion and Key Takeaways

In this problem, we covered a lot of ground. We started with two sets, A and B, defined by inequalities involving absolute values and basic algebra. We then solved for the range of real numbers in each set. After this, we found the union of the two sets, which included everything from negative infinity up to 7. Finally, we identified the integers that are members of this combined set. This problem illustrates the importance of understanding inequalities, the set of real numbers, the set of integers, and set operations like the union. Practicing these types of problems is crucial for strengthening your mathematical skills. By solving problems like these, you are practicing the fundamentals and developing strong problem-solving skills. The ability to break down a complex problem into smaller, manageable steps is a key skill in mathematics and in life. Remember to always double-check your work and to clearly understand the definitions and properties of the concepts involved. So, what are the main takeaways? Always remember the rules for dealing with inequalities, including those involving absolute values. Understand the concepts of union and intersection. And always be comfortable with the basic set of numbers (integers, real numbers, etc.). Keep practicing and don't be afraid to ask for help. Math might seem challenging at times, but with practice and understanding, it can be truly enjoyable. Keep up the good work, and you'll be a math whiz in no time! Good luck with your future math endeavors, and happy solving, guys!