Finding Integer Solutions: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem. We're gonna figure out the set A, which includes all the integers (x) that fit some specific conditions. It's like a treasure hunt, but instead of gold, we're looking for whole numbers! This kind of problem often pops up in algebra and number theory, and it's super important for building a solid math foundation. The core concept here is understanding inequalities and how they work with integers. We'll break down the problem step by step, making sure everything is clear and easy to follow. Get ready to flex those math muscles! We'll start by looking at the given conditions and translating them into a set of inequalities. Then, we'll solve each inequality separately to narrow down the possible values of x. Finally, we'll combine the solutions to find the exact set A. So, let's get started and unravel this mathematical puzzle together. This is a classic example of a problem where you need to use inequalities to find the range of possible values for a variable and then narrow it down to integers. This kind of problem is crucial for understanding more complex mathematical concepts later on, so pay close attention.

Decoding the Problem: Understanding the Basics

Alright, first things first, let's understand what the problem is asking. The set A is defined as all the integers (x) that satisfy the inequality: x + 3 < (2x + 13)/3 < x + 5. This means we're looking for whole numbers that, when plugged into this inequality, make it true. The notation x ∈ Z* means x belongs to the set of integers. Think of integers as whole numbers, both positive and negative, including zero.

So, our goal is to find all the integers that fit within the bounds set by this compound inequality. We have two main inequalities hidden in this: x + 3 < (2x + 13)/3 and (2x + 13)/3 < x + 5. We need to solve both of these and then find where their solutions overlap. The intersection of these solution sets will give us the members of set A. This is a common strategy in solving inequalities: break down the compound inequality into simpler parts, solve each part individually, and then combine the results. Remember, the key is to isolate x on one side of the inequality.

Before we jump into the calculations, let's talk strategy. We'll tackle each inequality one by one, making sure we apply the rules correctly (e.g., when you multiply or divide by a negative number, you flip the inequality sign). After solving each inequality, we'll graph the solutions on a number line to visualize the overlap and identify the integer solutions. This visual approach often makes it easier to understand the range of values that satisfy the compound inequality. So, let's break this down into manageable chunks and conquer this mathematical challenge together. It might seem daunting at first, but trust me, with a systematic approach, we'll get there.

Solving the Inequalities: Step-by-Step Breakdown

Now, let's get down to business and solve those inequalities! We'll take them one at a time and be super careful with our steps. This is where we get to apply our algebra skills and see how the rules of inequalities work.

First Inequality: x + 3 < (2x + 13)/3

Our first task is to solve x + 3 < (2x + 13)/3. To start, let's get rid of that fraction by multiplying both sides of the inequality by 3. This gives us: 3*(x + 3) < 2x + 13. Next, we distribute the 3 on the left side: 3x + 9 < 2x + 13. Now, let's isolate x. Subtract 2x from both sides: 3x - 2x + 9 < 13. This simplifies to x + 9 < 13. Finally, subtract 9 from both sides to get x < 4. So, the first inequality tells us that x must be less than 4.

Second Inequality: (2x + 13)/3 < x + 5

Alright, let's solve the second part of the compound inequality: (2x + 13)/3 < x + 5. Similar to the first one, let's start by getting rid of the fraction. Multiply both sides by 3: 2x + 13 < 3*(x + 5). Distribute the 3 on the right side: 2x + 13 < 3x + 15. Now, we want to isolate x. Subtract 2x from both sides: 13 < x + 15. Then, subtract 15 from both sides: 13 - 15 < x. This simplifies to -2 < x, or x > -2. So, the second inequality tells us that x must be greater than -2.

We've now solved both inequalities separately! The first one says x < 4, and the second one says x > -2. Now, we just need to find the integers that satisfy both conditions simultaneously. These steps demonstrate the basic algebraic manipulations needed to solve inequalities. Always remember to perform the same operation on both sides to keep the inequality balanced. Now, let's move on to the next step, where we will bring these solutions together to determine the elements of set A.

Combining the Solutions: Finding the Integer Set A

Alright, guys, we're in the home stretch now! We've successfully solved both individual inequalities. Now, it's time to put everything together and find the integers that satisfy both conditions. This is where we find the sweet spot where both inequalities are true.

We found that x < 4 from the first inequality and x > -2 from the second. This means x must be greater than -2 and less than 4. So, we're looking for integers that fall between -2 and 4. Let's list the integers that fit these conditions: -1, 0, 1, 2, and 3. These are the only integers that are both greater than -2 and less than 4. You can also visualize this on a number line. Mark the points -2 and 4. Then, shade the region between them. The integers within that shaded region are your solutions.

Therefore, the set A = {-1, 0, 1, 2, 3}. That’s it! We’ve found the set of all integers that satisfy the original inequality. It's a nice, neat set of numbers. Congratulations, we've successfully navigated the mathematical challenge! We started with a compound inequality, broke it down, solved each part, and then combined our solutions to find the set A. Understanding how to solve such problems is crucial for your math journey because it builds your ability to think logically and systematically. Remember, practice is key. Try some similar problems to reinforce your understanding and sharpen your skills. The ability to identify the range of integer solutions is a fundamental concept in many areas of mathematics and computer science.

Conclusion: Wrapping Up and Key Takeaways

And that's a wrap! We've successfully determined the set A = {-1, 0, 1, 2, 3}. This problem highlights the importance of understanding inequalities and how to manipulate them to find solutions. Remember, the key steps are to break down the compound inequality, solve each part separately, and then combine the solutions to find the final answer.

Key takeaways: Always remember to perform the same operation on both sides of the inequality to keep it balanced. Be extra careful when multiplying or dividing by a negative number, as it requires you to flip the inequality sign. Practice, practice, practice! The more you practice, the more comfortable and confident you'll become in solving these types of problems. Understanding inequalities and how to manipulate them is crucial for your math journey because it builds your ability to think logically and systematically.

This is just one example of how inequalities are used in mathematics. They pop up everywhere, from simple algebra problems to complex real-world applications. By mastering the basics, like we did here, you'll be well-prepared to tackle more advanced concepts. Keep up the great work, and don’t be afraid to ask for help if you get stuck. Math is a journey, and every problem you solve is a step forward! Keep exploring, keep learning, and keep challenging yourselves! Remember, the goal is not just to get the right answer, but to understand why the answer is correct. Happy math-ing, and keep up the great work, everyone! Now go out there and conquer some more math problems! You got this!