Solving Inequalities: Find X In ℕ* For (2x+5)/4 < X < (3x+13)/6

Hey guys! Today, we're diving deep into a fascinating math problem where we need to find the values of x that are positive integers (that's what ℕ* means, by the way) and satisfy a rather interesting double inequality: (2x+5)/4 < x < (3x+13)/6. Don't worry if it looks a bit intimidating at first; we'll break it down step by step, making sure everyone understands the process. So, grab your favorite beverage, maybe a notepad, and let’s get started!

Understanding the Problem

Before we jump into solving, let’s make sure we're all on the same page. The problem asks us to find the positive integer values of x that fit within the given range defined by the inequalities. Essentially, x needs to be larger than (2x+5)/4 and smaller than (3x+13)/6. This type of problem often pops up in algebra, and mastering it can seriously boost your problem-solving skills. We're not just looking for any number; it specifically needs to be a positive integer. This narrows down our search and adds a neat little twist to the challenge.

Breaking Down the Inequalities

Our main mission here is to isolate x. To do that effectively, we need to tackle each part of the double inequality separately. Think of it as solving two smaller puzzles that, when combined, give us the solution to the bigger one. This approach not only simplifies the problem but also makes it less prone to errors. So, let’s take a closer look at each inequality on its own.

First Inequality: (2x+5)/4 < x

Let's start with the first part: (2x+5)/4 < x. The presence of the fraction might make it look a bit scary, but fear not! Our first move is to eliminate that fraction. How do we do it? Simple! We multiply both sides of the inequality by 4. Remember, whatever we do on one side, we must do on the other to keep things balanced. This gives us: 2x + 5 < 4x. Now, we're getting somewhere! Our next goal is to gather all the x terms on one side and the constants on the other. A little algebraic maneuvering, and we'll have a clearer picture of what x needs to be.

Isolating x

To get the x terms together, let's subtract 2x from both sides. This keeps the inequality intact while moving the terms where we want them. So, 2x + 5 - 2x < 4x - 2x simplifies to 5 < 2x. We're almost there! Now, to completely isolate x, we need to get rid of that pesky 2 that's multiplying it. We do this by dividing both sides by 2. This gives us 5/2 < x, which can also be written as x > 2.5. This is a crucial piece of the puzzle. It tells us that x must be greater than 2.5.

Second Inequality: x < (3x+13)/6

Now, let’s shift our focus to the second part of the double inequality: x < (3x+13)/6. Just like before, our initial step is to get rid of the fraction. We’ll multiply both sides by 6 to clear it out. This gives us 6x < 3x + 13. See? It’s already looking more manageable! Our next step is the same as before: we want to isolate x. That means getting all the x terms on one side and the constants on the other. Time for some more algebraic action!

Isolating x (Again!)

To get the x terms together, we subtract 3x from both sides. This maintains the balance of the inequality while moving the terms where we need them. So, 6x - 3x < 3x + 13 - 3x simplifies to 3x < 13. We’re on the home stretch now! To completely isolate x, we divide both sides by 3. This gives us x < 13/3. If you prefer decimals, 13/3 is approximately 4.33. So, this inequality tells us that x must be less than 4.33.

Combining the Inequalities

We've successfully dissected the original problem into two simpler inequalities and found that x > 2.5 and x < 4.33. Now comes the fun part: putting these two pieces of information together to find our solution. We're looking for values of x that satisfy both conditions simultaneously. Think of it as finding the sweet spot where both inequalities overlap. This is where our understanding of number ranges and integers comes into play.

Finding the Overlapping Range

We know that x has to be greater than 2.5 and less than 4.33. If we visualize this on a number line, we're looking for the section where these two ranges overlap. This overlap gives us the possible values for x. But remember, we're not looking for just any numbers; we need positive integers. So, we need to consider only the whole numbers that fall within this range. This restriction makes our task a bit easier, as it narrows down the possibilities considerably.

Identifying the Integer Solutions

Okay, we know x must be greater than 2.5 and less than 4.33. So, what positive integers fit this bill? Let's think about it. The integers greater than 2.5 are 3, 4, 5, and so on. But x also needs to be less than 4.33. This means we can only consider integers up to 4. So, the possible values for x are 3 and 4. These are the positive integers that satisfy both inequalities. We've narrowed it down and found our potential solutions!

Verifying the Solutions

We've identified 3 and 4 as potential solutions, but it's always a good idea to double-check. Verification is a crucial step in problem-solving, especially in math. It ensures that our solutions are correct and that we haven't made any mistakes along the way. To verify, we simply plug each potential solution back into the original inequality and see if it holds true. This process gives us confidence in our answer and helps solidify our understanding of the problem.

Testing x = 3

Let's start by substituting x = 3 into the original inequality: (2x+5)/4 < x < (3x+13)/6. Plugging in 3, we get: (2(3)+5)/4 < 3 < (3(3)+13)/6. Simplifying this gives us (6+5)/4 < 3 < (9+13)/6, which further simplifies to 11/4 < 3 < 22/6. Converting these fractions to decimals (or mixed numbers) makes it easier to compare: 2.75 < 3 < 3.67. Is this true? Yes, it is! 3 is indeed greater than 2.75 and less than 3.67. So, x = 3 is definitely a solution.

Testing x = 4

Now, let's test x = 4. We substitute 4 into the original inequality: (2x+5)/4 < x < (3x+13)/6. Plugging in 4, we get: (2(4)+5)/4 < 4 < (3(4)+13)/6. Simplifying, we have (8+5)/4 < 4 < (12+13)/6, which becomes 13/4 < 4 < 25/6. Converting to decimals, we get 3.25 < 4 < 4.17. Again, this is true! 4 is greater than 3.25 and less than 4.17. So, x = 4 is also a solution.

Final Answer: The Solutions for x

We've gone through the entire process, from breaking down the inequalities to verifying our solutions. We found that the positive integer values of x that satisfy the inequality (2x+5)/4 < x < (3x+13)/6 are x = 3 and x = 4. These are our final answers! 🎉

Wrapping Up

So, there you have it! We've successfully solved a double inequality problem, step by step. Remember, the key to tackling these types of problems is to break them down into smaller, more manageable parts. By isolating x in each inequality and then combining the results, we can find the solutions. And don't forget to verify your answers! This problem demonstrates important algebraic techniques and logical reasoning, skills that are super valuable in mathematics and beyond. Keep practicing, and you'll become a pro at solving inequalities in no time!