Solving 24((2x-5)/3 - (x-3)/4) A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving deep into a fascinating equation that might seem a bit daunting at first glance, but trust me, it's totally conquerable. We're going to break down the steps to solve: 24((2x-5)/3 - (x-3)/4) =

The Initial Setup: Let's Get Organized

When you first look at this equation, the fractions and parentheses might make you want to run for the hills. But hold on! The key to taming any complex equation is to tackle it step by step, just like eating an elephant – one bite at a time! Our first order of business is to simplify the expression inside the parentheses. This means we need to deal with those pesky fractions. Remember, when adding or subtracting fractions, we need a common denominator. In this case, we're dealing with fractions that have denominators of 3 and 4. So, what's the least common multiple of 3 and 4? You guessed it, it's 12!

Now, we're going to rewrite each fraction with a denominator of 12. To do this, we'll multiply the first fraction, (2x-5)/3, by 4/4 (which is just a fancy way of saying 1), and the second fraction, (x-3)/4, by 3/3. This gives us:

(4 * (2x - 5)) / 12 - (3 * (x - 3)) / 12

See? We're already making progress! Now we have a common denominator, which means we can combine these fractions. This is where things start to get really interesting. We're not just manipulating numbers; we're revealing the hidden structure of the equation, like an archaeologist carefully brushing away dirt to uncover an ancient artifact. The beauty of mathematics lies in this process of discovery, of transforming the unfamiliar into something we can understand and work with. So, let's keep digging!

Distribute and Conquer: Taming the Parentheses

Okay, guys, we've got a common denominator, which is fantastic! But we still have those parentheses hanging around, and they're just begging to be dealt with. To get rid of them, we're going to use the distributive property. Remember, this means we multiply the number outside the parentheses by each term inside the parentheses. It's like we're sending a little mathematical messenger to deliver the multiplication to every corner of the expression. Let's start with the first part:

4 * (2x - 5) = 8x - 20

Easy peasy, right? Now, let's tackle the second part. This one's a little trickier because we have a negative sign in front of the 3. We need to remember to distribute that negative sign along with the 3. It's like the negative sign is a little gremlin that tags along and changes the signs of everything it touches. So, we have:

-3 * (x - 3) = -3x + 9

Notice how the -3 multiplied by the -3 becomes +9? Those negative signs can be sneaky, but we're too smart to let them trip us up! Now we can rewrite our expression inside the parentheses as:

(8x - 20) / 12 - (3x + 9) / 12 = (8x - 20 - 3x + 9) / 12

We're one step closer to simplifying this beast! Remember, math is like a puzzle – each step we take brings us closer to the solution. It's a journey of logical deduction and careful calculation, and the feeling of cracking the code is oh-so-satisfying.

Combine Like Terms: Bringing Order to the Chaos

Alright, team, we've distributed, we've conquered the parentheses, and now it's time to bring some order to the chaos! We have a bunch of terms jumbled together in the numerator of our fraction, and it's our job to sort them out. This is where combining like terms comes into play. Remember, like terms are those that have the same variable raised to the same power. In our case, we have 'x' terms and constant terms. It's like we're sorting socks – we want to pair up the ones that match!

Let's take a look at our numerator:

8x - 20 - 3x + 9

We have 8x and -3x, which are like terms. When we combine them, we get:

8x - 3x = 5x

Next, we have the constant terms, -20 and +9. Combining these gives us:

-20 + 9 = -11

So, our simplified numerator is 5x - 11. Now we can rewrite our expression inside the parentheses as:

(5x - 11) / 12

Doesn't that look so much cleaner and simpler than what we started with? We've taken a tangled mess and transformed it into something elegant and manageable. This is the power of algebraic manipulation! We're not just shuffling symbols around; we're revealing the underlying structure and relationships within the equation. It's like we're architects designing a building, carefully arranging the elements to create a strong and stable structure.

Multiplying Through: The Final Push

We've made incredible progress, guys! We've simplified the expression inside the parentheses, and now we're ready for the final push. Remember that 24 sitting outside the parentheses? It's time to bring it into the mix. We're going to multiply the entire expression inside the parentheses by 24. This is like unleashing the full force of our mathematical prowess! We've been building up to this moment, and now we're ready to deliver the knockout punch.

So, we have:

24 * ((5x - 11) / 12)

Now, before we go ahead and distribute that 24, let's see if we can simplify things a bit. Notice that 24 and 12 have a common factor? That's right, both are divisible by 12! We can simplify this fraction by dividing both the numerator (24) and the denominator (12) by 12:

(24 / 12) * ((5x - 11) / (12 / 12)) = 2 * (5x - 11)

Ah, much better! Simplifying before multiplying often makes our lives easier. It's like taking a shortcut on a hiking trail – we get to the destination with less effort. Now, we can distribute the 2:

2 * (5x - 11) = 10x - 22

And there you have it! We've successfully simplified the entire expression. Our final answer is:

10x - 22

Conclusion: A Mathematical Triumph

Wow, we did it! We took a complex-looking equation and, step by step, transformed it into a simple and elegant expression. We conquered fractions, tamed parentheses, combined like terms, and multiplied through to victory! This journey wasn't just about finding the answer; it was about learning the process of mathematical problem-solving. We learned how to break down a problem into smaller, manageable steps, how to apply key concepts like the distributive property and combining like terms, and how to persevere through challenges.

Remember, guys, math isn't about memorizing formulas; it's about understanding the underlying principles and developing the skills to think critically and solve problems. So, the next time you encounter a daunting equation, don't be afraid to dive in and start exploring. You might just surprise yourself with what you can achieve!