Simplifying Complex Fractions A Step-by-Step Guide

Hey guys! Let's dive into simplifying complex fractions. These might look intimidating at first, but with a systematic approach, they become quite manageable. We'll break down the given complex fraction step by step, making sure you understand each operation. So, let's get started!

Understanding Complex Fractions

Complex fractions, at their core, are fractions where the numerator, the denominator, or both contain fractions themselves. Think of it as a fraction within a fraction – a mathematical Matryoshka doll, if you will! To tackle these, our main goal is to simplify them into a single, manageable fraction. This involves a few key steps, including finding common denominators, combining fractions, and then performing the final division. Mastering complex fractions is super important because they pop up everywhere – from algebra to calculus, and even in real-world applications like physics and engineering. Seriously, these guys are the unsung heroes of mathematical problem-solving. When you encounter a complex fraction, don't freak out! Just remember the basic principles of fraction manipulation, and you'll be golden. First up, we'll identify the smaller fractions within the bigger one. This is like spotting the individual dolls in our Matryoshka – each one needs to be addressed before we can deal with the whole set. Then, we'll look for opportunities to simplify each part, like finding common denominators and combining terms. This is where your fraction skills really shine. Once we've cleaned up the numerator and the denominator, we'll be left with a simpler fraction division problem. And you know what that means, right? Flip the denominator and multiply! This is the big reveal, the moment when the complex fraction transforms into its sleek, simplified form. Along the way, keep an eye out for opportunities to cancel out common factors. This is like finding the secret shortcuts in a maze – it saves you time and effort. Simplifying complex fractions might seem like a daunting task at first, but with practice, it becomes second nature. Think of it as a mathematical puzzle – each step brings you closer to the satisfying solution. So, keep those fraction skills sharp, and you'll be a complex fraction master in no time!

The Given Complex Fraction

We're presented with the complex fraction:

$$\frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}}$$

This looks a bit tangled, right? But don't worry, we'll untangle it together! Our mission is to simplify this fraction into one of the given answer choices. The key here is to methodically work through the different parts of the fraction, step by step. We'll start by looking at the numerator and the denominator separately. Think of it like building a house – you need to lay the foundation (simplify the parts) before you can put on the roof (the final simplification). In the numerator, we have a fraction being subtracted from a whole number. And in the denominator, we have a similar situation. So, our first order of business is to deal with these subtractions. This means finding a common denominator for each part. Remember, a common denominator is like the universal language that allows us to add and subtract fractions. Once we have the common denominators, we can combine the terms in the numerator and the denominator, making them single fractions. This is like condensing a long sentence into a shorter, snappier one – it makes things clearer and easier to understand. After that, we'll have a fraction divided by another fraction. This is where the magic happens! Dividing by a fraction is the same as multiplying by its reciprocal. So, we'll flip the denominator and multiply it by the numerator. This step is crucial – it's the key that unlocks the simplified form of the complex fraction. Finally, we'll look for any common factors that we can cancel out. This is like the final polish, making sure our answer is in its simplest form. By following these steps, we'll transform this complex fraction into something much more elegant and manageable. So, let's roll up our sleeves and get started!

Step 1: Simplifying the Numerator

The numerator of our complex fraction is $\frac{3}{x-1}-4$. To combine these terms, we need a common denominator. Remember, to subtract fractions, they need to speak the same language – and that language is the common denominator. So, we need to rewrite the whole number 4 as a fraction with the same denominator as $\frac{3}{x-1}$. Think of it like converting currencies – we need to express both amounts in the same unit before we can compare them. The denominator we're aiming for is (x-1). So, we can rewrite 4 as $\frac{4(x-1)}{x-1}$. This might look a bit clunky, but it's a crucial step. Now, we have two fractions with the same denominator, ready to be combined. It's like having two ingredients measured in the same units – now we can mix them together! So, our numerator becomes $\frac{3}{x-1}-\frac{4(x-1)}{x-1}$. Now, we can perform the subtraction. We subtract the numerators and keep the denominator the same. This is like subtracting apples from apples – we end up with a certain number of apples. So, we have $\frac{3-4(x-1)}{x-1}$. Next, we need to simplify the numerator by distributing the -4. Remember the distributive property? It's like sharing the candy equally among your friends – each term inside the parentheses gets multiplied by the term outside. So, -4 multiplied by x gives us -4x, and -4 multiplied by -1 gives us +4. This means our numerator becomes $\frac{3-4x+4}{x-1}$. Finally, we can combine the constants in the numerator. We have 3 and 4, which add up to 7. So, our simplified numerator is $\frac{-4x+7}{x-1}$. Phew! That was a workout, but we've successfully tamed the numerator. Now it's time to turn our attention to the denominator and give it the same treatment.

Step 2: Simplifying the Denominator

Now let's tackle the denominator, which is $2-\frac{2}{x-1}$. Just like with the numerator, we need a common denominator to combine these terms. We're going to follow the same steps here, making sure we're consistent in our approach. Remember, consistency is key in mathematics – like following a recipe in baking, it ensures we get the right result. So, we need to rewrite the whole number 2 as a fraction with the same denominator as $\frac{2}{x-1}$. The denominator we're aiming for is, you guessed it, (x-1). So, we can rewrite 2 as $\frac{2(x-1)}{x-1}$. This is like putting on our fraction goggles, allowing us to see 2 in a new light – as a fraction with the desired denominator. Now, our denominator becomes $\frac{2(x-1)}{x-1}-\frac{2}{x-1}$. We have two fractions with the same denominator, ready to be combined. It's like having two puzzle pieces that fit together perfectly – now we can connect them. So, we can perform the subtraction. We subtract the numerators and keep the denominator the same. This gives us $\frac{2(x-1)-2}{x-1}$. Next, we need to simplify the numerator by distributing the 2. Remember, the distributive property is our friend here. It ensures we multiply each term inside the parentheses correctly. So, 2 multiplied by x gives us 2x, and 2 multiplied by -1 gives us -2. This means our numerator becomes $\frac{2x-2-2}{x-1}$. Finally, we can combine the constants in the numerator. We have -2 and -2, which add up to -4. So, our simplified denominator is $\frac{2x-4}{x-1}$. Awesome! We've conquered the denominator. Now we have a simplified numerator and a simplified denominator. The complex fraction is starting to look less complex, right? The next step is where we bring it all together and perform the final simplification.

Step 3: Dividing the Fractions

Okay, we've simplified the numerator to $\frac{-4x+7}{x-1}$ and the denominator to $\frac{2x-4}{x-1}$. Now, we're ready for the main event: dividing these fractions. Remember, dividing by a fraction is the same as multiplying by its reciprocal. This is like flipping a switch – we change the operation from division to multiplication and invert the second fraction. So, our complex fraction now looks like this:

$$\frac{-4x+7}{x-1} \div \frac{2x-4}{x-1} = \frac{-4x+7}{x-1} \cdot \frac{x-1}{2x-4}$$

See how we flipped the second fraction? This is the key step in dividing fractions. Now, we have a multiplication problem. And when we multiply fractions, we multiply the numerators together and the denominators together. This is like combining two recipes – we mix the ingredients from both to create something new. So, we have:

$$\frac{(-4x+7)(x-1)}{(x-1)(2x-4)}$$

Now, we look for opportunities to simplify. Notice that we have a common factor of (x-1) in both the numerator and the denominator. This is like spotting a duplicate ingredient in our combined recipe – we can remove it without changing the result. So, we can cancel out the (x-1) terms. This leaves us with:

$$\frac{-4x+7}{2x-4}$$

We're almost there! Now, let's see if we can simplify the denominator further. We can factor out a 2 from the denominator:

$$\frac{-4x+7}{2(x-2)}$$

And that's it! We've successfully simplified the complex fraction. Our final simplified form is $\frac{-4x+7}{2(x-2)}$.

Step 4: Matching the Answer Choices

Let's take a look at the answer choices provided. We have:

A. $\frac{2(x-2)}{-4 x+7}$ B. $\frac{-4 x+7}{2(x-2)}$ C. $\frac{-4 x+7}{2\left(x^2-2\right)}$ D.

Comparing our simplified form, $\frac{-4x+7}{2(x-2)}$, with the answer choices, we can see that it matches option B perfectly. It's like finding the missing piece of the puzzle – our simplified fraction fits right in! So, the correct answer is B. Yay, we did it!

Conclusion

Simplifying complex fractions might seem daunting at first, but by breaking them down into smaller, manageable steps, it becomes a lot easier. We tackled this problem by first simplifying the numerator and the denominator separately, then dividing the fractions, and finally, matching our result with the answer choices. Remember, practice makes perfect! The more you work with complex fractions, the more comfortable you'll become with them. So, keep practicing, and you'll be a complex fraction whiz in no time! And that's a wrap, guys! Hope this helped you understand how to simplify complex fractions. Keep up the great work, and remember, math can be fun!

Final Answer: The final answer is $\boxed{\frac{-4 x+7}{2(x-2)}}$