Solve The Blank: Simplify Complex Fractions In Equation

Hey guys! Today, we're diving into a super common type of math problem: filling in the blank in an equation involving complex fractions. Complex fractions can seem intimidating at first, but don't worry, we're going to break it down step by step and make it super clear. We will solve the following equation $\frac{\frac{1}{x+2}-\frac{1}{x}}{2}=\frac{x(x+2)}{x(x+2)} \cdot \frac{\left(\frac{1}{x+2}-\frac{1}{x}\right)}{2}=\frac{x(x+2) \cdot-x(x+2) \frac{1}{x}}{2 x(x+2)}=\frac{-(\square)}{2}$. So, buckle up, grab your pencils, and let's get started!

Understanding Complex Fractions

Before we jump into solving the equation, let's make sure we're all on the same page about what complex fractions are and how to handle them. A complex fraction is basically a fraction where the numerator, the denominator, or both contain fractions themselves. Think of it as a fraction within a fraction – a bit like those Russian nesting dolls, but with numbers! The key to simplifying complex fractions is to get rid of those inner fractions. There are a couple of ways to do this, but we'll focus on one method that's particularly useful for this problem: multiplying by a clever form of 1. This might sound a little abstract now, but it'll make sense once we see it in action.

Now, why do complex fractions exist, and why are they important? Well, they often pop up in more advanced math topics like calculus and rational functions. Being able to simplify them is a crucial skill for tackling these topics. Plus, understanding complex fractions helps solidify your understanding of basic fraction operations, which is always a good thing. So, even if they seem a bit tricky at first, mastering them is definitely worth the effort. Think of it as leveling up your math skills!

Walking Through the Equation Step-by-Step

Okay, let's get back to our specific problem. We have the equation $\frac{\frac{1}{x+2}-\frac{1}{x}}{2}=\frac{x(x+2)}{x(x+2)} \cdot \frac{\left(\frac{1}{x+2}-\frac{1}{x}\right)}{2}=\frac{x(x+2) \cdot-x(x+2) \frac{1}{x}}{2 x(x+2)}=\frac{-(\square)}{2}$, and our mission is to figure out what goes in that blank space to make the equation true. The first thing we notice is that the equation is already partially simplified. This is great because it gives us a roadmap to follow. Let's break down the steps that have already been taken, so we can see the pattern and figure out the next move. The initial expression has a complex fraction in the numerator, which is exactly what we discussed earlier. To get rid of this complex fraction, the next step involves multiplying both the numerator and the denominator by $x(x+2)$. Why this particular expression? Because $x(x+2)$ is the least common multiple (LCM) of the denominators within the complex fraction, namely $x+2$ and $x$. Multiplying by the LCM is a classic trick for clearing fractions within fractions. So, the equation shows this multiplication happening, which is a good sign we're on the right track. Now, let's focus on the next part of the equation.

We can see that $\frac{x(x+2)}{x(x+2)}$ is just 1. The equation highlights the multiplication by this "clever form of 1" to eliminate the complex fraction. This prepares the expression for further simplification. This step is crucial because it demonstrates the core strategy for dealing with complex fractions. Now, let's look at how the multiplication plays out in the numerator. We have $x(x+2)$ multiplied by the difference $\left(\frac{1}{x+2}-\frac{1}{x}\right)$. This is where the distributive property comes into play. We need to multiply $x(x+2)$ by each term inside the parentheses separately. This will help us get rid of the individual fractions within the numerator. The next part of the equation shows the result of this distribution, and it's where things get interesting. We need to carefully analyze how the terms cancel out and what's left behind. This is a critical step, so let's take our time and make sure we understand each part of the simplification.

Unraveling the Solution

Let's focus on the numerator of the third fraction in the equation: $x(x+2) \cdot \left(\frac{1}{x+2}-\frac{1}{x}\right)$. Remember, we need to distribute $x(x+2)$ to both terms inside the parentheses. When we multiply $x(x+2)$ by $\frac{1}{x+2}$, the $(x+2)$ terms cancel out, leaving us with just $x$. Similarly, when we multiply $x(x+2)$ by $\frac{1}{x}$, the $x$ terms cancel out, leaving us with $(x+2)$. So, after distributing, we have $x - (x+2)$. Notice the crucial minus sign between the terms! This is where a lot of mistakes can happen if we're not careful. We need to remember to distribute that negative sign to both terms inside the parentheses.

Now, let's simplify further. We have $x - (x+2)$, which becomes $x - x - 2$. The $x$ and $-x$ cancel each other out, leaving us with just $-2$. This is a significant simplification! We've managed to get rid of all the fractions and reduce the numerator to a simple number. So, the numerator of the third fraction simplifies to $-2$. This means the entire fraction becomes $\frac{-2}{2x(x+2)}$. Now, let's look at the last part of the equation. We have $\frac{-(\square)}{2}$. Comparing this to $\frac{-2}{2x(x+2)}$, we can see that the blank space must contain the expression that, when multiplied by -1, gives us -2. In other words, we need to figure out what expression, when placed in the blank, makes the numerators equal. To make the numerators match, we need to figure out what expression, when divided by $x(x+2)$, equals 1. This means the expression in the blank must be $\frac{2}{x(x+2)}$. Therefore, we can see that the missing expression in the blank is $\frac{2}{x(x+2)}$.

Putting It All Together

So, the final answer is $\frac{2}{x(x+2)}$. We found this by carefully simplifying the complex fraction step by step, using the distributive property, and paying close attention to the signs. Remember, the key to handling complex fractions is to multiply by a "clever form of 1" to eliminate the inner fractions. In this case, we multiplied by $\frac{x(x+2)}{x(x+2)}$, where $x(x+2)$ is the least common multiple of the denominators within the complex fraction. We then distributed, canceled terms, and simplified until we arrived at our final answer. This problem highlights the importance of understanding basic fraction operations and the distributive property. These are fundamental skills that will come up again and again in math, so it's worth taking the time to master them. And remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them logically. By breaking down the problem into smaller steps and thinking carefully about each step, we can tackle even the most challenging problems.

The answer to fill in the blank is $\frac{2}{x(x+2)}$.