Solving The Math Equation: Fill In The Blank!

Hey Plastik Magazine readers! Today, we're diving into a fun little math problem that involves filling in the blank. It might seem intimidating at first, but don't worry, we'll break it down step by step so it's super easy to follow. Math can be cool, especially when we tackle it together! We'll look at how to approach this equation, what steps to take, and how to simplify things to get to the answer. So, grab your thinking caps, and let's get started!

Breaking Down the Equation

Okay, guys, let's look at the equation we need to solve:

$$\frac{\frac{1}{x+5}-\frac{1}{x}}{5}=\frac{x(x+5)}{x(x+5)} \cdot \frac{\left(\frac{1}{x+5}-\frac{1}{x}\right)}{5}=\frac{x(x+5) \cdot-x(x+5) \frac{1}{x}}{5 x(x+5)}=\frac{-( }{5}$$

At first glance, it might look like a jumble of fractions and variables, but don't sweat it. The key here is to take it one step at a time. We have a complex fraction on the left side, and we need to simplify it to figure out what goes in that blank. Let's focus on the numerator of the main fraction first, which is $\frac{1}{x+5}-\frac{1}{x}$. To subtract these fractions, we need a common denominator. Remember how we used to find the least common multiple? It's kinda like that here. The common denominator for $x+5$ and $x$ is simply their product, which is $x(x+5)$. So, we need to rewrite each fraction with this new denominator. This means multiplying the first fraction, $\frac{1}{x+5}$, by $\frac{x}{x}$, and the second fraction, $\frac{1}{x}$, by $\frac{x+5}{x+5}$. When we do this, we get:

$$\frac{1}{x+5} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{x+5}{x+5} = \frac{x}{x(x+5)} - \frac{x+5}{x(x+5)}$$

See? We're making progress already! Now that we have a common denominator, we can easily subtract the numerators. We just combine the fractions and subtract the terms.

Simplifying the Numerator

Alright, let's keep moving! Now that we have a common denominator, we can subtract the numerators. So, we have:

$$\frac{x}{x(x+5)} - \frac{x+5}{x(x+5)}$$

When we subtract the numerators, we get:

$$\frac{x - (x+5)}{x(x+5)}$$

Careful with that negative sign, guys! Make sure you distribute it properly to both terms inside the parentheses. This gives us:

$$\frac{x - x - 5}{x(x+5)}$$

Notice anything cool? The $x$ and $-x$ cancel each other out! This leaves us with:

$$\frac{-5}{x(x+5)}$$

Awesome! We've simplified the numerator of the main fraction. Now we can plug this back into the original equation and see what else we can do. This is like building with LEGOs, right? We put together the pieces one by one. Keep this simplified numerator in mind, because it's a key part of solving the whole thing. Remember, the goal is to isolate what's in the blank, and we're getting closer with each step. So, let's keep going and see what the next part of the equation has in store for us!

Dealing with the Main Fraction

Okay, so we've simplified the numerator of the big fraction. Now let's bring it all together and tackle the main fraction. Remember, we had:

$$\frac{\frac{1}{x+5}-\frac{1}{x}}{5}$$

And we simplified the numerator to:

$$\frac{-5}{x(x+5)}$$

So, now we have:

$$\frac{\frac{-5}{x(x+5)}}{5}$$

This might look a bit funky, but remember that dividing by a number is the same as multiplying by its reciprocal. So, dividing by 5 is the same as multiplying by $\frac{1}{5}$. We can rewrite our fraction as:

$$\frac{-5}{x(x+5)} \cdot \frac{1}{5}$$

Now it's just a matter of multiplying the fractions. We multiply the numerators together and the denominators together:

$$\frac{-5 \cdot 1}{x(x+5) \cdot 5} = \frac{-5}{5x(x+5)}$$

Hey, check it out! We have a 5 in both the numerator and the denominator. What does that mean? We can simplify! Dividing both the numerator and denominator by 5, we get:

$$\frac{-1}{x(x+5)}$$

Wow! We've simplified the left side of the equation quite a bit. This is like cleaning up our workspace so we can see the next step more clearly. Now we can compare this simplified form with the right side of the original equation and see what else needs to be done. Remember, math is like a puzzle, and we're finding the pieces that fit together perfectly!

Connecting Both Sides of the Equation

Alright, let's bring in the other side of the equation and see how everything connects. We've simplified the left side to:

$$\frac{-1}{x(x+5)}$$

Now, let's look back at the original equation:

$$\frac{\frac{1}{x+5}-\frac{1}{x}}{5}=\frac{x(x+5)}{x(x+5)} \cdot \frac{\left(\frac{1}{x+5}-\frac{1}{x}\right)}{5}=\frac{x(x+5) \cdot-x(x+5) \frac{1}{x}}{5 x(x+5)}=\frac{-( }{5}$$

Notice how the equation leads us through steps to simplify the original complex fraction. We already did the hard work of simplifying the left side, so let's focus on how the right side gets there. The first step on the right side multiplies the complex fraction by $\frac{x(x+5)}{x(x+5)}$. This is just multiplying by 1, which doesn't change the value, but it sets us up to clear the fractions within the fraction. The next part shows the distribution (sort of) but has a small error that we can see. It seems to be trying to clear the denominators but doesn't quite get there.

Let's focus on the part:

$$\frac{x(x+5) \cdot-x(x+5) \frac{1}{x}}{5 x(x+5)}$$

It looks like there might be a typo or a misunderstanding in this step. It seems like the intention was to multiply $x(x+5)$ by the simplified numerator we found earlier, which was $\frac{-5}{x(x+5)}$. If we did that, we would get:

$$x(x+5) \cdot \frac{-5}{x(x+5)} = -5$$

This makes much more sense! The $x(x+5)$ terms cancel out, leaving us with just -5. So, if we replace the incorrect part with the correct simplification, the equation would look like this:

$$\frac{\frac{1}{x+5}-\frac{1}{x}}{5} = ... = \frac{-5}{5x(x+5)}$$

See how the pieces are fitting together now? We're not just solving a math problem; we're detectives uncovering the solution! By carefully looking at each step and simplifying along the way, we're able to make sense of what seemed complicated at first. Let's use this to finally fill in that blank!

Filling in the Blank and Final Answer

Okay, we're in the home stretch! We've done the simplifying, we've connected the two sides of the equation, and now it's time to fill in that blank. Let's recap where we are.

We started with:

$$\frac{\frac{1}{x+5}-\frac{1}{x}}{5}=\frac{x(x+5)}{x(x+5)} \cdot \frac{\left(\frac{1}{x+5}-\frac{1}{x}\right)}{5}=\frac{x(x+5) \cdot-x(x+5) \frac{1}{x}}{5 x(x+5)}=\frac{-( }{5}$$

We simplified the left side to:

$$\frac{-1}{x(x+5)}$$

And we corrected the right side to lead us to:

$$\frac{-5}{5x(x+5)}$$

But wait! We can simplify $\frac{-5}{5x(x+5)}$ further by dividing both the numerator and the denominator by 5, which gives us $\frac{-1}{x(x+5)}$.

Now, let's look at the final step in the original equation:

$$\frac{-( }{5}$$

We need to figure out what goes in the blank so that this fraction is equivalent to $\frac{-5}{5x(x+5)}$. Looking at the step before, it seems like the equation was trying to get rid of the fractions within the fraction by multiplying by $x(x+5)$. So, if we think about what happened when we multiplied $x(x+5)$ by the simplified numerator $\frac{-5}{x(x+5)}$, we got -5.

However, the question asks to fill the blank in $\frac{-( }{5}$. Looking at our work, the numerator we arrived at before dividing by 5 was -5. So, to fit the form $\frac{-( }{5}$, we need to consider what we had before simplifying the fraction $\frac{-5}{5x(x+5)}$. Before dividing by $x(x+5)$, we had -5 in the numerator. Therefore, the expression that goes in the blank should represent what was canceled out to leave the simplified form. Considering the error in the original steps, and working backward, the intended simplification within the parentheses of the original equation on the right side likely aimed to have $x - (x+5)$ after clearing denominators, which simplifies to -5. Thus, the blank should be filled with $x+5-x$, which equals 5. However, given the minus sign in front of the blank, we fill the blank with 5.

So, the final answer is 5, and the filled-in equation looks like this:

$$\frac{\frac{1}{x+5}-\frac{1}{x}}{5}=\frac{x(x+5)}{x(x+5)} \cdot \frac{\left(\frac{1}{x+5}-\frac{1}{x}\right)}{5}=\frac{x(x+5) \cdot-x(x+5) \frac{1}{x}}{5 x(x+5)}=\frac{-(5)}{5}$$

Great job, guys! We tackled a tricky equation together, step by step, and figured out the missing piece. Remember, math is all about breaking things down and taking it one step at a time. Keep practicing, and you'll be solving even the toughest problems in no time!