Solving Equations: A Step-by-Step Guide

Hey guys! Today, we're going to dive into solving the equation $\frac{6}{x+1}-\frac{1}{2}=\frac{1}{3 x+3}$. Don't worry, it might look a bit intimidating at first, but we'll break it down step by step so it becomes super clear. We'll walk through each stage, making sure you understand not just how to solve it, but why we're doing each step. Let's get started and turn this equation into something manageable!

Understanding the Equation

Before we jump into solving this, let's really understand what we're looking at. Our equation is $\frac{6}{x+1}-\frac{1}{2}=\frac{1}{3 x+3}$. The main goal here is to isolate x on one side of the equation. To do that, we need to get rid of those fractions. Fractions can sometimes make things look complex, but there are straightforward ways to deal with them.

First off, notice that we have x in the denominators of our fractions. This means that x cannot be certain values that would make the denominator zero because division by zero is a big no-no in mathematics! Specifically, x cannot be -1 because that would make the first denominator zero, and it also cannot be -1 in the third fraction since 3*(-1) + 3 = 0. Keeping these restrictions in mind is super important.

The presence of fractions indicates that we might need to find a common denominator to combine terms or eliminate the fractions altogether. Think of it like this: if you are adding or subtracting slices of a pie, you need to make sure the slices are of the same size (common denominator) before you can accurately count how many you have. Similarly, in our equation, we need to ensure all terms are on the same "footing" before we can start moving things around.

By understanding these basics – what our equation is, what we are trying to achieve, and any restrictions we need to be aware of – we set ourselves up for success in solving the problem. Remember, taking the time to understand the problem fully is often half the battle! With this foundation in place, let's move on to the next step: simplifying the equation.

Step 1: Simplifying the Equation

Alright, let's dive into simplifying our equation: $\frac{6}{x+1}-\frac{1}{2}=\frac{1}{3 x+3}$. The first thing we want to do is look for any common factors or ways to rewrite the terms to make them easier to work with. Notice that in the denominator of the third fraction, we have $3x + 3$. We can factor out a 3 from this, which gives us $3(x + 1)$. This is great because we now have a common factor of $(x + 1)$ in two of our denominators.

Rewriting the equation with this factored form, we get: $\frac{6}{x+1}-\frac{1}{2}=\frac{1}{3(x+1)}$. Seeing this, we realize that the least common denominator (LCD) for all the fractions is $6(x + 1)$. Why? Because it's the smallest expression that each denominator can divide into evenly. The first fraction's denominator $(x + 1)$ goes into $6(x + 1)$ six times, the second fraction's denominator 2 goes in $3(x + 1)$ times, and the third fraction’s denominator $3(x + 1)$ goes in twice.

Now, we're going to multiply every term in the equation by this LCD. This might sound like a big step, but it's a crucial one because it will eliminate all our fractions, making the equation much simpler to solve. When we multiply each term by $6(x + 1)$, we get:

$6(x + 1) \cdot \frac{6}{x+1} - 6(x + 1) \cdot \frac{1}{2} = 6(x + 1) \cdot \frac{1}{3(x+1)}$

Let's simplify each term individually. For the first term, the $(x + 1)$ in the numerator and denominator cancel out, leaving us with $6 \cdot 6 = 36$. For the second term, 6 divided by 2 is 3, so we have $3(x + 1)$. And for the third term, $3(x + 1)$ cancels out, leaving us with 2. So our equation now looks like this:

$36 - 3(x + 1) = 2$

See how much cleaner that looks? By finding the LCD and multiplying through, we've transformed a complex-looking fractional equation into a simple linear one. This is a common and incredibly useful technique in algebra. Next, we'll continue simplifying by distributing and combining like terms.

Step 2: Expanding and Combining Like Terms

Okay, so we've arrived at the simplified equation: $36 - 3(x + 1) = 2$. The next step is to expand the terms and combine any like terms we can find. This will help us further isolate x and get closer to our solution.

Let's focus on the term $-3(x + 1)$. To expand this, we need to distribute the -3 across both terms inside the parentheses. Remember, distributing means multiplying the term outside the parentheses by each term inside. So, $-3$ times x is $-3x$, and $-3$ times 1 is $-3$. Now our equation looks like this:

$36 - 3x - 3 = 2$

Great! We've expanded the expression. Now, let’s combine the like terms. In this equation, 36 and -3 are like terms because they are both constants (numbers without variables). When we combine them, we get $36 - 3 = 33$. So, our equation simplifies even further to:

$33 - 3x = 2$

Notice how much simpler the equation is becoming with each step. By expanding and combining like terms, we've reduced the complexity and made it easier to see how to isolate x. At this point, we have a straightforward linear equation. The next step involves moving all the constant terms to one side and the terms with x to the other side. This will bring us one step closer to solving for x. So, let’s move on to the next step and continue our journey to the solution!

Step 3: Isolating the Variable

Now that we have the equation $33 - 3x = 2$, it's time to isolate the variable x. This means we need to get x by itself on one side of the equation. To do this, we'll start by moving the constant term, which is 33, to the other side. Remember, whatever we do to one side of the equation, we must do to the other to keep things balanced.

To move 33, we need to subtract 33 from both sides of the equation. This gives us:

$33 - 3x - 33 = 2 - 33$

On the left side, $33 - 33$ cancels out, leaving us with just $-3x$. On the right side, $2 - 33$ equals $-31$. So, our equation now looks like this:

$-3x = -31$

We're getting closer! Now we have $-3x$ on one side. To get x completely by itself, we need to get rid of the -3 that's multiplying it. The opposite of multiplication is division, so we'll divide both sides of the equation by -3. This gives us:

$\frac{-3x}{-3} = \frac{-31}{-3}$

On the left side, the -3s cancel out, leaving us with just x. On the right side, a negative divided by a negative is a positive, so we have $\frac{31}{3}$. Thus, our solution is:

$x = \frac{31}{3}$

Woohoo! We've successfully isolated x and found its value. This was a crucial step in solving our equation. However, our work isn't quite done yet. The final step is to check our solution to make sure it's correct and doesn't violate any restrictions we identified earlier. So, let's move on to the final step: checking our solution.

Step 4: Checking the Solution

Alright, we've found a potential solution: $x = \frac{31}{3}$. But before we celebrate, it’s super important to check if this solution actually works in our original equation and if it violates any restrictions. Remember, we noted earlier that x cannot be -1 because that would make the denominators in our original equation zero. Since $\frac{31}{3}$ is definitely not -1, we're good on that front.

Now, let's plug $x = \frac{31}{3}$ back into our original equation, which is $\frac{6}{x+1}-\frac{1}{2}=\frac{1}{3 x+3}$, and see if both sides of the equation are equal. This process might seem a bit tedious, but it’s the best way to ensure our answer is correct.

First, let's substitute $\frac{31}{3}$ for x:

$\frac{6}{\frac{31}{3}+1}-\frac{1}{2}=\frac{1}{3(\frac{31}{3})+3}$

Now, we need to simplify each side. Starting with the left side, let's deal with the denominator in the first fraction. We need to add $\frac{31}{3}$ and 1. To do this, we'll rewrite 1 as $\frac{3}{3}$ so we have a common denominator:

$\frac{31}{3} + \frac{3}{3} = \frac{34}{3}$

So, our left side now looks like:

$\frac{6}{\frac{34}{3}}-\frac{1}{2}$

To divide by a fraction, we multiply by its reciprocal. So, we'll multiply 6 by $\frac{3}{34}$:

$6 \cdot \frac{3}{34} = \frac{18}{34}$

We can simplify $\frac{18}{34}$ by dividing both the numerator and the denominator by 2, which gives us $\frac{9}{17}$. Now, we subtract $\frac{1}{2}$:

$\frac{9}{17} - \frac{1}{2}$

To subtract these fractions, we need a common denominator, which is 34. So, we rewrite $\frac{1}{2}$ as $\frac{17}{34}$:

$\frac{9}{17} - \frac{1}{2} = \frac{18}{34} - \frac{17}{34} = \frac{1}{34}$

So, the left side of our equation simplifies to $\frac{1}{34}$. Now, let's tackle the right side of the equation:

$\frac{1}{3(\frac{31}{3})+3}$

First, we multiply 3 by $\frac{31}{3}$, which gives us 31. Then, we add 3:

$31 + 3 = 34$

So, the right side simplifies to:

$\frac{1}{34}$

Guess what? Both sides of the equation are equal! $\frac{1}{34} = \frac{1}{34}$. This means our solution, $x = \frac{31}{3}$, is correct. We've successfully solved the equation and verified our solution. Pat yourself on the back – you've earned it!

Conclusion

Wrapping things up, we've successfully solved the equation $\frac{6}{x+1}-\frac{1}{2}=\frac{1}{3 x+3}$ by following a step-by-step approach. We started by understanding the equation and identifying any restrictions. Then, we simplified the equation by finding the least common denominator and multiplying through. We expanded and combined like terms, isolated the variable, and finally, checked our solution. Our hard work paid off, and we found that $x = \frac{31}{3}$ is indeed the correct solution.

Remember, the key to solving algebraic equations is to break them down into manageable steps. Don't be intimidated by complex-looking problems. By simplifying, expanding, isolating, and checking, you can tackle just about any equation that comes your way. And always remember to double-check your work – it’s a lifesaver!

Solving equations is a fundamental skill in mathematics, and mastering it will open doors to more advanced topics. So, keep practicing, stay curious, and don't be afraid to ask questions. You've got this! Thanks for joining me on this mathematical journey. Until next time, keep solving!