Simplifying Rational Expressions A Step-by-Step Solution

Hey guys! Today, we're diving into a common algebraic challenge: adding rational expressions. Specifically, we're tackling the problem:

$$\frac{4x - 9}{x^2 - 9x + 18} + \frac{3 - 3x}{x^2 - 9x + 18}$$

We need to perform the indicated operation, which is addition in this case, and simplify the result as much as possible. Don't worry; we'll break it down step-by-step so it's super clear. Let's get started!

1. Understanding the Problem: Adding Rational Expressions

Before we jump into the solution, let's quickly recap what it means to add rational expressions. Rational expressions are essentially fractions where the numerator and denominator are polynomials. Think of them as algebraic fractions. Just like regular fractions, we can only add them directly if they have a common denominator. If they don't, we need to find one first.

In our case, we have two rational expressions:

• Expression 1: $\frac{4x - 9}{x^2 - 9x + 18}$
• Expression 2: $\frac{3 - 3x}{x^2 - 9x + 18}$

Notice anything special? That's right! Both expressions already share the same denominator: $x^2 - 9x + 18$. This is excellent news because it means we can skip the step of finding a common denominator and move straight to adding the numerators.

So, the key takeaway here is recognizing that we're dealing with algebraic fractions and that the common denominator simplifies our task significantly. Keep this in mind as we proceed. Understanding the structure of the problem is half the battle, guys!

2. Combining the Numerators: The Addition Process

Now that we've confirmed we have a common denominator, the next step is to combine the numerators. This is where the actual addition happens. Since both fractions have the same denominator ($x^2 - 9x + 18$), we can simply add the numerators together and keep the denominator as is. It's just like adding regular fractions with the same bottom number!

So, let's add the numerators:

$(4x - 9) + (3 - 3x)$

To simplify this, we'll combine like terms. Remember, like terms are terms that have the same variable raised to the same power. In this case, we have terms with 'x' and constant terms (numbers without variables).

Combining the 'x' terms:

$4x - 3x = x$

Combining the constant terms:

$-9 + 3 = -6$

So, the sum of the numerators is $x - 6$. Now, we put this result over our common denominator:

$\frac{x - 6}{x^2 - 9x + 18}$

We've successfully added the two rational expressions! But we're not done yet. The problem asks us to simplify if possible, so we need to check if our resulting fraction can be reduced further. This usually involves factoring, which we'll tackle in the next section. Keep going; you're doing great!

3. Factoring the Denominator: Unlocking Simplification

Alright, we've added the rational expressions, and we've got $\frac{x - 6}{x^2 - 9x + 18}$. Now comes the crucial step of simplifying. To do this, we need to see if there are any common factors between the numerator and the denominator. And to find those common factors, we often need to factor the expressions.

In this case, the numerator, $x - 6$, is already in its simplest form – it's a linear expression and can't be factored further. So, our focus shifts to the denominator: $x^2 - 9x + 18$. This is a quadratic expression, and quadratics can often be factored into two binomials.

So, how do we factor $x^2 - 9x + 18$? We're looking for two numbers that:

• Multiply to give the constant term (18)
• Add up to give the coefficient of the 'x' term (-9)

Think about the factors of 18: 1 and 18, 2 and 9, 3 and 6. Which pair could potentially add up to -9? Remember, we need a negative sum, so we'll need to consider negative factors as well.

After a bit of thought, you'll realize that -3 and -6 fit the bill:

• (-3) * (-6) = 18
• (-3) + (-6) = -9

Therefore, we can factor the denominator as:

$x^2 - 9x + 18 = (x - 3)(x - 6)$

Now our fraction looks like this:

$\frac{x - 6}{(x - 3)(x - 6)}$

We're so close to the final simplified answer! Factoring the denominator was the key to unlocking the simplification. In the next section, we'll see how this factorization allows us to cancel out common factors and arrive at our final result. Keep the momentum going!

4. Canceling Common Factors: The Final Simplification

We've reached the exciting part where we get to simplify our fraction! We've factored the denominator, and now our expression looks like this:

$\frac{x - 6}{(x - 3)(x - 6)}$

Do you see any common factors in the numerator and the denominator? Yes! We have $(x - 6)$ in both the top and the bottom of the fraction. This means we can cancel them out. Remember, canceling common factors is essentially dividing both the numerator and denominator by the same expression, which doesn't change the value of the fraction.

So, let's cancel out the $(x - 6)$ terms:

$\frac{\cancel{(x - 6)}}{(x - 3)\cancel{(x - 6)}}$$

After canceling, we're left with:

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

And that's it! We've simplified the expression as much as possible. There are no more common factors to cancel, and we've arrived at our final answer.

5. The Solution: Putting It All Together

Let's recap the entire process to make sure we've got it all down. We started with the problem:

$$\frac{4x - 9}{x^2 - 9x + 18} + \frac{3 - 3x}{x^2 - 9x + 18}$$

Here's what we did step-by-step:

1. Recognized the common denominator: We noticed that both fractions had the same denominator, $x^2 - 9x + 18$, which made our task easier.
2. Combined the numerators: We added the numerators $(4x - 9)$ and $(3 - 3x)$ to get $x - 6$.
3. Wrote the sum over the common denominator: This gave us the fraction $\frac{x - 6}{x^2 - 9x + 18}$.
4. Factored the denominator: We factored $x^2 - 9x + 18$ into $(x - 3)(x - 6)$.
5. Canceled common factors: We canceled the $(x - 6)$ term in both the numerator and the denominator.

This led us to our final simplified answer:

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

So, the correct answer is B. $\frac{1}{x-3}$

Key Takeaways for Adding and Simplifying Rational Expressions

Before we wrap up, let's highlight some key takeaways from this problem. These tips will help you tackle similar problems with confidence.

• Common Denominator is Key: Always check for a common denominator first. If you have one, you're already halfway there! If not, you'll need to find one before adding or subtracting.
• Combine Numerators Carefully: Pay close attention to signs when adding or subtracting numerators. Distribute negative signs properly if you're subtracting rational expressions.
• Factoring is Your Friend: Factoring is often essential for simplifying rational expressions. Look for opportunities to factor both the numerator and the denominator.
• Cancel Common Factors: Once you've factored, identify and cancel any common factors between the numerator and denominator.
• Simplify Completely: Make sure you've simplified the expression as much as possible. There shouldn't be any more common factors left to cancel.

Guys, mastering these steps will make you a pro at adding and simplifying rational expressions. Practice makes perfect, so try out some more examples. You've got this!

Practice Problems: Sharpen Your Skills

Now that we've walked through the solution step-by-step, it's time to put your knowledge to the test! Here are a couple of practice problems that are similar to the one we just solved. Try tackling them on your own, and then check your answers.

Practice Problem 1:

Simplify the following expression:

$$\frac{5x + 10}{x^2 - 4} + \frac{2x - 4}{x^2 - 4}$$

Practice Problem 2:

Perform the indicated operation and simplify:

$$\frac{3x - 6}{x^2 + x - 2} + \frac{5 - 2x}{x^2 + x - 2}$$

Remember to follow the steps we discussed: combine numerators, factor denominators, and cancel common factors. Working through these problems will solidify your understanding and build your confidence.

If you get stuck, don't hesitate to review the steps we covered earlier in this guide. And remember, the more you practice, the easier these problems will become. Good luck, and have fun!

Conclusion: You've Got This!

Alright, guys, we've reached the end of our journey through adding and simplifying rational expressions. You've learned how to tackle these problems step-by-step, from recognizing the common denominator to canceling out those common factors. You've even had a chance to practice your skills with a couple of example problems.

Remember, the key to success in algebra, and in math in general, is to break down complex problems into smaller, manageable steps. That's exactly what we did today. We took a seemingly complicated expression and systematically simplified it until we arrived at the solution.

So, the next time you encounter a problem like this, don't panic! Just follow the steps we've outlined, and you'll be well on your way to finding the answer. And if you ever get stuck, remember that there are plenty of resources available to help you, including this guide, your textbook, your teacher, and online resources.

Keep practicing, keep learning, and keep challenging yourself. You've got this!