Adding Rational Expressions: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever stumbled upon a math problem that looks a bit intimidating? Don't sweat it, because today we're diving into the world of adding rational expressions! We'll tackle this problem step-by-step, making sure you feel confident by the end. Our goal? To simplify the expression $\frac{6}{x+2}+\frac{7}{x-1}$ as much as humanly possible. Let's get started, shall we? This problem is super common in algebra, and understanding it unlocks a ton of other concepts. So, grab your pencils, and let's break it down! We'll make sure you understand the why behind each step, not just the how.

Before we begin, remember that rational expressions are just fractions with polynomials in the numerator and/or denominator. They might look scary at first glance, but they follow the same rules as regular fractions. The most important thing here is to find a common denominator before we can combine them. This is the cornerstone of adding or subtracting any fractions. Without a common denominator, you're trying to compare apples and oranges – it just doesn't work! So, buckle up; we're about to make these fractions match.

The Common Denominator and The First Step

Alright, guys, let's look at the problem again: $\frac{6}{x+2}+\frac{7}{x-1}$. The first thing we need to do is identify our denominators. We've got $(x+2)$ and $(x-1)$. Since these two expressions don't share any common factors, our least common denominator (LCD) will simply be their product. This means we'll multiply the two denominators together. Think of it like this: if you have two different ingredients and you want to make sure you have both of them in your final recipe, you need to include both. In this case, our ingredients are $(x+2)$ and $(x-1)$.

So, the LCD is $(x+2)(x-1)$. Now comes the fun part: We need to rewrite each fraction so that it has this common denominator. For the first fraction, $\frac{6}{x+2}$, we're missing the $(x-1)$ factor in the denominator. To fix this, we'll multiply both the numerator and the denominator by $(x-1)$. Remember, multiplying the numerator and denominator by the same expression is like multiplying by 1; it doesn't change the value of the fraction, just its appearance. We're simply getting a fancy new way to write the same thing.

On the other hand, the second fraction, $\frac{7}{x-1}$, is missing the $(x+2)$ factor. So, we'll multiply both the numerator and the denominator of this fraction by $(x+2)$. This ensures that both fractions have the same denominator, which is our ultimate goal. It's like having two puzzle pieces and making sure they both fit the same hole.

Rewriting the Fractions With the LCD

Let's put this into action! We have: $\frac{6}{x+2} * \frac{x-1}{x-1} + \frac{7}{x-1} * \frac{x+2}{x+2}$. This may seem a little busy, but it is a vital part. Let's handle the first fraction: $\frac{6}{x+2} * \frac{x-1}{x-1} = \frac{6(x-1)}{(x+2)(x-1)}$. Now, let's address the second fraction: $\frac{7}{x-1} * \frac{x+2}{x+2} = \frac{7(x+2)}{(x-1)(x+2)}$. Notice that both fractions now have the same denominator: $(x+2)(x-1)$.

We're making progress, right? We've successfully transformed our original fractions into equivalent fractions with a common denominator. This is a massive step forward! It sets the stage for the next phase, where we can finally add the numerators together. Think of it as preparing your ingredients before you start cooking. You wouldn't throw a bunch of random ingredients into a pot without knowing what you're making! We now know what we're making and how to add them. Remember to multiply both the numerator and the denominator by the factor missing from the original denominator. Now, the fractions look different, but they are equal in value; they are just written in a different form. You could say, we're remodeling them.

Combining and Simplifying

Now that we have the same denominator for both fractions, we can combine them! Our expression now looks like this: $\frac{6(x-1)}{(x+2)(x-1)} + \frac{7(x+2)}{(x-1)(x+2)}$. Since they have the same denominator, we can simply add the numerators and keep the common denominator. So, our combined fraction will be: $\frac{6(x-1) + 7(x+2)}{(x+2)(x-1)}$. See? It's all coming together! But we're not quite done; we need to simplify it even further.

The next step is to simplify the numerator. Let's distribute those numbers, shall we? We'll multiply the 6 by both terms inside the first set of parentheses, and the 7 by both terms inside the second set. So, $6(x-1)$ becomes $6x - 6$, and $7(x+2)$ becomes $7x + 14$. Now, the numerator looks like this: $6x - 6 + 7x + 14$.

Time to combine like terms! We have $6x$ and $7x$, which add up to $13x$. We also have $-6$ and $+14$, which combine to give us $+8$. So, our simplified numerator is $13x + 8$. Our combined fraction now looks like this: $\frac{13x + 8}{(x+2)(x-1)}$. We're on the home stretch now!

The Final Steps: Expanded Denominator and Simplification

Okay, team, we're in the final stretch! Our current expression is: $\frac{13x + 8}{(x+2)(x-1)}$. Now, let's take a look at the denominator. Often, you'll need to expand it and sometimes simplify further, but sometimes, like in our case, it's already in its simplest form. Let's do it and see what happens.

We'll multiply $(x+2)$ and $(x-1)$. This gives us $x^2 - x + 2x - 2$, which simplifies to $x^2 + x - 2$. So, our expression becomes $\frac{13x + 8}{x^2 + x - 2}$. It's tempting to think we can simplify this further, but we can't. The numerator, $13x + 8$, doesn't share any common factors with the denominator, $x^2 + x - 2$. There's no further simplification to be done here.

So, our final answer, the simplified form of $\frac{6}{x+2}+\frac{7}{x-1}$, is $\frac{13x + 8}{x^2 + x - 2}$. And there you have it, folks! We've successfully added and simplified our rational expressions. We've conquered those fractions, understood the process, and now you have the tools to handle similar problems with confidence. Keep practicing, and you'll be a pro in no time! Remember, the key is to find that common denominator, rewrite the fractions, combine, and simplify. You've got this!

Congratulations! You have completed the addition of rational expressions, and hopefully, you understood the process. Keep practicing, and you'll be a pro in no time! Remember, the key is to find that common denominator, rewrite the fractions, combine, and simplify. You've got this!