Adding Rational Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of rational expressions and figuring out how to add them. Specifically, we'll be tackling the expression $\frac{x}{4}+\frac{3}{5 x}$. Don't worry, it might seem a bit daunting at first, but I promise it's totally manageable. We'll break it down step by step to make sure everyone understands the process. So, grab your pencils, and let's get started!

Understanding Rational Expressions

Before we jump into the addition, let's quickly recap what rational expressions are. Essentially, they're fractions where the numerator and/or the denominator are polynomials. Remember, a polynomial is an expression made up of variables and coefficients, combined using addition, subtraction, and multiplication. Think of things like x, x², 3x + 2, etc. When we have a polynomial over another polynomial, we get a rational expression. In our case, $\frac{x}{4}$ and $\frac{3}{5 x}$ are rational expressions.

Adding rational expressions is similar to adding regular fractions, but with a slight twist. The key is to find a common denominator. This is the foundation of our operation. The common denominator allows us to combine the numerators. Once we find this, we can move forward. When we have a common denominator, we can add the numerators. In simpler terms, we must rewrite each fraction so that the bottom number (the denominator) is the same for both. This will allow us to combine the fractions easily.

Now, let's get into the specifics of our example, $\frac{x}{4}+\frac{3}{5 x}$. The goal is to make the denominators identical, so we can put our thinking caps on. The denominators are 4 and 5x. To get a common denominator, we need to find the least common multiple (LCM) of 4 and 5x. Do you remember how to do it? It's like a puzzle.

Finding the Common Denominator

The first step to adding any fractions is finding the common denominator. For our expression, $\frac{x}{4}+\frac{3}{5 x}$, the denominators are 4 and 5x. The common denominator has to be divisible by both 4 and 5x. So, how do we find this common denominator? We're going to think about the multiples of each denominator, so let's start with 4. The multiples of 4 are 4, 8, 12, 16, 20, 24... And the multiples of 5x are 5x, 10x, 15x, 20x, 25x... Notice something? We need the least common multiple of both. The smallest expression that appears in both lists is 20x. Therefore, the common denominator is 20x.

Now, you might be thinking, "Why 20x? Why not just multiply 4 and 5x together to get 20x?" Well, you are right. We could multiply 4 and 5x together to get 20x. However, by finding the least common multiple, we keep the numbers and the expressions in the numerator as simple as possible. It makes the entire process a little bit cleaner. It will help us avoid larger numbers later in the calculation. You could also multiply the two denominators together to get the common denominator, but using the least common denominator just simplifies things and often makes the calculation easier.

So, remember this important step: Always find the least common multiple to get the common denominator. Now that we have that figured out, we can get to the next step.

Transforming the Fractions

Okay, now that we've found our common denominator (20x), we need to rewrite each fraction with this new denominator. This is the part where we're going to change the fractions, but not their actual values. To do this, we'll multiply both the numerator and the denominator of each fraction by the appropriate factor.

Let's start with the first fraction, $\frac{x}{4}$. To get a denominator of 20x, we need to multiply the original denominator (4) by 5x. But remember, whatever we do to the bottom, we must do to the top! So, we'll also multiply the numerator (x) by 5x. This gives us $\frac{x * 5x}{4 * 5x}$, which simplifies to $\frac{5x^2}{20x}$. See how the bottom became 20x?

Now, let's move on to the second fraction, $\frac{3}{5 x}$. To get a denominator of 20x, we need to multiply the original denominator (5x) by 4. So, we'll also multiply the numerator (3) by 4. This gives us $\frac{3 * 4}{5x * 4}$, which simplifies to $\frac{12}{20x}$. Again, the bottom becomes 20x. Do you see a pattern? We multiply both the top and the bottom by the same number, and that doesn't change the value of the fraction.

So, now we have two new fractions: $\frac{5x^2}{20x}$ and $\frac{12}{20x}$. They have the same denominator, which means we can move to the next exciting step!

Adding the Numerators

Alright, we're almost there! We've found the common denominator, and we've rewritten our fractions with that common denominator. Now, the fun part: adding the numerators. Since both fractions now have the same denominator (20x), we can simply add their numerators and keep the common denominator. It's like magic!

So, we take our two new fractions: $\frac{5x^2}{20x}$ and $\frac{12}{20x}$. We add their numerators (5x² and 12) and keep the common denominator (20x). This gives us $\frac{5x^2 + 12}{20x}$. Easy, right?

This is the core of adding rational expressions. Once the fractions share a common denominator, you're just adding the top parts and keeping the bottom part the same. That is the whole shebang. So, just remember that: common denominator first, then add the numerators.

The Final Answer

We did it, guys! We successfully added the rational expressions. The sum of $\frac{x}{4}+\frac{3}{5 x}$ is $\frac{5x^2 + 12}{20x}$. Therefore, the correct answer from the multiple-choice options is C. $\frac{5 x^2+12}{20 x}$.

Great job! You've learned how to add rational expressions. Remember the key steps: find the common denominator, transform the fractions, and add the numerators. Keep practicing, and you'll become a pro in no time! Keep in mind that understanding the concept of a common denominator is critical to a wide range of mathematical operations. It is not just limited to adding fractions.

Simplifying Your Answer

Great job in adding your rational expressions! Now that we have the sum, we should always consider the next step: simplifying our answer. It's like the final polish on a great piece of work. Simplifying is crucial because it helps you to present your answer in the most concise and accurate form. However, for our expression $\frac{5x^2 + 12}{20x}$, there isn't any simplification possible. Why? Because the numerator, 5x² + 12, cannot be factored any further. Also, there are no common factors between the numerator and the denominator (20x). If there were, we'd divide both the top and bottom by the common factor to make the expression simpler.

Simplifying expressions is a fundamental skill in algebra. It helps in solving equations, understanding the behavior of functions, and making it easier to work with different mathematical problems. It often involves factoring, canceling out common terms, and applying the rules of exponents. For example, if we had $\frac{2x + 4}{2}$, we could simplify it to x + 2 by dividing both terms in the numerator by 2. It's always a good habit to look for these simplification opportunities. In some cases, simplification might also help in identifying the domain restrictions for a rational expression. Therefore, while our final answer may not need simplification, always keep an eye out for potential simplifications.

Additional Examples

Let's go through some additional examples to help you solidify your understanding. Here are some examples to help you out.

• Example 1: $\frac{2}{x} + \frac{1}{x^2}$ - The common denominator is x². Therefore, the result would be $\frac{2x + 1}{x^2}$. First multiply the first term by x/x to achieve a common denominator.
• Example 2: $\frac{x+1}{3} + \frac{x-2}{6}$ - The common denominator is 6. We multiply the first fraction by 2/2, then add. The result is $\frac{2x+2+x-2}{6} = \frac{3x}{6} = \frac{x}{2}$.
• Example 3: $\frac{1}{x+1} + \frac{1}{x-1}$ - The common denominator is (x + 1)(x - 1). Therefore, the result would be $\frac{(x-1) + (x+1)}{(x+1)(x-1)} = \frac{2x}{x^2 - 1}$.

As you can see, the process stays the same: find the common denominator, transform the fractions, add the numerators. These examples show how the same principles can be applied to different expressions. Always practice, and you will become proficient in this process. When you practice, you understand that math is about patterns and processes. By mastering the fundamentals, you'll be well-equipped to handle more complex problems. Also, you will build your confidence.

Practice Makes Perfect

I hope that was a helpful guide. Adding rational expressions can seem intimidating at first, but with practice, you'll master it. Remember to always find the common denominator, rewrite the fractions, and then add the numerators. Don't worry if you don't get it right away. Math is about the journey, not just the destination. Keep practicing, and you'll find it becomes easier over time. Good luck with your studies, and keep up the great work! If you have any questions, feel free to ask. Keep learning and growing, and remember, you got this!