Adding Rational Expressions: A Step-by-Step Guide

Hey guys! Today, we're going to tackle adding rational expressions. Specifically, we'll walk through how to add $\frac{3}{2x} + \frac{2}{x-5}$. Adding rational expressions might seem daunting at first, but don't worry, we'll break it down into manageable steps. The key to success here is finding a common denominator, and once we've got that, it's smooth sailing. We’ll go through each step meticulously, ensuring that you understand not just how to solve this particular problem, but also the general principles involved. This understanding will equip you to tackle a wide array of similar problems with confidence. Remember, practice makes perfect, so the more you work through these types of problems, the easier they will become. So, grab your pencils and let's dive in!

Finding the Common Denominator

The first crucial step in adding rational expressions is to find a common denominator. Think of it like adding fractions with different denominators – you can't directly add them until they share the same denominator. In our case, we have the expressions $\frac{3}{2x}$ and $\frac{2}{x-5}$. Our denominators are $2x$ and $(x-5)$. To find the least common denominator (LCD), we need to identify all the unique factors in each denominator and then multiply them together. In this example, the factors are $2$, $x$, and $(x-5)$. Since none of these factors are shared between the two denominators in their current form, the least common denominator is simply the product of all of them. Thus, the LCD is $2x(x-5)$. Now that we have our LCD, we need to rewrite each fraction so that it has this denominator. This involves multiplying the numerator and denominator of each fraction by whatever factor is needed to obtain the LCD. This process ensures that we're not changing the value of the original fractions, only their appearance. By ensuring both fractions have the same denominator, we set the stage for the straightforward addition of the numerators in the subsequent step.

Rewriting the Fractions

Now that we've identified our common denominator as $2x(x-5)$, let's rewrite each fraction with this denominator. For the first fraction, $\frac{3}{2x}$, we need to multiply both the numerator and the denominator by $(x-5)$. This gives us $\frac{3(x-5)}{2x(x-5)}$. Expanding the numerator, we get $\frac{3x - 15}{2x(x-5)}$. For the second fraction, $\frac{2}{x-5}$, we need to multiply both the numerator and the denominator by $2x$. This results in $\frac{2(2x)}{2x(x-5)}$, which simplifies to $\frac{4x}{2x(x-5)}$. Now, both fractions have the same denominator, $2x(x-5)$. This is a crucial step because it allows us to directly add the numerators while keeping the denominator the same. By ensuring that both fractions are expressed with the common denominator, we can accurately combine them into a single fraction. Remember to double-check your work at this stage to avoid any arithmetic errors, as these can propagate through the rest of the solution. This careful attention to detail will help ensure you arrive at the correct final answer.

Adding the Numerators

With both fractions now sharing the common denominator $2x(x-5)$, we can proceed to add the numerators. We have $\frac{3x - 15}{2x(x-5)} + \frac{4x}{2x(x-5)}$. To add these, we simply add the numerators together while keeping the denominator the same: $\frac{(3x - 15) + 4x}{2x(x-5)}$. Now, we combine like terms in the numerator. We have $3x$ and $4x$, which add up to $7x$. So, the numerator becomes $7x - 15$. Thus, our expression simplifies to $\frac{7x - 15}{2x(x-5)}$. At this point, it’s important to check whether the numerator can be factored. If it can, there might be an opportunity to simplify the expression further by canceling out common factors between the numerator and the denominator. However, in this particular case, $7x - 15$ cannot be factored in a way that would allow us to simplify with the denominator $2x(x-5)$. Therefore, we can confidently proceed to the next step, knowing that we've done everything we can to simplify the expression at this stage.

Simplifying the Expression

After adding the numerators, we have $\frac{7x - 15}{2x(x-5)}$. Now, let's see if we can simplify this expression further. First, we can expand the denominator: $2x(x-5) = 2x^2 - 10x$. So, the expression becomes $\frac{7x - 15}{2x^2 - 10x}$. Now, we look for any common factors between the numerator and the denominator. The numerator is $7x - 15$, and the denominator is $2x^2 - 10x$. We can factor out $2x$ from the denominator to get $2x(x-5)$. Unfortunately, $7x - 15$ cannot be factored easily, and it doesn't share any common factors with $2x$ or $(x-5)$. Therefore, the expression is already in its simplest form. So, our final answer is $\frac{7x - 15}{2x(x-5)}$ or $\frac{7x - 15}{2x^2 - 10x}$. Remember, always double-check your work and make sure you've simplified the expression as much as possible.

Final Answer

Alright, guys! After carefully working through each step, we've arrived at our final answer. The simplified expression for $\frac{3}{2x} + \frac{2}{x-5}$ is $\frac{7x - 15}{2x(x-5)}$ or, equivalently, $\frac{7x - 15}{2x^2 - 10x}$. We found the common denominator, rewrote the fractions, added the numerators, and simplified the resulting expression. Each step was crucial to ensuring we arrived at the correct final form. Remember that the process of adding rational expressions involves finding a common denominator, rewriting the fractions with that denominator, adding the numerators, and then simplifying the result. While this may seem like a lot of steps, with practice, it becomes a straightforward procedure. Keep practicing with different examples, and you’ll become more confident in your ability to solve these types of problems. I hope this guide has been helpful! Keep up the great work, and happy solving!