Subtracting Rational Expressions: A Step-by-Step Guide

Subtracting rational expressions might seem daunting at first, but don't worry, guys! It's all about finding a common denominator and then carefully combining the numerators. In this article, we'll break down the process step by step, using the example of subtracting $\frac{x}{8x-7}$ and $\frac{x+2}{9x}$. By the end, you'll be subtracting rational expressions like a pro.

Understanding Rational Expressions

Before diving into the subtraction, let's quickly recap what rational expressions are. A rational expression is simply a fraction where the numerator and the denominator are polynomials. Think of them as algebraic fractions. For instance, $\frac{x}{8x-7}$ and $\frac{x+2}{9x}$ are both rational expressions. The key to working with them is remembering the rules of fraction arithmetic, which we'll apply with a bit of algebraic flair. When dealing with rational expressions, always consider potential restrictions on the variable (x in our case). Restrictions occur when the denominator equals zero, as division by zero is undefined. In our example, $8x - 7 \neq 0$ and $9x \neq 0$, meaning $x \neq \frac{7}{8}$ and $x \neq 0$. These restrictions ensure that our expressions remain mathematically valid. When subtracting rational expressions, we aim to combine them into a single fraction. This involves finding a common denominator, adjusting the numerators accordingly, and then simplifying the resulting expression. The process is analogous to subtracting regular numerical fractions, but with the added complexity of algebraic terms. Therefore, understanding the underlying principles of fraction arithmetic is crucial for mastering rational expression subtraction. Always double-check your work, especially when distributing negative signs, to avoid common errors. Furthermore, make sure to simplify your final answer by factoring and canceling any common factors between the numerator and the denominator.

Finding the Least Common Denominator (LCD)

The most crucial step in subtracting fractions, whether they're numerical or algebraic, is finding the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. In our case, the denominators are $8x-7$ and $9x$. Since these expressions don't share any common factors, the LCD is simply their product: $9x(8x-7)$. Finding the LCD is like setting the stage for a smooth subtraction. It ensures that both fractions have the same 'size' of pieces, allowing us to directly compare and combine them. Without a common denominator, we'd be trying to add apples and oranges, which, as you know, doesn't work! There are a few methods to find the LCD, but for expressions like these, multiplying them together is often the most straightforward approach. However, always double-check to see if the denominators share any factors; if they do, you can simplify the LCD by only including those factors once. For example, if we were subtracting fractions with denominators $x(x+1)$ and $(x+1)(x+2)$, the LCD would be $x(x+1)(x+2)$, not $x(x+1)^2(x+2)$. Once you've found the LCD, you're halfway there! The next step is to rewrite each fraction with this new denominator, which we'll cover in the next section. Remember, accuracy is key when finding the LCD; a mistake here will propagate through the rest of the problem. So, take your time and double-check your work. With practice, finding the LCD will become second nature, making subtracting rational expressions a breeze.

Rewriting the Fractions with the LCD

Now that we've found the LCD, which is $9x(8x-7)$, we need to rewrite each fraction with this new denominator. To do this, we multiply the numerator and denominator of each fraction by the factor that makes its denominator equal to the LCD. For the first fraction, $\frac{x}{8x-7}$, we need to multiply both the numerator and the denominator by $9x$: $\frac{x}{8x-7} \cdot \frac{9x}{9x} = \frac{9x^2}{9x(8x-7)}$. For the second fraction, $\frac{x+2}{9x}$, we need to multiply both the numerator and the denominator by $8x-7$: $\frac{x+2}{9x} \cdot \frac{8x-7}{8x-7} = \frac{(x+2)(8x-7)}{9x(8x-7)}$. See what we did there? We're essentially multiplying each fraction by a form of '1', which doesn't change its value but allows us to combine them easily. Think of it like converting fractions to have a common unit before adding or subtracting them. This step is crucial because it ensures that we're working with equivalent fractions that can be directly combined. Accuracy is paramount here; double-check that you're multiplying both the numerator and the denominator by the correct factor. A common mistake is to only multiply the denominator, which changes the value of the fraction. Once you've rewritten both fractions with the LCD, you're ready to combine them. The next step is to subtract the numerators, which we'll cover in the following section. Remember, the goal is to create a single fraction with the LCD as the denominator and the combined numerators as the numerator. With practice, rewriting fractions with a common denominator will become a routine task, making subtracting rational expressions much easier.

Subtracting the Numerators

With both fractions now having the common denominator $9x(8x-7)$, we can subtract the numerators. This means we'll perform the operation: $\frac{9x^2}{9x(8x-7)} - \frac{(x+2)(8x-7)}{9x(8x-7)}$. Combine the numerators over the common denominator: $\frac{9x^2 - (x+2)(8x-7)}{9x(8x-7)}$. Now, we need to simplify the numerator by expanding the product $(x+2)(8x-7)$ and then subtracting it from $9x^2$. First, let's expand $(x+2)(8x-7)$: $(x+2)(8x-7) = 8x^2 - 7x + 16x - 14 = 8x^2 + 9x - 14$. Now, subtract this from $9x^2$: $9x^2 - (8x^2 + 9x - 14) = 9x^2 - 8x^2 - 9x + 14 = x^2 - 9x + 14$. So, our expression becomes: $\frac{x^2 - 9x + 14}{9x(8x-7)}$. Remember the importance of distributing the negative sign correctly when subtracting polynomials. A common mistake is to only subtract the first term and forget to change the signs of the other terms. This can lead to an incorrect result. Once you've simplified the numerator, the next step is to factor it, if possible. Factoring can help you identify common factors between the numerator and the denominator, which can be canceled to simplify the expression further. In our case, we can factor the numerator, which we'll do in the next section. So, keep your eyes peeled and stay focused on the details; subtracting rational expressions is all about precision and accuracy.

Simplifying the Result

After subtracting the numerators, we have the expression $\frac{x^2 - 9x + 14}{9x(8x-7)}$. Now, let's see if we can simplify this further. To do this, we'll try to factor the numerator, $x^2 - 9x + 14$. We're looking for two numbers that multiply to 14 and add up to -9. Those numbers are -2 and -7. So, we can factor the numerator as $(x-2)(x-7)$. Our expression now looks like this: $\frac{(x-2)(x-7)}{9x(8x-7)}$. Now, we check if there are any common factors between the numerator and the denominator that we can cancel out. In this case, there are no common factors. The numerator has factors of $(x-2)$ and $(x-7)$, while the denominator has factors of $9x$ and $(8x-7)$. None of these match, so we can't simplify the expression any further. Therefore, the simplified result is: $\frac{(x-2)(x-7)}{9x(8x-7)}$. And that's it! We've successfully subtracted the rational expressions and simplified the result. Remember, simplifying is a crucial step in any algebraic problem. It ensures that you're presenting your answer in the most concise and understandable form. Always look for opportunities to factor and cancel common factors; it can make a big difference. With practice, you'll become a master at simplifying rational expressions. So, keep practicing and don't be afraid to make mistakes; they're part of the learning process. With dedication and perseverance, you'll conquer even the most challenging algebraic problems.

Final Answer

Therefore, $\frac{x}{8x-7}-\frac{x+2}{9x} = \frac{(x-2)(x-7)}{9x(8x-7)}$.

Subtracting rational expressions involves several key steps: finding the LCD, rewriting the fractions with the LCD, subtracting the numerators, and simplifying the result. By following these steps carefully and paying attention to detail, you can master this important algebraic skill. Remember to always double-check your work and look for opportunities to simplify your answer. With practice, you'll become confident and proficient in subtracting rational expressions. So, keep practicing and don't give up! The world of algebra awaits your mastery!