Simplifying (x-4)/(x+2) - 6/(x+3): A Step-by-Step Guide

Hey guys! Today, we're diving into a common algebra problem: simplifying rational expressions. Specifically, we're going to tackle the expression (x-4)/(x+2) - 6/(x+3). Don't worry if it looks intimidating at first. We'll break it down step-by-step so you can master these types of problems. Think of it like this: we're not just solving an equation; we're building a foundation for more advanced math. Understanding how to simplify expressions like this one is crucial for calculus, pre-calculus, and even some real-world applications. So, let's grab our mathematical toolkits and get started!

Understanding Rational Expressions

Before we jump into the nitty-gritty, let's take a moment to understand what we're dealing with. A rational expression is simply a fraction where the numerator and denominator are polynomials. In our case, we have two rational expressions: (x-4)/(x+2) and 6/(x+3). The key to simplifying these expressions, especially when they are being added or subtracted, is to find a common denominator. This is just like working with regular fractions – you can't add or subtract them unless they have the same denominator. Finding this common denominator will be our first major step, so get ready to flex those fraction muscles!

Why Common Denominators Matter

Imagine trying to add 1/2 and 1/3 directly. You can't, right? You need to rewrite them with a common denominator (which is 6 in this case) to get 3/6 + 2/6. The same principle applies to rational expressions. The common denominator allows us to combine the fractions into a single, simplified expression. This not only makes the expression easier to work with but also helps in solving equations and analyzing functions. In the context of our problem, finding the common denominator for (x-4)/(x+2) and 6/(x+3) is the critical first step towards simplification. Understanding this fundamental principle is vital for any aspiring math whiz!

Step 1: Finding the Common Denominator

The denominators we have are (x+2) and (x+3). To find the least common denominator (LCD), we need to consider the factors of each denominator. In this case, the denominators don't share any common factors, so the LCD is simply the product of the two: (x+2)(x+3). This is a common scenario when dealing with linear expressions in the denominator. Think of it like finding the least common multiple (LCM) for numbers. If you have numbers like 3 and 5, the LCM is their product, 15. Similarly, for expressions like (x+2) and (x+3), we multiply them together to find the LCD. Once we have our LCD, the next step is to rewrite each fraction with this new denominator. This involves multiplying both the numerator and denominator of each fraction by the appropriate factor.

Rewriting the Fractions

Now, we need to rewrite each fraction with the common denominator (x+2)(x+3). For the first fraction, (x-4)/(x+2), we multiply both the numerator and denominator by (x+3). This gives us [(x-4)(x+3)] / [(x+2)(x+3)]. For the second fraction, 6/(x+3), we multiply both the numerator and denominator by (x+2), resulting in [6(x+2)] / [(x+3)(x+2)]. Notice that we're not changing the value of the fractions; we're just expressing them in an equivalent form with the common denominator. This manipulation is essential because it allows us to combine the fractions in the next step. Remember, multiplying both the top and bottom of a fraction by the same thing is like multiplying by 1 – it doesn't change the overall value.

Step 2: Combining the Fractions

With both fractions now sharing the common denominator (x+2)(x+3), we can combine them. This involves subtracting the numerators while keeping the denominator the same. So, we have [(x-4)(x+3) - 6(x+2)] / [(x+2)(x+3)]. Now, the fun part begins: expanding and simplifying the numerator. We'll need to use the distributive property (or the FOIL method) to multiply out the terms in the numerator. This step is crucial because it sets the stage for combining like terms and simplifying the entire expression. Mistakes in this expansion can throw off the entire solution, so it's important to be careful and methodical.

Expanding the Numerator

Let's expand the numerator step-by-step. First, we expand (x-4)(x+3) using the FOIL method (First, Outer, Inner, Last): xx + 3x - 4x - 43, which simplifies to x^2 - x - 12. Next, we expand 6(x+2): 6x + 62, which simplifies to 6x + 12. Now, we substitute these expanded forms back into the numerator: (x^2 - x - 12) - (6x + 12). Remember the subtraction sign! It's crucial to distribute it correctly across the second expression. This is a common area for errors, so pay close attention to the signs. With the expansions done, we're ready to combine like terms and further simplify the numerator.

Step 3: Simplifying the Numerator

Now, let's simplify the numerator: (x^2 - x - 12) - (6x + 12). Distributing the negative sign, we get x^2 - x - 12 - 6x - 12. Combining like terms, we have x^2 - 7x - 24. This quadratic expression is the simplified form of the numerator. Next, we'll need to see if we can factor this quadratic. Factoring can help us identify common factors with the denominator, which we can then cancel to further simplify the expression. Keep in mind that not all quadratics can be factored easily, but it's always worth checking. Factoring is a powerful tool in simplifying rational expressions and can often lead to a more elegant and understandable result.

Factoring the Quadratic (if possible)

We now have the quadratic expression x^2 - 7x - 24. Let's try to factor it. We're looking for two numbers that multiply to -24 and add up to -7. After a little thought, we find that -8 and 3 fit the bill. So, we can factor the quadratic as (x - 8)(x + 3). This is a key step because it allows us to see if there are any common factors with the denominator. If we find common factors, we can cancel them out, simplifying the expression even further. In our case, we do indeed have a common factor!

Step 4: Simplifying the Entire Expression

Our expression now looks like [(x - 8)(x + 3)] / [(x + 2)(x + 3)]. Notice that we have a common factor of (x + 3) in both the numerator and the denominator. We can cancel these out, which simplifies the expression to (x - 8) / (x + 2). This is the simplified form of the original expression! We've gone from a somewhat complex expression with fractions to a much cleaner and easier-to-understand form. Remember, the goal of simplifying is to make expressions more manageable and easier to work with in future calculations. The cancellation of common factors is a fundamental technique in simplifying rational expressions, so make sure you're comfortable with this step.

Final Simplified Expression

Therefore, the simplified form of (x-4)/(x+2) - 6/(x+3) is (x - 8) / (x + 2). We've successfully navigated the process of finding a common denominator, combining fractions, expanding and simplifying the numerator, factoring, and canceling common factors. This result is not just an answer; it's a demonstration of your understanding of rational expressions and algebraic manipulation. Keep practicing these types of problems, and you'll become a master of simplification in no time!

Key Takeaways

Let's recap the key steps we took to simplify this expression. First, we found the common denominator. Then, we rewrote the fractions with this common denominator and combined them. Next, we expanded and simplified the numerator. We then factored the numerator (when possible) and canceled out any common factors with the denominator. Finally, we arrived at our simplified expression. These steps form a general strategy for simplifying rational expressions, so keep them in mind as you tackle future problems. Remember, practice makes perfect, so don't be afraid to work through lots of examples. Each problem you solve will solidify your understanding and make you a more confident mathematician.

Tips for Success

• Always look for a common denominator first: This is the foundation for adding and subtracting rational expressions.
• Be careful with signs: Distributing negative signs correctly is crucial.
• Factor whenever possible: Factoring helps identify common factors that can be canceled.
• Double-check your work: It's easy to make small mistakes, so always review your steps.
• Practice, practice, practice! The more problems you solve, the better you'll become.

By mastering these techniques, you'll be well-equipped to handle a wide range of algebraic problems involving rational expressions. Keep up the great work, and happy simplifying!