Subtracting Fractions: A Step-by-Step Guide

Have you ever stared at a fraction problem and felt your brain do a little somersault? Don't worry, you're not alone! Fractions can seem intimidating, but with the right approach, they become much more manageable. In this article, we're going to break down a specific problem: finding the difference between the fractions 2/(x+10) and 3/(x+4). We'll walk through each step, explain the logic behind it, and by the end, you'll feel confident tackling similar problems. So, let's dive in and conquer those fractions together!

Understanding the Basics: Why Do We Need a Common Denominator?

Before we jump into the problem, let's quickly recap why we need a common denominator when adding or subtracting fractions. Imagine you have two slices of pizza. One is cut into 8 slices, and you have 2 of those slices (2/8 of the pizza). The other pizza is cut into 6 slices, and you have 3 of those (3/6 of the pizza). Can you easily tell how much pizza you have in total just by looking at the fractions 2/8 and 3/6? Probably not, right?

The slices are different sizes! To accurately add them, we need to express them in terms of the same size slices – that's where the common denominator comes in. It's like finding a common unit of measurement. We need to find a number that both denominators (the bottom numbers of the fractions) divide into evenly. This allows us to rewrite the fractions with the same denominator, making it easy to add or subtract the numerators (the top numbers).

In our pizza example, we could find a common denominator of 24 (both 8 and 6 divide into 24). We'd then convert 2/8 to 6/24 (multiply both numerator and denominator by 3) and 3/6 to 12/24 (multiply both numerator and denominator by 4). Now we can easily add: 6/24 + 12/24 = 18/24. See how much simpler it becomes with a common denominator? The same principle applies to algebraic fractions – we need that common ground to perform operations accurately. Remember, finding the common denominator is a critical first step in adding or subtracting fractions, whether they are simple numerical fractions or more complex algebraic expressions. This foundational understanding will make the rest of the process much smoother.

Step-by-Step Solution: Finding the Difference

Okay, let's get back to our original problem: finding the difference between 2/(x+10) and 3/(x+4). This means we need to subtract the second fraction from the first: $rac{2}{x+10} - rac{3}{x+4}$. Remember our earlier discussion about common denominators? That's our first hurdle here. The denominators are (x+10) and (x+4), which are different algebraic expressions. Just like we needed a common denominator for numerical fractions, we need one here too.

1. Finding the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that both denominators divide into evenly. In this case, since (x+10) and (x+4) don't share any common factors, the LCD is simply their product: (x+10)(x+4). Think of it like this: if you had denominators of 3 and 5, the LCD would be 3 * 5 = 15. It's the same idea with algebraic expressions.

2. Rewriting the Fractions with the LCD

Now we need to rewrite each fraction with the LCD as its denominator. This involves multiplying both the numerator and denominator of each fraction by the appropriate factor. For the first fraction, 2/(x+10), we need to multiply both the top and bottom by (x+4):

$$\frac{2}{x+10} * \frac{x+4}{x+4} = \frac{2(x+4)}{(x+10)(x+4)}$$

Notice that we're essentially multiplying by 1, so we're not changing the value of the fraction, just its form. For the second fraction, 3/(x+4), we need to multiply both the top and bottom by (x+10):

$$\frac{3}{x+4} * \frac{x+10}{x+10} = \frac{3(x+10)}{(x+4)(x+10)}$$

Now both fractions have the same denominator, (x+10)(x+4), which is our LCD! This is a crucial step – we've created a level playing field for our subtraction.

3. Performing the Subtraction

With a common denominator in place, we can now subtract the fractions. We subtract the numerators and keep the denominator the same:

$$\frac{2(x+4)}{(x+10)(x+4)} - \frac{3(x+10)}{(x+4)(x+10)} = \frac{2(x+4) - 3(x+10)}{(x+10)(x+4)}$$

This step is where we actually combine the fractions into one. Now we have a single fraction with a potentially messy numerator, but don't worry, we'll simplify it next.

4. Simplifying the Numerator

The next step is to simplify the numerator by distributing and combining like terms. Let's expand the numerator:

$$2(x+4) - 3(x+10) = 2x + 8 - 3x - 30$$

Now, combine the 'x' terms and the constant terms:

$$2x - 3x + 8 - 30 = -x - 22$$

So, our simplified numerator is -x - 22. Remember, simplifying the numerator is a key part of solving these types of problems. It helps us get to the most concise and understandable form of the answer.

5. Writing the Simplified Fraction

Now we can rewrite the entire fraction with the simplified numerator:

$$\frac{-x - 22}{(x+10)(x+4)}$$

This is a perfectly valid answer, but we can often go one step further by factoring out a -1 from the numerator to make it look a little cleaner:

$$\frac{-1(x + 22)}{(x+10)(x+4)}$$

While this is technically equivalent, it's often considered more aesthetically pleasing and can sometimes make further simplification easier if needed. This final step of writing the simplified fraction is where all our hard work comes together, giving us a clear and concise solution.

Alternative Forms and Further Simplification

Okay, we've arrived at the simplified fraction: $\frac{-1(x + 22)}{(x+10)(x+4)}$ or $\frac{-x - 22}{(x+10)(x+4)}$. But, are we completely done? Sometimes, there are alternative ways to express the answer, and it's good to be aware of them. Plus, there might be opportunities for further simplification that we haven't explored yet.

Expanding the Denominator (Sometimes)

In some cases, it might be beneficial to expand the denominator. While we left it in factored form, (x+10)(x+4), we could multiply it out:

$$(x+10)(x+4) = x^2 + 4x + 10x + 40 = x^2 + 14x + 40$$

This would give us an alternative form of the answer:

$$\frac{-x - 22}{x^2 + 14x + 40}$$

So, when would you choose to expand the denominator? It really depends on the context of the problem. Sometimes, expanding the denominator can reveal patterns or allow for further simplification, especially if you're dealing with more complex expressions later on. However, in many cases, leaving the denominator in factored form is perfectly acceptable and can even be preferable, as it makes it easier to identify potential restrictions on the variable 'x' (we'll touch on that in a bit).

Checking for Common Factors

Before we declare victory, we should always double-check if there are any common factors between the numerator and the denominator. This is like the final polish on our solution. In our case, the numerator is -1(x + 22), and the denominator is (x+10)(x+4) or x^2 + 14x + 40. Do you see any factors that they share? Nope! There's no way to simplify this fraction further by canceling out common factors. If we did find a common factor, we would divide both the numerator and denominator by that factor to get the most simplified form.

Identifying Restrictions on x

This is a crucial point that often gets overlooked: we need to consider any values of 'x' that would make our original expression undefined. Remember, we can't divide by zero! So, we need to look at our denominators and figure out what values of 'x' would make them equal to zero. In our original problem, we had denominators of (x+10) and (x+4). Setting each of these equal to zero, we get:

x + 10 = 0 => x = -10 x + 4 = 0 => x = -4

This means that x cannot be -10 or -4. If it were, we'd be dividing by zero, which is a big no-no in mathematics. We often express this by saying