Algebra Problem: Expressing An Expression As A Fraction

Hey guys! Let's dive into an algebra problem that many 8th graders might find a bit tricky. We're going to break down how to express the expression 3x + 1 - (3x² - 13x)/(x-4) as a single fraction. This involves some fundamental algebraic manipulations, and I'm here to guide you through each step. So, grab your pencils and notebooks, and let’s get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We have a mixed expression that includes a whole number part (3x + 1) and a fractional part (3x² - 13x)/(x-4). Our goal is to combine these into a single fraction. This means we need to find a common denominator, perform the necessary operations, and simplify the result. Sounds like fun, right? 😄

When you are tackling such problems, it's crucial to have a solid grasp of basic algebraic principles. These include understanding how to manipulate fractions, factor expressions, and simplify terms. Without these foundational skills, the problem may seem daunting. But don't worry! We'll take it one step at a time, and by the end of this article, you'll feel much more confident in your ability to handle similar algebraic challenges. Now, let’s get to the first step: finding that common denominator.

Step 1: Finding a Common Denominator

The first key step in combining these expressions is to find a common denominator. Remember, you can only add or subtract fractions if they have the same denominator. In our case, we have a whole number part (3x + 1) and a fractional part with a denominator of (x - 4). To combine them, we need to express 3x + 1 as a fraction with the same denominator.

So, how do we do that? Simple! We multiply (3x + 1) by (x - 4)/(x - 4). This might look a bit strange, but remember that multiplying by a fraction that equals 1 (in this case, (x - 4)/(x - 4)) doesn't change the value of the expression, only its form. This is a classic trick in algebra, and you’ll find it incredibly useful in many situations. Once we've done this, both parts of our expression will have the same denominator, and we can move on to combining them. Now, let's do the math:

(3x + 1) * (x - 4) / (x - 4)

This sets us up perfectly for the next step, which is to actually perform the multiplication and see what we get. Keep your eye on the prize – a single, simplified fraction! And remember, if you ever feel lost, just break the problem down into smaller, more manageable steps. That’s the secret to success in algebra (and, honestly, in life!). Let’s move on to step two and see how this multiplication plays out.

Step 2: Multiplying and Expanding

Now that we have a common denominator, let's multiply and expand the numerator of the whole number part. We have (3x + 1) * (x - 4). This requires us to use the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) to multiply each term in the first binomial by each term in the second binomial.

Let's break it down:

• First: 3x * x = 3x²
• Outer: 3x * -4 = -12x
• Inner: 1 * x = x
• Last: 1 * -4 = -4

Now, let’s combine these terms: 3x² - 12x + x - 4. We can simplify this further by combining the -12x and x terms, giving us 3x² - 11x - 4. So, the whole number part of our expression now looks like (3x² - 11x - 4) / (x - 4). See how we're slowly but surely transforming the original expression into something more manageable? This is the beauty of algebra – taking complex problems and breaking them down into simpler components.

With this expansion done, we're ready to combine this with the fractional part we had originally. Remember, the original problem was 3x + 1 - (3x² - 13x)/(x-4). We've just rewritten 3x + 1 as a fraction with the same denominator. Now we can subtract the two fractions. Are you feeling confident? You should be! Let's move on to the next step and actually perform the subtraction.

Step 3: Combining the Fractions

Alright, we're getting closer to our goal! We've rewritten the whole number part as a fraction with the same denominator as the other fraction. Now it's time to combine them. Remember, our expression looks like this:

(3x² - 11x - 4) / (x - 4) - (3x² - 13x) / (x - 4)

Since the denominators are the same, we can simply subtract the numerators. Be super careful with the signs here, guys! Subtracting a whole expression means we need to distribute the negative sign to each term in the second numerator.

So, we have:

3x² - 11x - 4 - (3x² - 13x)

Distributing the negative sign, we get:

3x² - 11x - 4 - 3x² + 13x

Now, let's combine like terms. We have 3x² and -3x², which cancel each other out. Then we have -11x and +13x, which combine to give 2x. And finally, we have the constant term -4. So, the numerator simplifies to 2x - 4.

Our expression now looks like (2x - 4) / (x - 4). We're almost there! The last step is to see if we can simplify this fraction any further. This usually involves factoring, which is what we'll tackle next. Stay with me – you're doing great!

Step 4: Simplifying the Fraction

We've reached the final step: simplifying the fraction. Our expression currently looks like this: (2x - 4) / (x - 4). To simplify, we need to see if we can factor anything out of the numerator or the denominator and then cancel out any common factors.

Let's look at the numerator, 2x - 4. Notice that both terms have a common factor of 2. We can factor out the 2, which gives us 2(x - 2). The denominator, x - 4, doesn't have any common factors that we can easily pull out.

So, our expression now looks like this:

2(x - 2) / (x - 4)

Now, we check if there are any common factors in the numerator and the denominator that we can cancel out. In this case, there aren't any. The term (x - 2) is different from (x - 4), so we can't cancel them.

This means that our simplified fraction is 2(x - 2) / (x - 4). We've successfully expressed the original expression as a single fraction and simplified it as much as possible. How cool is that?

Final Answer

So, after all that algebraic maneuvering, we've arrived at the final answer. The expression 3x + 1 - (3x² - 13x)/(x-4) can be expressed as the fraction:

2(x - 2) / (x - 4)

Isn't it satisfying to solve a complex problem like this? By breaking it down into smaller steps and carefully applying the rules of algebra, we were able to navigate through the problem and find the solution. Remember, algebra is all about practice, so the more you work on problems like this, the more comfortable and confident you'll become. Great job, everyone! If you tackled this problem with me, give yourself a pat on the back. 🎉

Key Takeaways

Before we wrap up, let's quickly recap the key steps we took to solve this problem. This will help reinforce what we've learned and give you a handy checklist for tackling similar problems in the future. Remember, the goal isn't just to get the right answer, but to understand the process and build your problem-solving skills.

1. Find a Common Denominator: This is crucial for adding or subtracting fractions. We multiplied the whole number part by a fraction equal to 1 to get the same denominator as the fractional part.
2. Multiply and Expand: We used the distributive property (FOIL) to expand the product of binomials in the numerator.
3. Combine the Fractions: We subtracted the numerators, being careful to distribute the negative sign correctly.
4. Simplify the Fraction: We factored the numerator to see if we could cancel out any common factors with the denominator.

By following these steps, you can confidently tackle similar algebraic problems. And remember, if you get stuck, don't hesitate to break the problem down into smaller pieces and tackle each one individually. You've got this!

Practice Makes Perfect

If you really want to master these kinds of algebraic manipulations, practice is key. Try working through similar problems, and don't be afraid to make mistakes – that's how we learn! Look for additional examples in your textbook or online, and challenge yourself to apply the steps we've discussed today.

Here are a couple of similar problems you might want to try:

1. Express 2x - 3 + (x² + 5x)/(x + 2) as a single fraction.
2. Simplify (4x² - 9)/(2x + 3) - x + 1.

Working through these examples will help solidify your understanding and build your confidence. And if you get stuck, remember the steps we've outlined, and don't be afraid to ask for help. Algebra can be challenging, but it's also incredibly rewarding when you crack a tough problem. Keep up the great work, guys, and I'll see you in the next algebra adventure! 🚀