Simplify (x^2-7)/(5x-4) - (2x+9)/(5x-4) A Step-by-Step Guide

Hey guys! Today, we are diving into the world of algebraic expressions, specifically focusing on how to simplify rational expressions. We'll break down the problem “Which expression is equivalent to (x^2-7)/(5x-4) - (2x+9)/(5x-4) ?” step by step, ensuring you grasp the underlying concepts and can tackle similar problems with confidence. This topic is super important in algebra, and getting it right can make a huge difference in your math journey. So, let’s jump right in and make math a little less scary and a lot more fun!

The Importance of Simplifying Rational Expressions

Before we jump into the problem, let's chat about why simplifying rational expressions is such a big deal. Simplifying rational expressions is like decluttering your room – it makes things easier to manage and understand. In math, it allows us to see the core relationships between variables and constants without being distracted by unnecessary complexity. Imagine trying to build a house with a blueprint that's filled with extra lines and scribbles; it's going to be confusing, right? Simplifying expressions is like cleaning up that blueprint so you can see the essential structure clearly. Plus, it's a fundamental skill that you'll use in higher-level math courses like calculus, where clean, simple expressions are crucial for solving complex problems. So, mastering this skill now will set you up for success down the road. Think of it as laying a solid foundation for your future math adventures!

Moreover, simplifying expressions isn't just about making things look neater; it's about making them more useful. A simplified expression can help you solve equations more easily, identify patterns, and make predictions. For instance, in physics, you might use rational expressions to describe the motion of objects or the behavior of electrical circuits. Simplifying these expressions can reveal key insights and make calculations much more manageable. In computer science, simplified expressions can help optimize algorithms and improve the efficiency of code. So, whether you're dreaming of becoming an engineer, a scientist, or a software developer, the ability to simplify rational expressions is a valuable asset. It's like having a superpower that allows you to see through the complexity and get to the heart of the matter. Keep this in mind as we work through the problem – you're not just learning a math trick; you're developing a skill that will serve you well in many different fields.

Lastly, remember that simplifying rational expressions often involves combining several algebraic skills, such as factoring, distributing, and combining like terms. This means that by practicing simplification, you're also reinforcing these other important skills. It's like going to the gym and working out different muscle groups – you're not just getting stronger in one area, but building overall strength and fitness. So, as we tackle this problem, pay attention to how these different skills come together. Notice how factoring can help you cancel out common factors, how distributing helps you remove parentheses, and how combining like terms helps you tidy up the expression. By understanding these connections, you'll become a more versatile and confident problem solver. Math is like a puzzle, and simplifying expressions is like finding the right pieces to make the picture clear. So, let's get started and put those puzzle-solving skills to work!

Breaking Down the Problem

Let's start by revisiting our problem: (x^2-7)/(5x-4) - (2x+9)/(5x-4). The first thing you’ll notice, and this is super important, is that we are subtracting two fractions, and these fractions have the same denominator: (5x-4). This is a huge win for us because it makes the subtraction process much simpler. When you have fractions with the same denominator, you can combine them directly by subtracting the numerators while keeping the denominator the same. Think of it like slicing a pizza – if you have two slices from the same pizza (same denominator), you can easily figure out how much pizza you have left by combining the slices (numerators). So, in our case, we can rewrite the expression as a single fraction with the denominator (5x-4), and the numerator will be the difference between the original numerators.

Now, let’s zoom in on the numerators. We have (x^2-7) as the first numerator and (2x+9) as the second. The key here is to remember that we are subtracting the second numerator from the first. This means we need to be extra careful with the signs. It’s like dealing with temperatures – subtracting a negative temperature is like adding heat, and subtracting a positive temperature is like cooling things down. In math, subtracting a group of terms means we need to distribute the negative sign to each term in the group. So, we're not just subtracting 2x; we're also subtracting 9. This is a common spot where people make mistakes, so it’s worth taking a moment to double-check your work. Think of it like paying for groceries – you need to make sure you subtract the cost of each item to get the correct total. The same principle applies here: we need to subtract each term in the second numerator to get the correct result. By paying close attention to these details, we can avoid common pitfalls and arrive at the correct answer with confidence.

To make this even clearer, let’s write it out explicitly: our new numerator will be (x^2-7) - (2x+9). See how we've put the second numerator in parentheses? This is a great habit to get into because it reminds us that we need to distribute the negative sign. It’s like putting a fence around something you want to protect – the parentheses help us keep the terms together and ensure we handle them correctly. Now, the next step is to distribute that negative sign, which means we’ll be changing the signs of the terms inside the parentheses. So, +2x becomes -2x, and +9 becomes -9. This is a crucial step, and it’s where the magic happens. By distributing the negative sign correctly, we’re essentially “undoing” the parentheses and preparing the expression for further simplification. Think of it like unwrapping a present – once you’ve removed the wrapping paper (parentheses), you can see what’s inside and start playing with it. In this case, what’s “inside” is the individual terms that we can now combine and simplify. So, let’s move on to the next step and see how we can tidy things up and get to the final answer!

Step-by-Step Solution

Okay, guys, let's walk through the solution step by step. Remember, our expression is (x^2-7)/(5x-4) - (2x+9)/(5x-4). Since the denominators are the same, we can combine the numerators. This gives us our first crucial step:

Step 1: Combine the Numerators

We rewrite the expression as a single fraction: [(x^2-7) - (2x+9)] / (5x-4). This step is all about setting the stage for the real simplification. It's like gathering your ingredients before you start cooking – you need everything in one place to make the process smooth and efficient. By combining the numerators, we've essentially brought all the pieces of the puzzle together and prepared them for the next step. Now, we need to tackle the subtraction in the numerator, which means dealing with those parentheses.

Step 2: Distribute the Negative Sign

This is where we distribute the negative sign in the numerator: (x^2-7 - 2x - 9) / (5x-4). Remember, subtracting (2x+9) is the same as adding (-2x-9). This is a super common trick in algebra, and mastering it will save you a lot of headaches. It’s like knowing a shortcut on your commute – it makes the journey faster and less stressful. By distributing the negative sign, we've removed the parentheses and made the numerator a single, continuous expression. Now, we can move on to the next step, which is all about tidying things up and making them as simple as possible.

Step 3: Combine Like Terms

Now, we combine like terms in the numerator. We have x^2, -2x, -7, and -9. The like terms here are -7 and -9, which combine to -16. So, our expression becomes: (x^2 - 2x - 16) / (5x-4). This step is like organizing your closet – you're putting similar items together to make things easier to find. By combining like terms, we've simplified the numerator and made it more manageable. Now, we have a clean and concise expression that we can easily compare to the given options. This is the final touch, the cherry on top, that brings our solution to completion.

So, the equivalent expression is (x^2 - 2x - 16) / (5x-4). Hooray! We made it through the maze of algebra and emerged victorious. Remember, the key to success in math is to break down complex problems into smaller, more manageable steps. By taking things one step at a time and paying attention to the details, you can conquer even the most challenging equations. And with that, we've not only solved the problem but also reinforced some essential algebraic skills. Keep practicing, keep exploring, and keep having fun with math!

Analyzing the Options

Now that we've found the simplified expression, (x^2 - 2x - 16) / (5x-4), let's compare it to the given options and see which one matches. This is a crucial step in problem-solving because it ensures that we haven't made any silly mistakes along the way. It’s like proofreading your essay – you're giving your work a final check to catch any errors and make sure everything is perfect. So, let's put on our detective hats and carefully examine each option.

Option 1: (x^2-7+2x+9)/(5x-4)

This option incorrectly adds the numerators instead of subtracting them. Remember, the original problem involves subtraction, so we need to subtract the entire second numerator, not add it. This option is like trying to bake a cake with the wrong recipe – you might end up with something, but it won't be the delicious treat you were hoping for. In math, using the wrong operation can lead to completely different results, so it’s essential to pay close attention to the signs and operations involved. This option serves as a good reminder of the importance of careful reading and attention to detail. It's a classic example of how a small mistake can lead to a wrong answer, so always double-check your work!

Option 2: (x^2-7-2x-9)/(5x-4)

This option correctly subtracts the second numerator by distributing the negative sign. When we simplify the numerator, we get x^2 - 7 - 2x - 9, which combines to x^2 - 2x - 16. This perfectly matches our simplified expression. We have a winner! This option is like finding the missing piece of a puzzle – it fits perfectly and completes the picture. In math, finding the correct option is a satisfying moment, a validation of all the hard work and careful steps you've taken. This option demonstrates the importance of distributing the negative sign correctly and combining like terms accurately. It’s a testament to the power of following the rules and paying attention to the details.

Option 3: (x^2-7+2x-9)/(5x-4)

This option makes a mistake in distributing the negative sign. It adds 2x instead of subtracting it. This is another common error, highlighting the importance of carefully distributing the negative sign across all terms in the second numerator. This option is like getting a flat tire on your journey – it’s a setback, but it’s also a chance to learn and improve. In math, mistakes are opportunities for growth. By analyzing where we went wrong, we can strengthen our understanding and avoid similar errors in the future. This option serves as a reminder of the importance of being meticulous and double-checking each step.

Option 4: (x^2-7-2x+9)/(5x-4)

This option also makes a mistake in distributing the negative sign. It subtracts 2x correctly but adds 9 instead of subtracting it. This reinforces the need to be extra careful when distributing negative signs. This option is like a false alarm – it might seem right at first, but a closer look reveals the error. In math, it’s crucial to be skeptical and always verify your answers. This option highlights the importance of thoroughness and attention to detail. It’s a reminder that even if most of the steps are correct, a single mistake can lead to the wrong conclusion. So, always take the time to review your work and ensure everything is accurate.

By analyzing each option, we’ve not only confirmed that Option 2 is the correct answer but also reinforced the common mistakes to avoid. This process of elimination and comparison is a valuable skill in problem-solving. It’s like being a detective, piecing together the evidence and ruling out suspects until you find the culprit. In math, this approach can help you build confidence and accuracy. So, remember to always take the time to analyze the options and double-check your work. It’s the key to unlocking the correct answer and mastering the art of problem-solving!

Key Takeaways and Common Mistakes

Alright, guys, let's wrap up our discussion by highlighting some key takeaways and common mistakes to watch out for when simplifying rational expressions. This is like the post-game analysis – we're reviewing the highlights and learning from our experiences so we can play even better next time. So, let's dive in and make sure we've got all the important points covered.

Key Takeaways:

  • Same Denominator is Your Friend: When subtracting (or adding) rational expressions, having the same denominator makes life so much easier. You can directly combine the numerators, which simplifies the process significantly. It's like having a common language – when everyone speaks the same language, communication is smooth and efficient. In math, a common denominator is like that common language, allowing us to combine fractions seamlessly.

  • Distribute the Negative Sign Carefully: This is the golden rule! When subtracting a group of terms, remember to distribute the negative sign to each term in the group. This is where many mistakes happen, so be extra cautious. It’s like handling fragile items – you need to be gentle and careful to avoid breaking anything. In math, the negative sign is like a fragile item, and distributing it correctly is crucial to preserving the integrity of the expression.

  • Combine Like Terms to Simplify: After distributing, combine like terms in the numerator to simplify the expression as much as possible. This makes the expression cleaner and easier to work with. It’s like organizing your workspace – a tidy space leads to a clear mind and efficient work. In math, combining like terms is like tidying up the expression, making it more manageable and understandable.

Common Mistakes to Avoid:

  • Forgetting to Distribute the Negative Sign: This is the most frequent error. Always remember to distribute the negative sign to every term in the numerator you are subtracting. It’s like forgetting your keys – it can throw off your whole day. In math, forgetting to distribute the negative sign can lead to a completely wrong answer, so make it a habit to double-check.

  • Adding Numerators Instead of Subtracting: If the problem involves subtraction, make sure you are subtracting the numerators, not adding them. This seems obvious, but it’s easy to slip up, especially under pressure. It’s like mixing up left and right – it can lead you in the wrong direction. In math, using the wrong operation can have significant consequences, so always double-check the problem and make sure you’re performing the correct operation.

  • Incorrectly Combining Like Terms: Double-check that you are only combining terms that are actually “like” – that is, they have the same variable raised to the same power. It’s like sorting socks – you need to match the pairs correctly. In math, combining unlike terms is like mixing up the socks, leading to a nonsensical result. So, always take a moment to ensure you’re combining the right terms.

By keeping these key takeaways and common mistakes in mind, you'll be well-equipped to tackle similar problems with confidence and accuracy. It’s like having a checklist before a big trip – you're making sure you have everything you need and are prepared for any challenges along the way. In math, these takeaways and reminders are your checklist, helping you stay on track and avoid common pitfalls. So, remember these tips, practice regularly, and you'll be simplifying rational expressions like a pro in no time!

Practice Problems

To really nail this skill, guys, let's look at some practice problems. Working through examples is the best way to solidify your understanding and build confidence. It's like practicing a musical instrument – the more you play, the better you get. So, grab a pencil and some paper, and let's get to work!

Practice Problem 1:

Simplify the expression: (3x^2 + 5x - 2) / (2x + 1) - (x^2 - 3x + 1) / (2x + 1)

This problem is similar to the one we just solved, but it has slightly different coefficients and terms. The key is to follow the same steps: combine the numerators, distribute the negative sign, and combine like terms. Remember, the goal is to simplify the expression as much as possible. It’s like climbing a mountain – you break it down into smaller, manageable steps and focus on reaching the next milestone. So, take your time, work through each step carefully, and see if you can arrive at the simplified expression.

Practice Problem 2:

Simplify the expression: (4x^3 - 7x + 3) / (x - 2) - (2x^3 + 5x - 1) / (x - 2)

This problem involves higher powers of x, but the same principles apply. Don't let the x^3 terms intimidate you – just focus on combining like terms correctly. This problem is like tackling a more complex recipe – it might have more ingredients and steps, but the fundamental techniques are the same. So, stay calm, follow the instructions, and you'll be amazed at what you can create.

Practice Problem 3:

Simplify the expression: (x^2 + 6x - 4) / (3x - 2) - (2x^2 - x + 5) / (3x - 2)

This problem provides a good opportunity to practice distributing the negative sign and combining like terms. Pay close attention to the signs of each term as you work through the problem. This problem is like solving a puzzle – you need to fit the pieces together correctly to see the complete picture. So, pay attention to the details, think strategically, and you'll find the solution.

Working through these practice problems will help you master the skill of simplifying rational expressions. Remember, the more you practice, the more comfortable and confident you'll become. It's like learning a new language – it might seem daunting at first, but with consistent effort and practice, you'll be fluent in no time. So, keep practicing, keep exploring, and keep pushing yourself to improve. You've got this!

By tackling these problems, you'll not only improve your algebraic skills but also develop your problem-solving abilities. Math is like a muscle – the more you exercise it, the stronger it gets. So, keep challenging yourself, keep learning, and keep growing. You're on a journey of discovery, and every problem you solve is a step forward. So, let's keep moving forward and unlock the amazing world of mathematics!

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Title: Simplify (x^2-7)/(5x-4) - (2x+9)/(5x-4) A Step-by-Step Guide