Simplify 3(2x - 8) - 11x: Step-by-Step Algebraic Guide

by Mei Lin 55 views

Hey there, math enthusiasts! Ever stumbled upon an algebraic expression that looked like a tangled mess? Don't worry, we've all been there. Today, we're going to break down one such expression step by step, making it super clear and easy to understand. Our mission: to simplify the expression 3(2x - 8) - 11x and figure out which of the given options is the correct equivalent.

Unraveling the Algebraic Puzzle

Before we dive into the nitty-gritty, let's take a moment to appreciate the beauty of algebra. It's like a secret code that helps us solve problems and understand relationships between numbers and variables. In this case, we have an expression that combines numbers, variables (that's the 'x'), and operations like multiplication and subtraction. Our goal is to simplify it, which means to rewrite it in a cleaner, more concise form. So, let’s understand the expression 3(2x - 8) - 11x. This expression involves a combination of multiplication, subtraction, and the distributive property. We need to simplify it by performing the operations in the correct order and combining like terms. We'll start by tackling the distributive property, which is our key to unlocking this algebraic puzzle. The distributive property states that a(b + c) = ab + ac. In simpler terms, it means we need to multiply the term outside the parentheses (in our case, '3') by each term inside the parentheses (which are '2x' and '-8'). This is a crucial step, so let's do it carefully.

Step 1: Distribute Like a Pro

Okay, guys, let's get this done using the distributive property. We need to multiply 3 by both terms inside the parentheses: 2x and -8.

  • 3 * (2x) = 6x
  • 3 * (-8) = -24

So, 3(2x - 8) becomes 6x - 24. Now our expression looks a bit simpler: 6x - 24 - 11x. We've successfully distributed the '3', and the expression is starting to take a more manageable shape. But we're not done yet! We still have those like terms lurking around, waiting to be combined. Combining like terms is a fundamental concept in algebra. Like terms are those that have the same variable raised to the same power. In our expression, '6x' and '-11x' are like terms because they both have 'x' raised to the power of 1. The '-24' is a constant term, meaning it doesn't have any variables attached to it. We can only combine like terms, so we'll focus on the '6x' and '-11x'.

Step 2: Combine Like Terms with Finesse

Now, let's gather the like terms and combine them. We have 6x and -11x. Think of it like having 6 apples and then owing 11 apples. If you give away your 6 apples, you'll still owe 5 apples. Mathematically, it's like this: 6x - 11x = -5x. So, our expression now looks like this: -5x - 24. We've combined the 'x' terms, and we're left with a much simpler expression. This is the simplified form of the original expression. But wait, we still need to compare this with the given options to find the correct answer. Let's take a look at those options again and see which one matches our simplified expression.

Step 3: Identify the Equivalent Expression

Alright, we've arrived at the simplified expression: -5x - 24. Now, let's compare it with the answer choices provided:

  • A. -17x - 24
  • B. -17x + 24
  • C. -5x - 8
  • D. -5x - 24

Can you spot the match? Option D, -5x - 24, is exactly what we got! So, we've successfully navigated through the algebraic maze and found the correct answer. Option D perfectly matches our simplified expression. But before we celebrate, let's quickly glance at the other options to understand why they are incorrect. Option A, -17x - 24, is incorrect because it seems like there might have been an error in combining the 'x' terms. Perhaps the numbers were added instead of subtracted. Option B, -17x + 24, has a similar issue with the 'x' term and also has the wrong sign for the constant term. Option C, -5x - 8, correctly combines the 'x' terms but makes a mistake with the constant term. It looks like the distributive property might not have been applied correctly in this case. Understanding why the other options are wrong is just as important as finding the correct answer. It helps solidify our understanding of the concepts and prevents us from making similar mistakes in the future.

Common Pitfalls to Avoid

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Let's talk about some common pitfalls to watch out for. One of the most frequent errors is messing up the distributive property. Remember, you need to multiply the term outside the parentheses by every term inside. Forgetting to multiply by one of the terms is a classic mistake. Another common pitfall is combining unlike terms. You can only combine terms that have the same variable raised to the same power. Trying to combine 'x' terms with constant terms, for example, will lead to an incorrect answer. Sign errors are also a major culprit. Be extra careful with negative signs, especially when distributing or combining terms. A misplaced negative sign can completely change the answer. To avoid these pitfalls, it's helpful to double-check your work at each step. Write down each step clearly and methodically, so you can easily spot any errors. Practice makes perfect, so the more you work with algebraic expressions, the better you'll become at avoiding these mistakes.

The Significance of Algebraic Simplification

You might be wondering, why bother simplifying algebraic expressions anyway? Well, simplifying expressions is a fundamental skill in algebra and has a wide range of applications. Simplified expressions are easier to work with. They make it easier to solve equations, graph functions, and perform other algebraic operations. Think of it like decluttering your desk – a clean and organized desk makes it easier to find what you need and get your work done. Simplifying expressions also helps us understand the underlying relationships between variables. By simplifying, we can often reveal patterns and insights that might not be obvious in the original, more complex expression. Algebraic simplification is used extensively in various fields, including physics, engineering, computer science, and economics. It's a crucial tool for modeling real-world phenomena and solving practical problems. So, the skills you develop in simplifying expressions will serve you well in many different areas of study and work. Moreover, mastering algebraic simplification builds a strong foundation for more advanced math topics. It's like learning the alphabet before you can write sentences – it's a necessary building block for future success. As you progress in your math journey, you'll encounter increasingly complex expressions, and the ability to simplify them will become even more critical.

Wrapping Up

So, there you have it! We've successfully simplified the expression 3(2x - 8) - 11x and identified the equivalent expression as -5x - 24 (Option D). We've walked through the steps, discussed common pitfalls, and highlighted the importance of algebraic simplification. Remember, practice is key to mastering these skills. The more you work with algebraic expressions, the more confident and proficient you'll become. So, keep practicing, keep exploring, and keep unraveling those algebraic puzzles! You've got this, guys! Algebra might seem daunting at first, but with a systematic approach and a bit of practice, you can conquer any expression that comes your way. Remember the key steps: distribute, combine like terms, and double-check your work. And don't forget to celebrate your successes along the way! Each expression you simplify is a step forward in your mathematical journey. Happy simplifying!