Solving For The Rectangle Width The Equation 2x + 2(x+2) = 24
Hey guys! Let's dive into a fun math problem today! We're going to figure out the width of a rectangle using some algebra and a bit of logical thinking. This is a classic problem that pops up in many math courses, so let's break it down together. We will rewrite the equation in a human-friendly tone to make it easier to understand and then proceed to solve it by checking the options. We will also explore alternative methods for solving this problem to provide a comprehensive understanding. So, let’s get started and make math a little less intimidating!
Understanding the Problem
So, here's the deal: we have a rectangle. This rectangle's length is 2 inches longer than its width. We also know that the perimeter of this rectangle is 24 inches. Now, we have a cool equation that represents this situation: 2x + 2(x + 2) = 24. In this equation, x stands for the width of the rectangle in inches. Our mission, should we choose to accept it, is to find out which value of x from the set {1, 3, 5, 7} makes this equation true. Basically, we need to figure out which one of these numbers could be the width of our rectangle. Let's get our detective hats on and solve this puzzle together! We'll start by really understanding what each part of the problem means and then move on to the fun part: finding the answer. Are you ready? Let's go!
Breaking Down the Rectangle Problem
Okay, let's really understand what's going on here. We are dealing with a rectangle, right? Rectangles are those shapes we all know and love, with four sides where the opposite sides are equal in length and all angles are 90 degrees. Now, this particular rectangle has a special relationship between its sides. The length is 2 inches more than the width. Think of it like this: if the width is a certain amount, the length is that same amount plus an extra 2 inches. This is crucial because it sets up the foundation for our equation. Then, we have the perimeter. Remember, the perimeter is the total distance around the outside of the rectangle. Imagine walking all the way around the edge; that's the perimeter. And we know this distance is 24 inches for our rectangle. So, the problem gives us these clues: the length's relationship to the width and the total perimeter. Our job is to use these clues to nail down exactly what the width is. It's like a little puzzle where all the pieces fit together perfectly. Understanding these basics is the key to solving the problem, so let's move on to how we can use this knowledge to crack the equation!
The Equation: 2x + 2(x + 2) = 24
Alright, let's break down this equation like pros. 2x + 2(x + 2) = 24 might look a bit intimidating at first, but trust me, it's just a bunch of symbols telling a story. The x here is our mystery number – it's the width of the rectangle that we're trying to find. Now, let's see what the equation is saying. The term 2x represents two times the width. Why two times? Because in a rectangle, there are two sides that have the same width. Make sense? Next up, we have 2(x + 2). This part is about the length. Remember, the length is 2 inches more than the width, so we write it as x + 2. And just like the width, there are two sides with the same length, so we multiply the whole thing by 2. So far, so good? Now, we're adding these two parts together: 2x (the total width) plus 2(x + 2) (the total length). And what does that equal? The total perimeter, which we know is 24 inches. This equation is basically a mathematical way of saying, "If you add up all the sides of this rectangle, you get 24 inches." Pretty neat, huh? Now that we understand the equation, let's get into the fun part – solving it!
Time to Solve: Plugging in the Values
Okay, team, let's get down to business! We have our equation, and we have a set of possible answers: {1, 3, 5, 7}. Our mission is to figure out which one of these numbers, when we plug it in for x, makes the equation 2x + 2(x + 2) = 24 true. This is like a detective game, where we try each suspect (each number) and see if they fit the crime (the equation). Let's take each number one by one and see what happens. It's all about trial and error, but in a super organized, mathematical way. We'll go through each option carefully, do the calculations, and see if the left side of the equation equals the right side. If it does, we've found our culprit! If not, we move on to the next number. This is a straightforward way to solve the problem, and it's perfect for when you have a limited set of possible answers. So, let's start plugging and chugging!
Testing x = 1
Alright, let's start with our first suspect: x = 1. We're going to plug this into our equation and see what happens. Our equation is 2x + 2(x + 2) = 24. So, everywhere we see an x, we're going to replace it with a 1. Let's do it! We get 2(1) + 2(1 + 2). Now, we need to simplify this and see if it equals 24. First, 2(1) is just 2. Then, inside the parentheses, 1 + 2 is 3, so we have 2(3), which is 6. So, our equation now looks like 2 + 6. What's 2 + 6? It's 8. So, when x = 1, the left side of our equation equals 8. But we need it to equal 24! Since 8 is definitely not 24, we know that x = 1 is not the width of our rectangle. It's like trying on a shoe that's way too small – it just doesn't fit. So, we can cross 1 off our list and move on to the next number. Let's keep going!
Trying x = 3
Okay, next up on our list is x = 3. Let's see if this one fits the bill. We're doing the same thing as before: plugging x = 3 into our equation 2x + 2(x + 2) = 24. So, we replace every x with a 3, which gives us 2(3) + 2(3 + 2). Time to simplify! First, 2(3) is 6. Inside the parentheses, 3 + 2 is 5, so we have 2(5), which is 10. Now our equation looks like 6 + 10. What's 6 + 10? It's 16. So, when x = 3, the left side of our equation equals 16. But remember, we need it to equal 24! 16 and 24 are different numbers, so x = 3 is not our answer either. It's like trying on a shoe that's the right style but the wrong size. We're getting closer, though! We've eliminated two possibilities, so let's keep going and try the next number.
Evaluating x = 5
Alright, let's give x = 5 a shot. We're going to plug this into our trusty equation and see if it works. So, we're replacing x with 5 in 2x + 2(x + 2) = 24, which gives us 2(5) + 2(5 + 2). Time to crunch some numbers! 2(5) is 10. Inside the parentheses, 5 + 2 is 7, so we have 2(7), which is 14. Now our equation looks like 10 + 14. What does that add up to? It's 24! Bingo! When x = 5, the left side of our equation equals 24, which is exactly what we wanted. This means that x = 5 is the solution to our equation. We've found the width of the rectangle! It's like finding the missing puzzle piece that fits perfectly. We can confidently say that 5 is the correct answer, but just to be thorough, let's quickly check the last option to make sure it doesn't work.
Checking x = 7
Just to be super sure, let's check x = 7. Even though we've already found a solution, it's good practice to make sure the other options don't work. We're plugging x = 7 into our equation 2x + 2(x + 2) = 24, so we get 2(7) + 2(7 + 2). Let's simplify. 2(7) is 14. Inside the parentheses, 7 + 2 is 9, so we have 2(9), which is 18. Now our equation looks like 14 + 18. What's 14 + 18? It's 32. So, when x = 7, the left side of our equation equals 32. But we need it to equal 24! 32 is way more than 24, so x = 7 is definitely not the width of our rectangle. This confirms that our solution of x = 5 is the only one that works from the set of numbers we were given. We've officially cracked the case! We found the right number, and we double-checked our work. High five!
Alternative Method: Solving the Equation Algebraically
Hey, guess what? There's another way to solve this problem! Instead of plugging in numbers, we can use algebra to solve the equation directly. This is a super useful skill that will help you in all sorts of math problems. We're going to take our equation, 2x + 2(x + 2) = 24, and use some algebraic techniques to isolate x and find its value. This method might seem a bit more abstract at first, but it's incredibly powerful. It's like having a secret code that unlocks the answer. We'll walk through each step carefully, explaining the logic behind it, so you can see how it works. By the end of this, you'll have another tool in your math toolbox! So, let's dive into the world of algebra and see how we can solve this equation in a different way.
Step-by-Step Algebraic Solution
Okay, let's get started with our algebraic adventure! Our goal is to isolate x in the equation 2x + 2(x + 2) = 24. This means we want to get x all by itself on one side of the equation. First, we need to deal with those parentheses. We'll use the distributive property, which means we multiply the 2 outside the parentheses by each term inside. So, 2(x + 2) becomes 2 * x + 2 * 2, which simplifies to 2x + 4. Now our equation looks like 2x + 2x + 4 = 24. See how we're making progress? Next, let's combine like terms. We have two 2x terms, so we can add them together: 2x + 2x is 4x. Our equation is now 4x + 4 = 24. We're getting closer! Now, we want to get the 4x term by itself, so we need to get rid of that + 4. We do this by subtracting 4 from both sides of the equation. This keeps the equation balanced. So, 4x + 4 - 4 = 24 - 4 simplifies to 4x = 20. Almost there! Finally, we need to get x all by itself. It's currently being multiplied by 4, so we'll do the opposite operation: divide both sides by 4. This gives us 4x / 4 = 20 / 4, which simplifies to x = 5. Woo-hoo! We did it! We solved the equation and found that x = 5, which is exactly what we got when we plugged in the numbers. This algebraic method is a powerful way to solve equations, and it's awesome to have in your math toolkit. Now you've seen two different ways to solve this problem – plugging in values and using algebra. You're a math whiz!
Conclusion: x = 5 is the Width!
So, there you have it, folks! We've successfully solved the mystery of the rectangle's width. Whether we plugged in the numbers or used our algebra skills, we arrived at the same answer: x = 5 inches. This means the width of our rectangle is 5 inches. And since the length is 2 inches more than the width, the length is 7 inches. We can even double-check our work by calculating the perimeter: 2(5) + 2(7) = 10 + 14 = 24 inches, which is exactly what the problem told us. We nailed it! This problem was a great example of how we can use equations to represent real-world situations and how we can solve them using different methods. We explored plugging in values, which is a great strategy when you have a limited set of options, and we dove into algebra, which is a powerful tool for solving equations directly. By understanding both approaches, you're well-equipped to tackle similar problems in the future. Keep up the awesome work, and remember, math is just another puzzle waiting to be solved!