Hey everyone! Let's dive into a fun math problem about sewing pillows. We've got Margot, who's a whiz with a needle and thread, and she's working on some square pillows. This isn't just about sewing; it's about using algebra to figure out the perimeter of these pillows. The perimeter, as you might remember, is the total distance around the outside of a shape. In this case, since we're dealing with square pillows, we just need to add up the lengths of all four sides. So, let's break down the problem step-by-step.
Decoding the Dimensions of Margot's Pillows
Margot's first pillow has sides that measure 2x² + 1 inches. Now, that might look a little intimidating with the x² and all, but don't worry, we'll tackle it together. Remember, in algebra, 'x' is just a variable, a placeholder for a number we might not know yet. The important thing here is that all four sides of this square pillow are the same length, which is 2x² + 1 inches. To find the perimeter, we need to add this length four times (since a square has four sides). This can be written as 4 * (2x² + 1). When we distribute the 4, we will be multiplying each term inside the parentheses by 4. Four times 2x² is 8x², and four times 1 is 4. So, the perimeter of the first pillow is 8x² + 4 inches. This expression tells us the total length of ribbon Margot needs for the first pillow, depending on the value of 'x'. If x was 2, for example, we could substitute 2 everywhere we see an x in the expression to get the numerical perimeter. This is the beauty of algebraic expressions; they give us a general formula that works for any value of the variable.
Now, Margot is planning to make another pillow, also square, but with a different side length. This time, the side length is 4x - 7 inches. Again, this is an algebraic expression, meaning the length of the side depends on the value of 'x'. But the principle is the same: a square has four equal sides, so to find the perimeter, we need to multiply this expression by 4. This gives us 4 * (4x - 7). Let's distribute the 4 again. Four times 4x is 16x, and four times -7 is -28. So, the perimeter of the second pillow is 16x - 28 inches. This expression represents the amount of ribbon Margot will need for her second pillow, and just like before, if we knew the value of 'x', we could calculate the exact length of ribbon.
The Key Question: Finding the Expression
So, what's the main question here? We're asked to find an expression that can be used to determine something about these pillows, which, after analyzing the given information, is about the perimeter and how to calculate it given the side lengths expressed algebraically. The question is implicitly asking for the expressions that represent the perimeters of both pillows. We have already found those expressions. For the first pillow, the perimeter is 8x² + 4 inches, and for the second pillow, it's 16x - 28 inches. These expressions are the answer to the core question. Understanding how to derive these expressions is key, it shows you how algebraic principles are used in everyday problems like sewing. The beauty of algebra is that it allows us to represent quantities and relationships in a concise and general way.
Comparing Pillow Sizes: A Deeper Dive into Expressions
But what if we wanted to compare the amount of ribbon Margot needs for each pillow? What if we wanted to know how much more ribbon she needs for one pillow compared to the other? This is where things get even more interesting! To compare the perimeters, we can subtract one expression from the other. Let's say we want to find out how much more ribbon Margot needs for the first pillow (8x² + 4 inches) compared to the second pillow (16x - 28 inches). We would subtract the second expression from the first: (8x² + 4) - (16x - 28). Remember, when subtracting expressions, we need to be careful with the signs. Subtracting a negative number is the same as adding a positive number. So, we can rewrite the expression as 8x² + 4 - 16x + 28. Now, we can combine like terms. We have 8x², which doesn't have any like terms, so it stays as is. Then we have -16x, which also doesn't have any like terms. Finally, we have the constant terms, 4 and 28. Adding those together gives us 32. So, the expression for the difference in ribbon needed is 8x² - 16x + 32. This expression tells us exactly how much more ribbon the first pillow needs compared to the second pillow, depending on the value of 'x'. If this expression results in a negative number, it indicates that the second pillow needs more ribbon.
This kind of comparison is super useful in many situations, not just sewing. Imagine you're comparing the cost of two different materials, or the amount of paint needed for two walls of different sizes. Algebraic expressions allow us to make these kinds of comparisons easily.
The Power of Algebraic Expressions
So, there you have it! We've not only found the expressions for the perimeters of Margot's pillows (8x² + 4 inches and 16x - 28 inches), but we've also seen how to compare those expressions to find the difference in ribbon needed (8x² - 16x + 32 inches). This problem might seem simple on the surface, but it highlights the power of algebraic expressions. They allow us to represent real-world situations mathematically, making it easier to solve problems and make comparisons. Whether you're sewing pillows, building a house, or even just figuring out the best deal at the grocery store, understanding algebra can be a huge help.
Keep practicing, guys, and you'll be solving all kinds of problems in no time! Remember, math is just another language, and with a little effort, you can become fluent.
Let's jump into an engaging scenario involving Margot, a creative seamstress, and her delightful square pillow project! This isn't your typical sewing task; it's a fantastic journey into the world of algebraic expressions and how they relate to real-life situations. The heart of our discussion lies in understanding how to express the perimeter of these pillows using algebra. The perimeter is the distance all the way around an object. Imagine walking around the edge of the pillow; the total distance you walk is the perimeter. Since Margot is making square pillows, each pillow has four equal sides, which simplifies things a bit. We're given the side lengths as algebraic expressions, which means they contain variables (like 'x'). Don't let that scare you! We'll break it down and make it super clear. Let’s get started!
Decoding Pillow Dimensions with Algebra
Margot's initial pillow boasts a side length of 2x² + 1 inches. At first glance, the term 2x² might seem a bit complex, but let's demystify it. In algebra, 'x' is a variable, a symbol representing a value that can change. It's like a placeholder for a number. The exponent, the small 2 above the 'x', means we're squaring 'x' (multiplying it by itself). The coefficient, the 2 in front of the x², tells us we're multiplying the squared value by 2. So, 2x² is just a way of saying