Comparing Expressions: <, >, Or =

Hey guys! Today, we're diving into the exciting world of comparing mathematical expressions. We'll be using the symbols '$<$' (less than), '$>$' (greater than), and '$=$' (equal to) to figure out which expression is bigger, smaller, or if they're the same. Think of it as a mathematical showdown! We'll break down each comparison step-by-step, so you'll become a pro at this in no time. Let's get started!

(a) $49-7+8$ $\square$ $49-7+8$

In this first example, we're comparing the expression 49 - 7 + 8 with itself. Now, this might seem super obvious, and that's because it is! We're essentially asking if something is equal to itself. Remember, the fundamental principle here is the identity property of equality. This property states that any quantity is always equal to itself. It's like looking in a mirror – you see your exact reflection, right? There's no room for debate; it's a direct match.

When we break down the expression 49 - 7 + 8, we first perform the subtraction: 49 minus 7, which gives us 42. Then, we add 8 to that result: 42 plus 8 equals 50. So, the expression simplifies to 50. Now, let’s compare this to the other side. Guess what? It's also 49 - 7 + 8, which, as we just calculated, is also 50. Therefore, we are comparing 50 with 50.

So, what symbol do we use to show that 50 is the same as 50? That's right, it's the equals sign '$=$'. This means that the two expressions are exactly the same in value. There’s no difference, no trickery, just a straightforward equality. This is a foundational concept in mathematics, guys, and understanding it is crucial for tackling more complex problems later on. Think of it as the bedrock upon which many other mathematical ideas are built. If you grasp this simple concept, you'll be well-prepared for the challenges ahead.

Therefore, we can confidently fill in the blank with '$=$', showing that 49 - 7 + 8 is indeed equal to 49 - 7 + 8. This exercise might seem simple, but it reinforces a fundamental mathematical principle: anything is equal to itself. Keep this in mind as we move on to more complex comparisons. Remember, even the most intricate mathematical problems often boil down to these basic truths. So, embrace the simplicity and build your understanding from there!

(b) $83 \times 42-18$ $\square$ $83 \times 40-18$

Okay, guys, let's tackle the next comparison: 83 × 42 - 18 versus 83 × 40 - 18. This one's a bit trickier, but don't worry, we'll break it down. The key here is to focus on the order of operations and how changing one number in a multiplication can impact the final result. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? We always multiply before we subtract.

Let's analyze the first expression, 83 × 42 - 18. We start by multiplying 83 by 42. Now, we don't need to calculate the exact answer just yet; we just need to understand that this multiplication will result in a certain value. Then, we'll subtract 18 from that value. So, we have a product (83 × 42) and we're reducing it by 18.

Now, let's look at the second expression, 83 × 40 - 18. Again, we start with the multiplication: 83 multiplied by 40. Notice something important here? We're multiplying 83 by a smaller number (40) than in the first expression (42). This means the product in the second expression will be smaller than the product in the first expression. Think about it like this: if you have 83 groups of 42 items, that's going to be more than 83 groups of 40 items.

Since 83 multiplied by 42 will result in a larger number than 83 multiplied by 40, and we're subtracting the same amount (18) from both results, the expression with the larger product will ultimately be greater. In simpler terms, if you start with a bigger number and subtract the same amount, you'll end up with a bigger result. It’s like having more money in your bank account before paying the same bill – you’ll have more left over.

Therefore, 83 × 42 - 18 is greater than 83 × 40 - 18. We can confidently fill in the blank with the '$>$' symbol. This comparison highlights the importance of understanding how multiplication affects the magnitude of a number and how that impact carries through subsequent operations. It's not just about crunching numbers; it's about understanding the relationships between them. Great job, guys! Let’s keep going!

(c) $145-17 \times 8$ $\square$ $145-17 \times 6$

Alright, let's move on to the third comparison: 145 - 17 × 8 versus 145 - 17 × 6. Just like before, the order of operations is our guiding principle here. Remember PEMDAS? Multiplication comes before subtraction, so we need to focus on those multiplication parts first. This is where things get interesting, guys, because we're subtracting different amounts from the same starting number.

In the first expression, 145 - 17 × 8, we first multiply 17 by 8. This gives us a certain value that we're going to subtract from 145. In the second expression, 145 - 17 × 6, we multiply 17 by 6. Now, think carefully: which product is larger? 17 multiplied by 8, or 17 multiplied by 6? Since we're multiplying by a larger number (8 versus 6), the product 17 × 8 will be greater than the product 17 × 6. It's like buying more of the same item – the more you buy, the higher the total cost.

Here's where the crucial insight comes in: we're subtracting these products from the same initial number, 145. If you subtract a larger amount from 145, you'll end up with a smaller result. Conversely, if you subtract a smaller amount from 145, you'll end up with a larger result. It's like having a cake and giving away slices – the bigger the slice you give away, the less cake you have left for yourself. It’s a concept related to inverse relationships in mathematics.

So, since 17 × 8 is greater than 17 × 6, subtracting 17 × 8 from 145 will result in a smaller number than subtracting 17 × 6 from 145. Therefore, 145 - 17 × 8 is less than 145 - 17 × 6. We can confidently fill in the blank with the '$<$' symbol. This comparison really drives home the importance of thinking about the relationships between operations and how they affect the final outcome. You're doing awesome, guys! Let's tackle the last one.

(d) $23 \times 48-35$ $\square$ $23 \times 48-39$

Okay, team, let’s dive into our final comparison: 23 × 48 - 35 versus 23 × 48 - 39. This one’s another great example of how a small change can impact the overall result. As always, we need to keep the order of operations in mind (PEMDAS, remember?). Multiplication before subtraction is the name of the game.

In both expressions, we start with the same multiplication: 23 multiplied by 48. This is important, guys, because it means both expressions begin with the exact same value. The difference lies in what we subtract afterward. It’s like having the same initial amount of money but spending different amounts – who will have more left?

In the first expression, 23 × 48 - 35, we subtract 35 from the product of 23 and 48. In the second expression, 23 × 48 - 39, we subtract 39 from the same product. Now, think carefully: which number is larger, 35 or 39? Clearly, 39 is greater than 35. This is the crucial piece of the puzzle. The amount you subtract from the same initial value makes all the difference.

Since we're subtracting a larger number (39) in the second expression, the result will be smaller than in the first expression, where we subtract a smaller number (35). Think of it like this: if you have a certain amount of candy and you give away 39 pieces, you'll have less candy left than if you only gave away 35 pieces. It’s all about understanding how subtraction works in relation to the starting value.

Therefore, 23 × 48 - 35 is greater than 23 × 48 - 39. We can confidently fill in the blank with the '$>$' symbol. This final comparison really highlights the concept that when subtracting from the same number, subtracting a larger value results in a smaller final number. It’s a simple idea, but it’s incredibly powerful in mathematics. You’ve nailed it, guys!

By working through these examples, you've gained a solid understanding of how to compare mathematical expressions using '$<$', '$>$', and '$=$'. Remember to always keep the order of operations in mind and think about how each operation affects the overall value of the expression. Keep practicing, and you'll become mathematical expression comparison masters in no time!