Positive Integer Identification Without Calculation: Math Problem

Hey guys! Let's dive into a cool math problem where we need to figure out which expressions will give us positive integers, all without actually doing the full calculation. This is super handy for building our number sense and understanding how positive and negative numbers interact. We've got six expressions to look at, so let's break them down step by step.

Understanding the Basics

Before we jump into the expressions, it's important to nail down the basics. Remember, a positive integer is a whole number greater than zero. So, 1, 2, 3, and so on, are positive integers. Zero isn't positive or negative, and negative numbers are less than zero.

The key to solving this without calculating is understanding the rules of addition, subtraction, and multiplication with negative numbers. Here's a quick refresher:

• Subtracting a negative is like adding a positive: Think of it like canceling out a debt. For example, (-7) - (-2) is the same as (-7) + 2. Subtracting a negative number essentially moves you to the right on the number line, which is the same as addition.
• Multiplying two negatives gives a positive: When you multiply two negative numbers, the result is always positive. This is because you're essentially reversing the direction twice. For example, (-3) * (-5) will be a positive number.
• Multiplying a positive and a negative gives a negative: Multiplying a positive number by a negative number always results in a negative number. This is because you are changing the sign of the positive number.
• Addition and subtraction of positives and negatives: This is all about direction on the number line. Adding a positive moves you right, adding a negative moves you left. Subtracting a positive moves you left, and subtracting a negative moves you right.

With these rules in mind, we can analyze each expression and determine if the result will be a positive integer without needing to perform the exact calculations. It's all about predicting the sign of the result!

Analyzing the Expressions

Now, let’s roll up our sleeves and get into the core of the problem: figuring out which of the expressions will yield a positive integer result. We'll go through each one, applying those rules we just brushed up on, and see if we can predict the outcome. Remember, the goal here is not to solve for the exact number, but to determine the sign—whether it's going to be positive or negative.

Expression a: (-7) - (-2)

In this expression, we have (-7) - (-2). The key thing to notice here is that we are subtracting a negative number. As we discussed, subtracting a negative is the same as adding a positive. So, this expression can be rewritten as (-7) + 2. Now, we have a negative number plus a positive number. Think of it as starting at -7 on the number line and moving 2 steps to the right. Will we end up on the positive side of zero? No, we won't. We'll still be in the negatives. Therefore, the result will be negative, and we can confidently say this is not a positive integer.

Expression b: (+5) - (+6)

Moving on to the next one, we have (+5) - (+6). Here, we're subtracting a positive number from another positive number. Imagine you have 5 apples, and someone takes away 6. You're going to end up with a negative amount, right? So, the result here will definitely be negative. We're starting at +5 on the number line and moving 6 steps to the left, which lands us in the negative zone. So, this one is also not a positive integer.

Expression c: (+4) · (-4)

Now let's consider (+4) · (-4). This expression involves multiplication. We're multiplying a positive number by a negative number. Remember our rule: a positive times a negative equals a negative. So, the result of this multiplication will be a negative number. We don't need to calculate the exact number to know that it won't be a positive integer. Negative all the way!

Expression d: (-3) · (-5) · (+2)

Expression d is (-3) · (-5) · (+2). This one is interesting because it involves multiplying three numbers. First, we see two negative numbers being multiplied: (-3) · (-5). We know that a negative times a negative gives us a positive. So, (-3) · (-5) will result in a positive number. Now, we have a positive number multiplied by (+2), which is another positive number. A positive times a positive is always positive. So, the final result of this expression will be positive. This is our first potential positive integer!

Expression e: (-33) - (-22) - (+11)

Let's tackle (-33) - (-22) - (+11). This expression involves both subtraction and negative numbers. First, let's deal with the subtractions of negatives. We can rewrite (-33) - (-22) as (-33) + 22. This is like starting at -33 and moving 22 steps to the right. We'll still be in the negative territory, but closer to zero. Now, we have the result of that (which is negative) minus (+11). Subtracting a positive from a negative will only make the number more negative. So, the final result will definitely be negative. Not a positive integer here.

Expression f: (-1) - (-3) - (-2)

Last but not least, we have (-1) - (-3) - (-2). This expression looks tricky, but let’s break it down. Remember, subtracting a negative is the same as adding a positive. So, we can rewrite this as (-1) + 3 + 2. Now, we're starting at -1 and adding 3, which gets us to +2. Then we add another 2, which gets us to +4. The final result is positive! This expression will give us a positive integer.

Conclusion: The Positive Integer Expressions

Alright, guys, we've cracked the code! By analyzing each expression without actually calculating the final answer, we were able to determine which ones would result in a positive integer. So, let’s recap our findings:

• Expression d: (−3) · (−5) · (+2) will result in a positive integer because the product of two negatives is positive, and then multiplying by a positive keeps the result positive.
• Expression f: (-1) - (-3) - (-2) will also result in a positive integer because subtracting negative numbers is the same as adding positive numbers, leading to a positive result.

Expressions a, b, c, and e, on the other hand, will result in negative numbers. It's awesome how we can use these rules to predict the outcome without needing to crunch the numbers directly. This is a fantastic skill to have in math, as it helps you develop a deeper understanding of how numbers work together. Keep practicing, and you'll become a pro at identifying positive and negative outcomes in no time!