Comparing Expressions: >, <, Or =?
Hey guys! Let's dive into some math problems where we need to figure out if one expression is greater than, less than, or equal to another. We're going to be working with integers and basic operations like addition and multiplication. So, grab your thinking caps, and let's get started!
Understanding the Basics
Before we jump into solving these problems, let's quickly review the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order is super important because it tells us which operations to perform first to get the correct answer. Remember, multiplication comes before addition and subtraction!
When we're dealing with positive and negative numbers, it's also crucial to understand how these numbers interact with each other. Multiplying or dividing two negative numbers results in a positive number, while multiplying or dividing a positive and a negative number gives you a negative result. Also, adding a negative number is the same as subtracting its positive counterpart.
Keeping these fundamentals in mind will make solving these expressions much easier. So, let's break down each problem step by step and figure out which sign (>, <, or =) fits in the blank.
Problem 1: -10 + 10 * 0
Alright, let's tackle the first one: -10 + 10 * 0. Following the order of operations, we need to do the multiplication first. So, what's 10 multiplied by 0? It's 0, of course! Any number multiplied by zero is zero. Now our expression looks like this: -10 + 0.
What happens when we add 0 to -10? Well, adding zero doesn't change the value, so we're left with -10. There's nothing to compare it to in this isolated expression, but understanding that the result simplifies to -10 is our first step. This problem highlights how the multiplicative property of zero simplifies calculations. Remember, guys, zero is a powerful number in math – it can make things disappear!
Problem 2: -90 + 99 * 8
Next up, we have -90 + 99 * 8. Again, we start with multiplication. 99 times 8... hmm, let's break it down. 99 is very close to 100, so we can think of it as (100 * 8) - (1 * 8), which is 800 - 8, giving us 792. Now our expression is -90 + 792.
Now we have an addition problem with a negative and a positive number. Think of it as starting at -90 on the number line and moving 792 spaces to the right. Since 792 is much larger than 90, we know the result will be positive. To find the exact value, we can subtract 90 from 792: 792 - 90 = 702. So, -90 + 99 * 8 equals 702. This one shows us how breaking down multiplication and thinking of numbers in chunks can make calculations easier. Also, remember to keep track of those positive and negative signs!
Problem 3: 51 + (-54) * 0
Okay, let's move on to 51 + (-54) * 0. Just like before, we tackle the multiplication first. This time, we have -54 multiplied by 0. And as we learned in Problem 1, anything times zero is zero. So, (-54) * 0 = 0. Our expression now simplifies to 51 + 0.
Adding zero to any number doesn't change its value, so 51 + 0 = 51. The result is simply 51. Once again, the multiplicative property of zero makes the calculation straightforward. See how important it is to spot these opportunities to simplify!
Problem 4: 27 + (-69) * -10
Now for 27 + (-69) * -10. Multiplication first, guys! We're multiplying two negative numbers here: -69 and -10. Remember, a negative times a negative equals a positive. So, (-69) * -10 = 690. Our expression becomes 27 + 690.
This is a simple addition problem. 27 + 690 = 717. So, the answer is 717. This problem really emphasizes the rule about multiplying negative numbers. Always remember: two negatives make a positive!
Problem 5: 7 + (-8) + (-7) * 0
Let's try 7 + (-8) + (-7) * 0. You know the drill – multiplication first! We have (-7) * 0, which, as we've seen several times now, equals 0. The expression simplifies to 7 + (-8) + 0.
Now we're left with addition. We can rewrite 7 + (-8) as 7 - 8. What's 7 minus 8? It's -1. Then we add 0, which doesn't change the value, so the final answer is -1. This one combines the zero property with the rules for adding positive and negative numbers. It's a good reminder to take things one step at a time!
Problem 6: 12 + (-10) + (-1) * 0
Last but not least, we have 12 + (-10) + (-1) * 0. Same as before, multiplication is our first stop. (-1) * 0 = 0. Our expression now reads 12 + (-10) + 0.
Again, we're left with addition. 12 + (-10) is the same as 12 - 10, which equals 2. Adding 0 doesn't change the result, so the final answer is 2. This problem reinforces the importance of following the order of operations and recognizing how the multiplicative property of zero can simplify our calculations.
Wrapping Up
So there you have it! We've worked through six expressions, carefully applying the order of operations and the rules for positive and negative numbers. The key takeaways here are to always remember PEMDAS, pay attention to signs, and look for opportunities to simplify using the properties of zero. I hope this helps, guys! Keep practicing, and you'll become math whizzes in no time!