Number Line Integers: Solving For A - B

Let's break down this math problem step by step, guys! We're dealing with integers on a number line, distances, and a little bit of algebra. It might seem tricky at first, but don't worry, we'll get through it together. We'll focus on understanding each part of the problem, from the number line concept to the final calculation. So, grab your thinking caps, and let's dive in!

Understanding the Problem

In this section, we're going to dissect the problem statement and make sure we understand exactly what it's asking. The key here is to visualize what's happening on the number line. We have two integers, 'a' and 'b', that are the same distance away from -2. Think of -2 as the midpoint between 'a' and 'b'. Since 'a' is the smaller integer and 'b' is the larger, 'a' will be to the left of -2 on the number line, and 'b' will be to the right. It's like having a teeter-totter balanced at -2, with 'a' and 'b' being the same distance from the center. This understanding of visualizing the integers helps make the problem easy to solve.

Next, we have the equation a x b = -12. This tells us that when we multiply 'a' and 'b', we get a negative result. What does this mean? Well, it means that one of the integers must be negative, and the other must be positive. This fits perfectly with our understanding of 'a' being smaller than -2 (negative) and 'b' being larger than -2 (likely positive). This part of the statement gives us the key to finding the exact numbers in the problem. This connection between number line positioning and the resulting algebraic equations gives us a better insight.

Finally, the question asks us to find the value of a - b. This means we need to figure out the specific values of 'a' and 'b' first, and then subtract 'b' from 'a'. It's important to pay attention to the order here, as subtraction is not commutative (a - b is not the same as b - a). The goal is now much clearer with the expression in our sight. We will solve for 'a' and 'b' and then compute the subtraction result.

Finding the Integers a and b

Okay, let's get down to brass tacks and find those integers, 'a' and 'b'! We know they're equidistant from -2, and that a x b = -12. This is where we can use a little bit of trial and error, but in a smart way. Since 'a' is negative and 'b' is likely positive, let's think about pairs of factors of 12 that have opposite signs. The pairs of factors of 12 are (1, 12), (2, 6), and (3, 4). Now, we need to consider the negative versions as well.

Let’s try a few possibilities. What if a = -1? To get a product of -12, b would have to be 12. But are -1 and 12 the same distance from -2? Nope! -1 is 1 unit away from -2, while 12 is a whopping 14 units away. So, this pair doesn't work. We are using the equidistant property to check our hypothesis. This is a very efficient way to narrow down possibilities.

Let's try a = -3. Then b would have to be 4 to get a product of -12. How far is -3 from -2? It's 1 unit away. How far is 4 from -2? It's 6 units away. Again, these are not equidistant. Thus, we move on to the next possible pair. We are methodically eliminating possibilities and honing our strategy.

Now, let's test a = -4. This means b would be 3. The distance between -4 and -2 is 2 units. The distance between 3 and -2 is 5 units. These are not equidistant either, so this pair doesn't work.

Finally, let's look at a = -6. If a = -6, then b = 2. The distance between -6 and -2 is 4 units. The distance between 2 and -2 is also 4 units! Bingo! We found our pair. We have confirmed that -6 and 2 satisfy both conditions: their product is -12, and they are equidistant from -2. Our methodical approach has paid off.

So, we've determined that a = -6 and b = 2. Now we're ready for the final step: finding a - b.

Calculating a - b

Alright, we've identified that a = -6 and b = 2. Now comes the super-simple part: calculating a - b. Remember, subtraction order matters! We need to subtract 'b' from 'a', not the other way around. So, we have:

a - b = -6 - 2

Think of this as starting at -6 on the number line and then moving 2 units further to the left (in the negative direction). This gives us:

-6 - 2 = -8

So, the value of a - b is -8. And that's it! We've successfully solved the problem. This shows us the simple calculation step once we have found the correct values for 'a' and 'b'.

Answer and Wrap-up

The answer to the question, “What is the value of a - b?” is -8 (Option B). We got there by carefully understanding the problem, visualizing the integers on a number line, using the given equation to narrow down possibilities, and finally, performing the subtraction. Remember, breaking down complex problems into smaller, manageable steps is a powerful strategy in math, and in life! Understanding the core concepts, such as the number line and the properties of integers, is key to successfully tackling these types of problems.

We started by decoding the problem statement, identifying the critical information, such as the equidistant nature of the integers and the product relationship. Then, we systematically explored possible integer pairs, using the distance from -2 as a crucial check. This methodical approach led us to the correct values for 'a' and 'b'. Finally, we carefully executed the subtraction, ensuring we maintained the correct order. This illustrates the importance of attention to detail in mathematical calculations. Hope you guys found this explanation helpful! Keep practicing, and you'll become number-line ninjas in no time! The ability to solve math problems requires patience, understanding and a logical approach.