Finding The Sum Of Integers: A Math Problem

Hey guys! Let's dive into a cool math problem. This one is all about finding the sum of integers. Specifically, we're looking at the integers that are greater than -6 and less than 5. Don't worry, it's not as scary as it sounds! We'll break it down step by step to make sure you totally get it. Understanding integers is super important in math, and this is a perfect example of how they work. We'll go through the process of identifying the relevant integers and then adding them up to find the total. Ready to get started?

Understanding the Problem: Integers and Boundaries

Alright, first things first. What exactly are we dealing with here? We're talking about integers. Remember those? Integers are whole numbers – no fractions or decimals allowed! They can be positive, negative, or even zero. The problem asks us to find the sum of integers that fall within a specific range: greater than -6 and less than 5. This means we need to consider all the whole numbers that fit between these two boundaries, excluding -6 and 5 themselves. The range is crucial here because it sets the limits for the numbers we'll be working with. So, before we start adding things up, let's make sure we clearly understand what numbers are in our set. We want all the whole numbers that are bigger than -6, which means we start with -5. Then we keep going up until we get to 4, because we can't include 5 since the question says 'less than 5'. This part is all about being precise and not missing any numbers. Being precise is especially important in mathematics to make sure your answer is correct. Remember: if we were to include -6, we would have included a number that is not greater than -6, so that's why we don't include it. Similarly, we don't include 5 because we need the numbers to be less than 5. Got it? Awesome. Let's move on to the fun part!

To really nail this down, let's think about it visually. Imagine a number line. Picture -6 and 5 marked on that line. We're only interested in the numbers that sit between those two points, but not including the points themselves. That helps us visualize the range and avoid any confusion about which numbers to include. Number lines are super helpful tools when we're working with integers and ranges. They make it easier to see how numbers relate to each other and to quickly identify the numbers we need. Making use of tools, such as the number line, will help you become a mathematical thinker. It's a great strategy to employ when dealing with similar problems in the future. Now, we are ready to figure out those numbers. Once we have the numbers, we just add them up. Easy, right? Let's take a look at the actual numbers that fit the bill. The list will start with -5, then -4, and so on. We should end up with the integer 4, as this is the biggest number that is still less than 5. Once we know those numbers, the rest is super simple. Let's see how it looks.

The Integers in Our Range

Okay, let's list out the integers that are greater than -6 and less than 5. This is the heart of the problem, so let's make sure we get it right! Our list starts with -5, because that's the smallest integer that meets our criteria (it's greater than -6). Then, we have -4, -3, -2, -1, 0, 1, 2, 3, and 4. Did you notice the pattern? Each number is one more than the last. We stopped at 4 because 4 is less than 5, so it fits within our range. This list is super important, because these are the numbers that we'll add up to find our answer. Being organized and writing down the numbers in an ordered list is a smart move because it helps us avoid mistakes. Double-checking that we haven't missed any numbers or included any incorrect ones can make a big difference in the final answer.

We're not including -6 because the problem said greater than -6, not greater than or equal to. Similarly, we're not including 5 because the problem said less than 5, not less than or equal to. Make sure you really pay attention to the details of each question! The wording is important, and even slight differences can change the solution entirely. The list we've created is the foundation for solving the problem. So, take a look, make sure you understand it, and then we're good to go. It is good practice to repeat this list a few times, so that it becomes second nature to you.

Calculating the Sum: Adding the Integers

Now that we've identified all the integers that fit the criteria, it's time to add them up! This is where we put our mathematical skills to work and find the final answer. We're going to add all the numbers in our list: -5, -4, -3, -2, -1, 0, 1, 2, 3, and 4. There are a couple of ways to do this. You can add them one by one, or you can look for shortcuts to make the process easier and faster.

One neat trick is to look for pairs of numbers that add up to zero. For example, -1 and 1 cancel each other out, as do -2 and 2, and -3 and 3, and -4 and 4. This simplifies the addition considerably. So, after canceling out these pairs, what are we left with? Well, we still have -5 and 0. When we add them together, we get -5. See? It wasn't as hard as it might have seemed at first! Recognizing these kinds of patterns can save us a lot of time and effort, and it’s a great skill to develop in math. Another way to do it is by simply adding the numbers from left to right: -5 + -4 = -9; -9 + -3 = -12; -12 + -2 = -14; -14 + -1 = -15; -15 + 0 = -15; -15 + 1 = -14; -14 + 2 = -12; -12 + 3 = -9; -9 + 4 = -5. Of course, you should always do it in a way that feels comfortable to you! There’s no single right way to do it. The important thing is that you arrive at the correct answer. The more you practice these kinds of problems, the easier and faster it will become. And, of course, the more confident you'll be when it comes to tackling more complex math challenges in the future.

Step-by-Step Calculation

Let's break down the addition step by step to make sure we don't miss anything. First, we start by adding the negative numbers: -5 + -4 + -3 + -2 + -1 = -15. Then we add the positive numbers: 1 + 2 + 3 + 4 = 10. Finally, we add 0: 0. Now we can put them all together: -15 + 10 + 0 = -5. This is how we get to our final answer. It really helps to write down each step, so you can clearly see how you got the answer. Going through the steps also makes it easier to spot any mistakes, so you can correct them. When you check your work, you will also notice any patterns you can use to make the process easier.

By adding the integers this way, we're making sure we consider all the numbers and avoid making any mistakes in our calculations. Remember to double-check your work to be sure! This structured approach also makes it easier to catch any errors. Did you make the same calculation? Did you get the same result? If you didn't, don't worry! Go back and see where the mistake happened. Learning from your mistakes is one of the most effective ways to improve your math skills. Being careful with the addition and subtraction of integers is very important! You can also double check your calculation using a calculator. This allows you to verify your answer and build your confidence in your own skills.

The Answer and Conclusion

So, what's the final answer, guys? The sum of the integers greater than -6 and less than 5 is -5. Easy peasy, right? We've successfully navigated the problem by carefully identifying the relevant integers and then adding them up. The key takeaways here are understanding integer ranges and using simple addition to find the sum. We learned how to identify the numbers, and then we learned how to add them. Remember the number line? It's a great tool for visualizing these kinds of problems.

The Correct Answer

So, looking back at the original question and the answer choices, the correct answer is (C) -5. Congratulations! You've successfully solved the problem! Give yourself a pat on the back! Solving these kinds of problems builds a strong foundation in math, and that will help you tackle more complicated problems later on. You are now better prepared for tackling other math problems involving integers. The ability to do so will come in handy when you learn more math concepts, such as algebra, calculus, etc. So, keep up the great work and keep practicing! Being patient and practicing regularly is the secret to getting better and better at math. Keep an eye out for other problems like these, and you will become a master of integers in no time. The most important thing is that you stick with it and you don't give up! We are all capable of doing mathematics, and we can all grow to appreciate its beauty. With a little practice, you'll be able to solve similar problems with ease.