Solving Integer Expression 6-(-4) - 15+7 A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of numbers and symbols? Don't worry, we've all been there! Today, we're going to break down a seemingly tricky integer expression and solve it together, step-by-step. Our mission is to conquer this expression: 6 - (-4) - 15 + 7. Sounds intimidating? Trust me, it's not! We'll tackle it with a friendly, conversational approach, ensuring you not only get the answer but also understand the process behind it. So, grab your pencils (or your favorite note-taking app!) and let's dive into the world of integers!

Understanding the Basics of Integer Expressions

Before we jump into solving our specific problem, let's quickly refresh our understanding of integers and how they behave in mathematical expressions. Integers, as you probably know, are whole numbers (no fractions or decimals!) that can be positive, negative, or zero. Think of them as numbers that represent both quantity and direction – like temperature above or below zero, or money earned or owed. When we're dealing with integer expressions, we're essentially combining these numbers using basic arithmetic operations: addition, subtraction, multiplication, and division. The key to solving these expressions lies in following the correct order of operations and paying close attention to the signs (positive or negative) of the integers involved.

The order of operations, often remembered by the acronym PEMDAS (or BODMAS in some regions), is our guiding principle. It tells us the sequence in which we should perform operations: Parentheses/Brackets first, then Exponents/Orders, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Ignoring this order can lead to incorrect answers, so it's crucial to keep it in mind. Another important aspect is understanding how negative signs interact with different operations. Subtracting a negative number is the same as adding its positive counterpart, and multiplying or dividing numbers with the same sign (both positive or both negative) results in a positive answer, while different signs result in a negative answer. These are the fundamental rules we'll be using to crack our expression!

Step 1: Dealing with the Double Negative

Okay, let's get our hands dirty with the actual problem: 6 - (-4) - 15 + 7. The first thing that probably jumps out at you is the double negative: -(-4). This is a classic situation in integer arithmetic, and it's actually quite simple to resolve. Remember what we discussed earlier? Subtracting a negative number is the same as adding its positive counterpart. So, -(-4) is the same as +4. Let's rewrite our expression, replacing -(-4) with +4: 6 + 4 - 15 + 7. See? We've already made the expression a bit cleaner and easier to work with. This step is crucial because it eliminates a potential source of confusion and sets us up for the next stage of solving. It's all about breaking down the problem into manageable chunks, and we've just conquered the first one!

This principle of transforming subtraction of a negative into addition is a cornerstone of integer manipulation. It might seem like a small detail, but mastering it is essential for accurately solving more complex expressions. Think of it as a secret weapon in your mathematical arsenal! So, let's take a moment to appreciate the power of this simple rule before we move on. Remember, subtracting a negative is like adding a positive. Keep that in mind, and you'll be well on your way to becoming an integer-solving pro!

Step 2: Performing Addition and Subtraction from Left to Right

Now that we've simplified our expression to 6 + 4 - 15 + 7, it's time to tackle the remaining operations. Remember PEMDAS (or BODMAS)? We've already taken care of any parentheses or exponents, and there's no multiplication or division in this expression. That leaves us with addition and subtraction. The rule here is to perform these operations from left to right, just like you read a sentence. This is super important because changing the order can change the answer! So, let's start with the leftmost operation: 6 + 4. This is straightforward addition, and the result is 10. Now, let's rewrite our expression again, replacing 6 + 4 with 10: 10 - 15 + 7.

We're making good progress! We've reduced the expression to just three terms. Next up is 10 - 15. This is where things get a little trickier, as we're subtracting a larger number from a smaller one, resulting in a negative number. If you think of a number line, you're starting at 10 and moving 15 units to the left. This lands you at -5. So, 10 - 15 = -5. Let's update our expression one more time: -5 + 7. We're almost there!

This step highlights the importance of paying attention to the signs of the integers. Subtracting a larger number from a smaller one always results in a negative number, and it's crucial to get this right. Visualizing a number line can be a helpful strategy here, especially when you're first learning about integer arithmetic. It provides a concrete way to understand how addition and subtraction work with negative numbers. Keep practicing these types of calculations, and they'll become second nature in no time!

Step 3: The Final Calculation

We've reached the final stage of our integer-solving journey! Our expression is now beautifully simplified to -5 + 7. This is a relatively simple addition problem involving a negative and a positive integer. Again, thinking of the number line can be helpful. You're starting at -5 and moving 7 units to the right. Where do you end up? You end up at 2! Therefore, -5 + 7 = 2. And there you have it! We've successfully solved the integer expression 6 - (-4) - 15 + 7, and the answer is 2.

This final step demonstrates how all our previous work has led us to a clear and concise solution. By breaking down the problem into smaller, manageable steps, we've avoided confusion and ensured accuracy. Remember, the key to solving complex problems is to approach them systematically and patiently. Don't try to do everything at once. Instead, focus on each step individually, and you'll be amazed at what you can achieve. We've conquered this expression together, and you've learned valuable skills that you can apply to many other mathematical challenges. Congratulations!

Conclusion: Mastering Integer Expressions

So, guys, we've successfully navigated the integer expression 6 - (-4) - 15 + 7 and arrived at the solution: 2. We've not only found the answer but also explored the underlying principles of integer arithmetic and the importance of following the order of operations. Remember, solving these expressions is like building a house – you need a solid foundation (understanding the basics) and a clear plan (following the steps in the correct order). We started by dealing with the double negative, then performed addition and subtraction from left to right, and finally arrived at our final answer.

Hopefully, this step-by-step guide has demystified integer expressions and equipped you with the tools and confidence to tackle similar problems. The key takeaways are: understanding the order of operations (PEMDAS/BODMAS), knowing how to handle negative signs, and breaking down complex problems into smaller, manageable steps. Practice makes perfect, so keep working on these types of expressions, and you'll become a true integer-solving master! Math might seem daunting at times, but with a little patience and the right approach, you can conquer any challenge. Keep exploring, keep learning, and most importantly, keep having fun with numbers!