Solving (-3) + 6 + (-7) - (-9) Step-by-Step

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Hey guys! Today, we're going to break down a common type of math problem: solving expressions with positive and negative numbers. Specifically, we'll tackle the expression (-3) + 6 + (-7) - (-9) step by step. This kind of problem is fundamental in algebra and arithmetic, and mastering it will help you build a strong foundation for more complex calculations. We'll go through each operation, explaining the rules and logic behind them, so you not only get the correct answer but also understand why it's the correct answer. No more math mysteries, just clear and simple steps! We'll start by understanding the basic principles of adding and subtracting negative numbers, which can sometimes be a bit tricky. Then, we'll apply these principles to our specific problem, making sure each step is crystal clear. Remember, math isn't about memorizing formulas, it's about understanding the concepts. So, grab your pencils and let's dive in! By the end of this guide, you'll be solving similar expressions with confidence. We will also discuss common mistakes and how to avoid them, ensuring you develop a solid understanding of the underlying mathematical principles. This guide is designed for students of all levels, whether you're just starting out with algebra or need a refresher on basic arithmetic. So, let's get started and make math a little less intimidating and a lot more fun!

Understanding the Basics: Positive and Negative Numbers

Before we jump into our problem, let's quickly review the basics of positive and negative numbers. Think of a number line: zero is in the middle, positive numbers are to the right, and negative numbers are to the left. Adding a positive number moves you to the right, while adding a negative number moves you to the left. Subtraction is the opposite: subtracting a positive number moves you left, and subtracting a negative number moves you right. This mental picture is super helpful when dealing with these kinds of calculations. For instance, imagine you're standing at -3 on the number line. If you add 6, you move 6 steps to the right, which will land you at 3. Now, if you add -7, you move 7 steps to the left from 3, which takes you to -4. Understanding this movement along the number line can make the process of adding and subtracting negative numbers much more intuitive. It's also crucial to remember the concept of additive inverses. Every number has an additive inverse, which is the number that, when added to the original number, results in zero. For example, the additive inverse of 5 is -5, and vice versa. Recognizing additive inverses can simplify complex expressions and make calculations easier. We'll see how this applies in our problem as we go through the steps. Furthermore, understanding the properties of positive and negative numbers extends beyond simple addition and subtraction. It's also essential for multiplication and division, where the rules for handling signs are slightly different. For example, multiplying two negative numbers results in a positive number, while multiplying a positive number by a negative number results in a negative number. These foundational concepts are the building blocks for more advanced mathematics, so having a solid grasp of them is crucial for your mathematical journey. Let's keep these principles in mind as we tackle our problem step by step.

Step 1: Rewrite the Expression

The first thing we're going to do is rewrite our expression (-3) + 6 + (-7) - (-9) to make it a little easier to work with. Remember that subtracting a negative number is the same as adding its positive counterpart. So, "- (-9)" becomes "+ 9". Our expression now looks like this: (-3) + 6 + (-7) + 9. This simple change can significantly reduce confusion and make the subsequent calculations smoother. It's like turning a complex sentence into simpler phrases; it makes the whole thing easier to digest. By changing the subtraction of a negative number into addition, we're essentially applying the rule that subtracting a negative is the same as adding a positive. This rule is a cornerstone of arithmetic and algebra, and understanding it is key to mastering mathematical operations with signed numbers. Think of it this way: if you're taking away a debt (a negative), you're effectively increasing your assets (adding a positive). This real-world analogy can help solidify the concept in your mind. Additionally, rewriting the expression in this way allows us to group positive and negative numbers together more easily in the next steps. This grouping strategy is a common technique used to simplify complex calculations and make them more manageable. It's all about breaking down the problem into smaller, more digestible parts. So, by rewriting the expression, we've already taken a significant step towards solving it. Now, we're ready to move on to the next stage, where we'll start combining the numbers and simplifying the expression further. Remember, each step we take is a piece of the puzzle, and by breaking down the problem into smaller steps, we're making it much easier to solve.

Step 2: Combine the Numbers from Left to Right

Now that we have our rewritten expression, (-3) + 6 + (-7) + 9, we can start combining the numbers. We'll work from left to right, just like reading a sentence. First, let's tackle (-3) + 6. Imagine you're at -3 on the number line and you move 6 steps to the right. Where do you end up? You'll land at 3. So, (-3) + 6 = 3. Now, our expression looks like this: 3 + (-7) + 9. See how we're slowly but surely making progress? It's like building with LEGO bricks; each step adds to the final structure. By performing the operations in a step-by-step manner, we minimize the chances of making errors and ensure that we're following the correct order of operations. This approach is particularly helpful when dealing with multiple numbers and different signs, as it allows us to focus on one operation at a time. Now, let's move on to the next part of the expression: 3 + (-7). Again, think of the number line. You're at 3, and you need to move 7 steps to the left. This will take you into the negative territory. How far left will you go? You'll end up at -4. So, 3 + (-7) = -4. Our expression is now simplified to -4 + 9. We're almost there! By continuing to work from left to right, we're ensuring that we're following the correct mathematical convention. This consistency is key to accurate calculations and problem-solving. Remember, math is like a language, and just like in any language, there are rules that we need to follow to communicate effectively. So, let's keep going and finish this problem strong!

Step 3: Final Calculation

We've reached the final step! Our expression has been simplified to -4 + 9. This is a straightforward calculation. If you're at -4 on the number line and you move 9 steps to the right, where do you end up? You'll land at 5. Therefore, -4 + 9 = 5. And that's our answer! We've successfully solved the expression (-3) + 6 + (-7) - (-9). High five! It's amazing how a seemingly complex problem can be broken down into simple steps, isn't it? This final calculation highlights the importance of understanding how positive and negative numbers interact. By visualizing the number line and thinking about movements to the left and right, we can easily perform these calculations without getting bogged down in complex rules. It's all about making math intuitive and relatable. Now that we've reached the solution, it's a good idea to take a moment to review the entire process. We started by rewriting the expression to make it easier to work with, then we combined the numbers step by step from left to right, and finally, we arrived at our answer. This methodical approach is not only effective for solving this specific problem but also applicable to a wide range of mathematical challenges. Remember, practice makes perfect, so the more you work through problems like this, the more confident you'll become in your mathematical abilities. So, congratulations on reaching the end of this guide! You've not only solved a math problem but also learned valuable problem-solving strategies that will serve you well in your future mathematical endeavors.

Common Mistakes and How to Avoid Them

When dealing with expressions like (-3) + 6 + (-7) - (-9), there are a few common mistakes that students often make. Let's discuss these and how to avoid them. One frequent error is misinterpreting the subtraction of a negative number. Remember, subtracting a negative is the same as adding a positive. For example, - (-9) becomes + 9. Forgetting this rule can lead to incorrect calculations. To avoid this, always rewrite the expression first to convert subtractions of negatives into additions. Another common mistake is not paying attention to the order of operations. While we solved this problem from left to right, it's crucial to remember the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) for more complex expressions. In our case, there were no parentheses or exponents, but in other problems, these would need to be addressed first. To prevent errors related to the order of operations, make sure to carefully analyze the expression and identify the operations that need to be performed in the correct sequence. Additionally, sign errors are a common pitfall when working with positive and negative numbers. For example, adding a negative number might be confused with subtracting a positive number, or vice versa. To avoid these mistakes, it's helpful to visualize the number line and think about the direction of movement. Adding a negative number moves you to the left, while subtracting a positive number also moves you to the left. Conversely, adding a positive number moves you to the right, and subtracting a negative number moves you to the right. Furthermore, rushing through the calculations can also lead to errors. It's essential to take your time, double-check each step, and ensure that you're applying the correct operations and rules. A methodical approach, where you break down the problem into smaller, more manageable steps, can significantly reduce the chances of making mistakes. Finally, it's always a good idea to practice regularly. The more you work through problems involving positive and negative numbers, the more comfortable and confident you'll become in your ability to solve them accurately. So, keep practicing, and don't be afraid to make mistakes – they're a natural part of the learning process!

Practice Problems

To solidify your understanding of solving expressions with positive and negative numbers, let's try a few practice problems. These problems will give you the chance to apply the steps we've discussed and build your confidence. Remember, practice is key to mastering any mathematical concept. Here are a few problems to get you started:

  1. (-5) + 8 + (-2) - (-4)
  2. 7 + (-9) - (-3) + (-1)
  3. (-10) - (-6) + 4 + (-5)
  4. 2 - 5 + (-8) - (-11)

For each problem, follow the steps we've outlined in this guide. First, rewrite the expression to convert subtractions of negatives into additions. Then, combine the numbers from left to right, one step at a time. Finally, double-check your answer to ensure that you haven't made any sign errors or other mistakes. Don't be afraid to use the number line visualization to help you with the calculations. Imagine yourself moving along the number line as you add and subtract the numbers. This can make the process more intuitive and less prone to errors. If you get stuck on a problem, go back and review the steps we've discussed in this guide. Pay particular attention to the rules for adding and subtracting negative numbers, as these are often the source of confusion. Remember, it's okay to make mistakes – they're a valuable learning opportunity. The important thing is to learn from your mistakes and keep practicing. You can also try creating your own practice problems by varying the numbers and signs in the expression. This will help you develop a deeper understanding of the underlying concepts and improve your problem-solving skills. And if you're still struggling, don't hesitate to seek help from a teacher, tutor, or classmate. Math is a collaborative effort, and there are plenty of resources available to support your learning journey. So, grab your pencil, put on your thinking cap, and let's tackle these practice problems! With consistent effort and practice, you'll become a master of solving expressions with positive and negative numbers.

Conclusion

Alright guys, we've reached the end of our step-by-step guide to solving (-3) + 6 + (-7) - (-9)! We've covered the basics of positive and negative numbers, learned how to rewrite expressions, combined numbers systematically, and even discussed common mistakes to avoid. You've come a long way! The key takeaway here is that complex math problems can be broken down into smaller, manageable steps. By understanding the underlying principles and following a logical approach, you can tackle even the trickiest expressions with confidence. Remember the importance of rewriting expressions to eliminate the subtraction of negative numbers, and always work from left to right to maintain consistency. Visualize the number line to help you understand how positive and negative numbers interact, and don't be afraid to take your time and double-check your work. Math isn't about speed; it's about accuracy and understanding. Now that you've mastered this specific problem, you have a solid foundation for tackling other similar expressions. The skills you've learned here are transferable and will serve you well in your future mathematical endeavors. So, keep practicing, keep exploring, and keep challenging yourself. Math is a journey, not a destination, and there's always something new to learn and discover. And remember, if you ever feel stuck or confused, don't hesitate to ask for help. There's a whole community of mathematicians out there ready to support you on your journey. So, go forth and conquer those math problems! You've got this! Congratulations on completing this guide, and I hope you found it helpful and informative. Keep up the great work, and I look forward to seeing all the amazing things you'll achieve in math!