Simplifying Expressions: -42 - 19 - (-24)

by ADMIN 42 views

Hey math enthusiasts! Today, we're diving into a straightforward expression: -42 - 19 - (-24). Don't worry, it looks a bit intimidating at first glance, but trust me, it's totally manageable. We're going to break it down step by step, making sure everyone understands how to solve these kinds of problems. This is a fundamental concept in arithmetic, and mastering it will set you up for success in more complex mathematical adventures down the road. So, grab your pencils (or your favorite digital devices) and let's get started. Understanding how to handle negative numbers and double negatives is key to acing this. By the end, you'll be able to confidently tackle similar problems with ease. Let's make math fun and less scary, alright?

Breaking Down the Expression: The Fundamentals

Alright, guys, let's start with the basics. Our expression is -42 - 19 - (-24). The first thing to remember is that subtracting a positive number is the same as adding a negative number. So, the '-19' is really like adding a negative nineteen. And the most crucial part? The double negative, ' -(-24)'. Do you know what happens when you subtract a negative number? Yep, it's the same as adding a positive number. So, '-(-24)' becomes '+ 24'. Think of it like this: two negatives cancel each other out, like magic! Now, we can rewrite our expression to make it a little easier to digest: -42 - 19 + 24. See? It's already looking friendlier. Now, let's consider the signs. When dealing with addition and subtraction of integers, the signs are critical. Always pay attention to whether a number is positive or negative, because that determines how it affects the overall value of the expression. This meticulous approach helps prevent silly mistakes and ensures accurate solutions. Also, remember the order of operations (PEMDAS/BODMAS) doesn't explicitly come into play here because we're just dealing with addition and subtraction, but the concept of working left to right still holds. Let's work through it step by step, keeping an eye on those signs.

Now, let's talk about the order in which we solve this. We'll work from left to right, because that's the simplest way to go about it when you only have addition and subtraction. First, we'll deal with -42 - 19. If you're struggling with this, just think of it like adding the absolute values and keeping the negative sign. In other words, you add 42 and 19 to get 61, and since both numbers were negative, the result is -61. So, our expression now becomes -61 + 24. We're getting closer to the solution, aren't we? Remember, the aim is to simplify, simplify, simplify, until you arrive at a single number, which is our final answer. It may seem like a long process, but it's important to remember that each step brings you closer to the correct answer. The more you practice, the faster and more comfortable you'll become with this process.

Step-by-Step Solution: Unveiling the Answer

Alright, let's get down to the nitty-gritty and work this problem out step by step. Our simplified expression is -42 - 19 + 24. First things first: let's tackle -42 - 19. As we discussed, subtracting 19 from -42 is the same as adding -19 to -42. When we add two negative numbers, we add their absolute values and keep the negative sign. So, 42 + 19 = 61. Therefore, -42 - 19 = -61. Now, our expression becomes -61 + 24. We've effectively reduced the problem to this simpler form. When we have a negative number and a positive number being added, we subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value. In this case, 61 is larger than 24, and 61 is negative, so our answer will be negative. We subtract 24 from 61, which gives us 37. Therefore, -61 + 24 = -37. That's it, guys! We have successfully simplified the expression. The final answer is -37. We've solved the expression by breaking it down into smaller, more manageable steps, and we made sure to pay close attention to the signs. Always remember: practice makes perfect, and with each problem you solve, you'll gain more confidence and a deeper understanding of these concepts.

Here’s a quick recap of the steps:

  1. Original Expression: -42 - 19 - (-24)
  2. Simplify the double negative: -42 - 19 + 24
  3. Combine the first two terms: -42 - 19 = -61
  4. Rewrite the expression: -61 + 24
  5. Calculate the final answer: -61 + 24 = -37

Common Mistakes and How to Avoid Them

Let's talk about some common pitfalls when dealing with these types of expressions. The most frequent mistake? Mismanaging those negative signs. Seriously, it's super easy to get them mixed up, especially when you're in a hurry. Here's a tip: always, always, double-check your signs. Write them down, circle them, whatever it takes to keep track. Another common issue is forgetting the order of operations. While our particular problem doesn't involve multiplication or division, it's a good habit to keep PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) in mind. This will help you in more complex problems later on. Sometimes, students will incorrectly perform the subtraction or addition. For example, they might add -42 and 19 incorrectly or subtract the wrong way. To avoid this, always make sure you're adding and subtracting the correct numbers, and pay attention to whether the result should be positive or negative. And finally, rushing can lead to errors. Take your time. Write out each step clearly. It's better to be slow and accurate than fast and wrong. Trust me on this one. Rushing through a math problem can lead to careless errors, and the extra few seconds you take to double-check can save you from a lot of frustration and wasted effort. So, take a deep breath, stay focused, and you’ll do great! Remember, math is like building a house; you need a solid foundation.

To avoid common mistakes:

  • Double-check signs: Negative signs are tricky; always verify.
  • Take your time: Don't rush; accuracy matters.
  • Practice regularly: The more you practice, the better you'll get.

Conclusion: Mastering the Basics

Alright, folks, we've reached the end of our journey through this expression. You've seen how to simplify -42 - 19 - (-24) step by step. We've tackled the double negative, combined the numbers, and arrived at the correct answer: -37. Congratulations! You've successfully navigated a fundamental concept in arithmetic. Remember, mastering these basic skills is crucial for your future mathematical endeavors. Whether you're planning on tackling algebra, calculus, or even just balancing your checkbook, understanding how to work with integers is essential. So, keep practicing, keep learning, and don't be afraid to ask for help when you need it. Math is a journey, not a destination, and every problem you solve brings you closer to becoming a math whiz. Consider this as a small victory, a stepping stone on your path to mathematical prowess. Every expression you solve increases your confidence and understanding. Keep challenging yourself, and remember that even the most complicated problems can be broken down into manageable steps. Keep practicing, and you'll be amazed at how quickly you improve. Now go forth and conquer more expressions! You've got this!