Solving $-2--5=?$ A Step-by-Step Guide

Introduction: Mastering Integer Subtraction

In the realm of mathematics, understanding integer subtraction is a fundamental skill, crucial for navigating various mathematical concepts and real-world applications. This article delves into the intricacies of the given problem, $-2--5=?$, aiming to provide a comprehensive explanation of the solution while reinforcing the core principles of integer subtraction. We'll explore the rules governing subtraction with negative numbers, dissect the problem step-by-step, and arrive at the correct answer, accompanied by clear explanations and illustrative examples.

The core concept at play here is the subtraction of a negative number, which can often be confusing for learners. We will demystify this concept by demonstrating how subtracting a negative number is equivalent to adding its positive counterpart. This understanding is critical not only for solving this specific problem but also for building a solid foundation in algebra and other advanced mathematical topics. Integer subtraction is not just an abstract mathematical exercise; it has practical implications in various fields, such as finance, physics, and computer science. For example, calculating temperature differences, tracking financial transactions, or determining changes in altitude all involve integer subtraction. Therefore, mastering this skill is essential for both academic success and real-world problem-solving.

Understanding the rules of integer subtraction also lays the groundwork for more complex algebraic manipulations. When solving equations, simplifying expressions, or graphing functions, the ability to confidently handle negative numbers and their operations is paramount. This article aims to empower readers with a clear and concise understanding of integer subtraction, equipping them with the skills necessary to tackle a wide range of mathematical challenges. By the end of this discussion, you will not only be able to solve problems like $-2--5=?$ but also gain a deeper appreciation for the elegance and logic of mathematics.

Deconstructing the Problem: $-2--5=?$

The problem at hand, $-2--5=?$, involves subtracting a negative number from another negative number. To solve this, we must recall the fundamental rule: subtracting a negative number is the same as adding its positive counterpart. This concept is crucial for understanding how to manipulate expressions with negative numbers accurately. Let's break down the equation step by step to see how this rule applies in practice.

Initially, we have $-2--5$. The double negative (--) indicates that we are subtracting a negative five. According to the rule we just discussed, this is equivalent to adding positive five. Therefore, we can rewrite the expression as $-2 + 5$. Now, the problem becomes a simple addition of two integers with opposite signs. To solve this, we find the difference between the absolute values of the two numbers and assign the sign of the number with the larger absolute value.

In this case, the absolute value of -2 is 2, and the absolute value of 5 is 5. The difference between 5 and 2 is 3. Since 5 has a larger absolute value and is positive, the result will be positive. Therefore, $-2 + 5 = 3$. This demonstrates how the initial subtraction problem transforms into an addition problem, making it easier to solve. This step-by-step approach not only helps in finding the correct answer but also reinforces the understanding of the underlying mathematical principles.

This transformation of subtraction into addition is a key technique in handling integer operations. It simplifies the process and reduces the chances of making errors. By understanding this rule, you can confidently tackle similar problems involving negative numbers. Moreover, this concept is not limited to simple arithmetic; it extends to more complex algebraic expressions and equations, making it a foundational skill for advanced mathematics. By mastering this technique, you'll find that many seemingly daunting problems become much more manageable.

Solving the Equation: Step-by-Step Solution

To definitively solve the equation $-2--5=?$, let's walk through each step meticulously. This detailed approach ensures a clear understanding of the process and minimizes the possibility of errors. As we've established, subtracting a negative number is equivalent to adding its positive counterpart. This principle forms the basis of our solution.

Step 1: Rewrite the Subtraction as Addition The first and most crucial step is to transform the subtraction of a negative number into addition. The expression $-2--5$ can be rewritten as $-2 + 5$. This transformation is based on the fundamental mathematical rule that subtracting a negative is the same as adding a positive. This single step significantly simplifies the problem and makes it easier to solve. By rewriting the equation, we eliminate the confusion often associated with double negative signs and convert the problem into a more familiar addition format.

Step 2: Perform the Addition Now that we have the expression $-2 + 5$, we can proceed with the addition. We are adding two integers with different signs, one negative (-2) and one positive (5). To perform this addition, we find the difference between their absolute values and assign the sign of the number with the larger absolute value. The absolute value of -2 is 2, and the absolute value of 5 is 5. The difference between 5 and 2 is 3. Since 5 has a larger absolute value and is positive, the result will be positive. Therefore, $-2 + 5 = 3$.

Step 3: Verify the Solution To ensure the accuracy of our solution, it's always a good practice to verify the result. We found that $-2--5 = 3$. We can confirm this by thinking of a number line. Starting at -2 and moving 5 units to the right (because we are subtracting a negative, which is equivalent to adding) lands us at 3. This mental verification strengthens our confidence in the solution.

By breaking down the problem into these three steps, we can see how a seemingly complex expression is simplified and solved. This method not only provides the correct answer but also enhances understanding of the underlying mathematical principles. This step-by-step approach is a valuable tool for tackling various mathematical problems, ensuring accuracy and clarity in the solution process.

The Answer and Options Analysis

After carefully solving the equation $-2--5=?$, we have arrived at the solution: 3. Now, let's analyze the provided options to identify the correct answer and understand why the other options are incorrect. This analysis will reinforce our understanding of the problem and the solution process.

The given options are:

• F) 3
• G) -3
• H) -7
• J) 7
• K) None of these

By comparing our solution (3) with the options, it is clear that option F) 3 is the correct answer. This confirms our step-by-step calculation and ensures that we have accurately applied the rules of integer subtraction.

Now, let's examine why the other options are incorrect:

• G) -3: This answer would be obtained if we incorrectly treated the subtraction of a negative as a simple subtraction. If we were to calculate -2 - 5, we would get -7, not -3. This highlights the importance of understanding that subtracting a negative is equivalent to adding a positive.
• H) -7: This answer is the result of adding the two negative numbers without properly addressing the double negative. It shows a misunderstanding of the order of operations and the rules for subtracting negative numbers.
• J) 7: This option is incorrect because it seems to add the absolute values of the numbers without considering their signs. It fails to account for the fundamental rule that subtracting a negative is adding a positive.
• K) None of these: This option would be correct only if none of the other provided options matched the correct solution. Since we have already determined that option F is correct, this option is incorrect.

This thorough analysis of the options not only confirms our solution but also provides valuable insights into common errors and misunderstandings related to integer subtraction. By understanding why the incorrect options are wrong, we strengthen our grasp of the underlying mathematical principles and improve our problem-solving skills.

Conclusion: Solidifying Understanding of Integer Subtraction

In conclusion, we have successfully solved the equation $-2--5=?$, demonstrating the crucial principle of subtracting a negative number. Through a step-by-step approach, we transformed the subtraction of a negative into the addition of a positive, arriving at the correct answer of 3. This process not only provides the solution but also reinforces a fundamental concept in mathematics.

Throughout this discussion, we have emphasized the importance of understanding the rules governing integer subtraction. Subtracting a negative number is not just a mathematical trick; it is a logical operation with a solid mathematical basis. By recognizing that subtracting a negative is equivalent to adding its positive counterpart, we can confidently tackle a wide range of problems involving negative numbers.

Furthermore, we analyzed the given options, identifying the correct answer (F) 3 and explaining why the other options were incorrect. This option analysis serves as a valuable learning tool, highlighting common mistakes and reinforcing the correct application of mathematical principles. Understanding why incorrect answers are wrong is as important as knowing why the correct answer is right.

The skills and concepts discussed in this article are not limited to this specific problem. Integer subtraction is a foundational skill that underpins various mathematical concepts, including algebra, calculus, and beyond. Mastering this skill will undoubtedly enhance your problem-solving abilities and provide a solid foundation for future mathematical endeavors.

By carefully deconstructing the problem, solving it step-by-step, and analyzing the options, we have not only found the correct answer but also deepened our understanding of integer subtraction. This comprehensive approach is key to building confidence and proficiency in mathematics. Remember, practice and a clear understanding of the underlying principles are essential for success in mathematics. This article serves as a stepping stone towards achieving that goal.