Simplify The Expression (3-4|2(-3)|)/|-3|.

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In this comprehensive guide, we will break down the process of simplifying the mathematical expression 3−4∣2(−3)∣∣−3∣\frac{3-4|2(-3)|}{|-3|}. This problem involves absolute values and order of operations, making it a great exercise for honing your arithmetic skills. Whether you're a student looking to improve your math grade or simply someone who enjoys problem-solving, this detailed explanation will help you understand each step clearly. We will explore the concepts of absolute values, multiplication, subtraction, and division, providing a thorough walkthrough that ensures you grasp the underlying principles. By the end of this guide, you'll not only be able to solve this specific problem but also tackle similar expressions with confidence.

Understanding Absolute Values

Before we dive into the simplification process, it's crucial to understand absolute values. The absolute value of a number is its distance from zero on the number line. This means that the absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart. For example, the absolute value of 5, denoted as |5|, is 5, and the absolute value of -5, denoted as |-5|, is also 5. This concept is essential because it dictates how we handle negative signs within the expression. Understanding the definition of absolute value is the cornerstone for solving expressions like 3−4∣2(−3)∣∣−3∣\frac{3-4|2(-3)|}{|-3|}. The absolute value ensures that we are dealing with the magnitude of the number, irrespective of its sign. This is particularly important in complex expressions where negative signs can significantly alter the outcome if not handled correctly. When we see absolute value bars, we must first evaluate the expression inside them and then take the absolute value of the result. This step-by-step approach ensures accuracy and avoids common pitfalls. In our given expression, we encounter absolute values in two places: |2(-3)| and |-3|. We need to address these first to simplify the numerator and the denominator, respectively. Ignoring the absolute value would lead to a fundamentally different problem, so understanding this concept is paramount. By grasping the essence of absolute values, we set the stage for a smooth and accurate simplification process. Remember, the key is to treat the absolute value as a way to strip away the negative sign, leaving us with the positive magnitude of the number. This principle guides us through the subsequent steps of the problem, ensuring a correct final answer.

Step-by-Step Simplification

To simplify the expression 3−4∣2(−3)∣∣−3∣\frac{3-4|2(-3)|}{|-3|}, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). First, we'll address the absolute values. Inside the numerator, we have |2(-3)|. Multiplying 2 by -3 gives us -6. The absolute value of -6, |-6|, is 6. In the denominator, we have |-3|, which is simply 3. So, the expression now becomes 3−4(6)3\frac{3-4(6)}{3}. Next, we perform the multiplication in the numerator: 4 multiplied by 6 is 24. Our expression is now 3−243\frac{3-24}{3}. Then, we subtract 24 from 3 in the numerator, which gives us -21. So, the expression simplifies to −213\frac{-21}{3}. Finally, we divide -21 by 3, which results in -7. Therefore, the simplified form of the expression is -7. Following the order of operations meticulously is crucial for arriving at the correct answer. Each step builds upon the previous one, and any error in one step can cascade through the rest of the solution. This methodical approach not only ensures accuracy but also enhances understanding of the mathematical principles involved. In this particular problem, the presence of absolute values adds an extra layer of complexity, but by addressing them early in the process, we streamline the subsequent calculations. Remember, absolute values always yield non-negative results, which simplifies the arithmetic. The step-by-step simplification process highlights the importance of breaking down a complex problem into smaller, manageable steps. This strategy is applicable not only to mathematical expressions but also to various problem-solving scenarios in different fields. By mastering the art of simplification, we enhance our ability to tackle challenges effectively and efficiently. The final answer, -7, is the culmination of this careful and systematic approach, demonstrating the power of precision in mathematics. This method provides a clear and concise pathway to the solution, making it easier to understand and replicate in similar problems.

Detailed Breakdown of the Numerator

Let's take a closer look at simplifying the numerator of the expression, which is 3 - 4|2(-3)|. The key to handling this part of the expression lies in the correct application of the order of operations and the understanding of absolute values. First, we focus on the innermost part of the expression within the absolute value bars: 2 multiplied by -3. This multiplication results in -6. So, we now have 3 - 4|-6|. Next, we need to evaluate the absolute value of -6. As we discussed earlier, the absolute value of a number is its distance from zero, so |-6| is 6. This simplifies our numerator to 3 - 4(6). Now, we perform the multiplication: 4 multiplied by 6 is 24. The expression becomes 3 - 24. Finally, we subtract 24 from 3, which gives us -21. Therefore, the simplified form of the numerator is -21. A meticulous breakdown of the numerator is essential because it contains the bulk of the operations that need to be performed. This step-by-step approach ensures that we don't overlook any crucial details and that we handle the absolute value and multiplication correctly before moving on to subtraction. The order in which we perform these operations is critical; deviating from PEMDAS can lead to an incorrect result. The absolute value, in particular, plays a pivotal role in this process. By understanding that it transforms any negative number into its positive counterpart, we can proceed with the subsequent calculations with confidence. The detailed analysis of the numerator not only helps in solving this specific problem but also enhances our understanding of how complex expressions are handled in general. This skill is invaluable in more advanced mathematical contexts. Breaking down the numerator into manageable steps makes the entire process less daunting and more accessible. This approach allows us to focus on each operation individually, minimizing the chances of making errors. The result, -21, is a crucial component in the overall simplification of the expression, and its accurate calculation is paramount to arriving at the correct final answer. This thorough methodology is a testament to the importance of precision and clarity in mathematical problem-solving.

Simplifying the Denominator

The denominator of the expression 3−4∣2(−3)∣∣−3∣\frac{3-4|2(-3)|}{|-3|} is simply |-3|. Simplifying this part is straightforward and crucial for the overall solution. The absolute value of a number, as we've discussed, is its distance from zero. In this case, the number is -3, and its distance from zero is 3. Therefore, |-3| simplifies to 3. This step might seem simple, but it's an essential part of the process. Simplifying the denominator is often a quicker task compared to the numerator, but its importance should not be underestimated. A mistake in the denominator can lead to a completely incorrect final answer. In this instance, the denominator consists of a single absolute value expression, making the simplification relatively direct. However, in more complex problems, the denominator might involve multiple operations or nested expressions, requiring a similar step-by-step approach as the numerator. The key is to identify the components of the denominator and apply the relevant mathematical principles to simplify them. Understanding the absolute value concept is paramount here. The absolute value transforms any negative number into its positive equivalent, ensuring that we are dealing with the magnitude of the number. This transformation is essential for subsequent calculations. The simplified denominator, 3, serves as the divisor in the final step of the problem. Its accurate calculation sets the stage for the final division, which will yield the solution to the expression. By addressing the denominator separately, we ensure that each part of the expression is handled with precision, contributing to the overall accuracy of the solution. This methodical approach is a hallmark of effective problem-solving in mathematics and other disciplines.

Final Calculation and the Answer

After simplifying the numerator to -21 and the denominator to 3, we now have the expression −213\frac{-21}{3}. The final step is to perform the division: -21 divided by 3. This division results in -7. Therefore, the simplified form of the original expression 3−4∣2(−3)∣∣−3∣\frac{3-4|2(-3)|}{|-3|} is -7. This is our final answer. The final calculation is the culmination of all the previous steps. It brings together the simplified numerator and denominator to produce the solution. This step requires a solid understanding of division and how to handle negative numbers. The negative sign in the numerator indicates that the result will be negative, which is a crucial detail to keep in mind. The division itself is straightforward, but its accuracy is paramount. A mistake in this final step would negate all the correct work done previously. The final answer, -7, represents the simplified value of the entire expression. It is the culmination of a methodical and precise approach, highlighting the importance of each step along the way. From understanding absolute values to following the order of operations, each component played a critical role in arriving at this solution. The ability to simplify complex expressions is a valuable skill in mathematics and beyond. It demonstrates a strong understanding of fundamental principles and the ability to apply them effectively. This step-by-step guide has illustrated how to break down a seemingly complicated problem into manageable parts, ensuring accuracy and clarity. The final result, -7, stands as a testament to the power of careful calculation and attention to detail in mathematical problem-solving. This thorough process not only provides the correct answer but also reinforces the underlying concepts, making it easier to tackle similar problems in the future.

Conclusion

In conclusion, we have successfully simplified the expression 3−4∣2(−3)∣∣−3∣\frac{3-4|2(-3)|}{|-3|} to -7. This process involved a thorough understanding of absolute values, the order of operations, and careful arithmetic. By breaking down the problem into manageable steps, we were able to tackle each part with precision and accuracy. The key steps included simplifying the absolute value expressions, performing the multiplication and subtraction in the numerator, and finally, dividing the simplified numerator by the simplified denominator. This step-by-step approach not only led us to the correct answer but also enhanced our understanding of the underlying mathematical principles. The conclusion of this guide reinforces the importance of a systematic approach to problem-solving in mathematics. By revisiting the key steps and highlighting the concepts involved, we solidify the learning experience. The ability to simplify complex expressions is a valuable skill that can be applied in various contexts. It demonstrates a mastery of fundamental mathematical principles and the ability to apply them effectively. This guide has provided a clear and concise pathway to the solution, making it easier to understand and replicate in similar problems. The final answer, -7, is a testament to the power of careful calculation and attention to detail. By following the outlined steps, anyone can confidently tackle similar expressions and arrive at the correct answer. The process of simplification is not just about finding the solution; it's also about developing a deeper understanding of the mathematical concepts involved. This guide has aimed to achieve both, providing a comprehensive resource for anyone looking to improve their problem-solving skills in mathematics. The systematic approach and detailed explanations make this guide a valuable tool for students, educators, and anyone with an interest in mathematical problem-solving.