Mastering Absolute Value And Arithmetic Calculations Step By Step Solutions

In the realm of mathematics, absolute value represents a number's distance from zero, regardless of its sign. Mastering absolute value calculations is crucial for various mathematical concepts and applications. This comprehensive guide will walk you through the process of solving complex expressions involving absolute values, providing clear explanations and step-by-step solutions. We will dissect problems involving multiple operations within absolute value signs, ensuring you grasp the underlying principles and can confidently tackle similar challenges. Understanding how to accurately compute these expressions is a foundational skill in mathematics.

a) |-12 - 25 - 6| - |54 - 62 + (-23)|

This problem involves calculating the absolute value of two separate expressions and then finding the difference between them. To solve this, we must first simplify the expressions inside each absolute value bracket. Let's begin with the first absolute value: |-12 - 25 - 6|. Inside this absolute value, we have a series of subtractions. Combining these numbers, we get -12 - 25 - 6 = -43. Thus, the absolute value of -43 is |-43| = 43. The absolute value of a number is always non-negative, so we take the magnitude without considering the sign. Next, let's focus on the second absolute value: |54 - 62 + (-23)|. Here, we have a combination of subtraction and addition. First, subtract 62 from 54, which yields 54 - 62 = -8. Then, add -23 to this result: -8 + (-23) = -31. The absolute value of -31 is |-31| = 31. Now that we have computed both absolute values, we need to find the difference between them. The original problem asks for |-12 - 25 - 6| - |54 - 62 + (-23)|, which simplifies to 43 - 31. Subtracting 31 from 43 gives us 12. Therefore, the final answer is 12.

In summary, the key to solving this type of problem is to meticulously simplify the expressions inside each absolute value bracket before taking the absolute value. This involves correctly applying the rules of addition and subtraction. Once the values inside the absolute values are simplified, finding their absolute values becomes straightforward. The final step is to perform any remaining operations between the absolute values, in this case, subtraction. By breaking down the problem into these steps, it becomes manageable and less prone to errors. Remember, the absolute value always yields a non-negative result, reflecting the distance from zero on the number line.

b) |45 + (-11) - (-15) - 2 - (-18) - 6|

This expression requires careful handling of addition and subtraction within the absolute value. The absolute value expression given is |45 + (-11) - (-15) - 2 - (-18) - 6|. Our initial step involves simplifying the expression inside the absolute value bars. We begin by addressing the additions and subtractions sequentially. First, we have 45 + (-11), which is the same as 45 - 11, resulting in 34. Next, we subtract -15, which is equivalent to adding 15, so we get 34 + 15 = 49. Continuing the simplification, we subtract 2 from 49, giving us 49 - 2 = 47. Then, we subtract -18, which is the same as adding 18, so we have 47 + 18 = 65. Finally, we subtract 6 from 65, resulting in 65 - 6 = 59. Thus, the expression inside the absolute value simplifies to 59. Now, we take the absolute value of 59, which is |59|. Since 59 is already a positive number, its absolute value is simply 59.

To reiterate, the methodical approach is critical for such problems. We tackled the operations step by step, carefully managing the signs. Subtracting a negative number is the same as adding its positive counterpart, and this understanding is essential to avoiding mistakes. Each operation was performed in sequence, ensuring we maintained accuracy throughout the simplification process. The final step of taking the absolute value is straightforward once the expression inside has been fully simplified. This problem highlights the importance of attention to detail and the proper application of arithmetic rules. The ability to correctly manipulate these operations is a fundamental skill in mathematics, and practice in this area leads to greater confidence and proficiency.

c) |-75 + 57 - (-24) + (-15) - 17 - 9|

In this part, we are presented with another absolute value expression that requires careful simplification. The expression is |-75 + 57 - (-24) + (-15) - 17 - 9|. As before, the first step is to simplify the arithmetic inside the absolute value bars. We begin by adding -75 and 57. This gives us -75 + 57 = -18. Next, we subtract -24, which is the same as adding 24. So, we have -18 + 24 = 6. Then, we add -15, which is the same as subtracting 15, resulting in 6 - 15 = -9. Continuing, we subtract 17 from -9, giving us -9 - 17 = -26. Finally, we subtract 9 from -26, which yields -26 - 9 = -35. Therefore, the simplified expression inside the absolute value is -35. Now, we take the absolute value of -35, which is |-35| = 35.

To summarize the solution process, we methodically performed each addition and subtraction operation, paying close attention to the signs. Subtracting a negative number is treated as adding the positive number, and adding a negative number is treated as subtraction. This careful handling of signs is crucial for achieving the correct result. After simplifying the expression inside the absolute value bars, we arrived at -35. The absolute value of any number is its distance from zero, and thus it is always non-negative. Therefore, the absolute value of -35 is 35. This problem underscores the necessity of a step-by-step approach when dealing with multiple arithmetic operations, ensuring accuracy and clarity in the solution.

d) |-128 - (-144)| - |-65 + 61 + (-80)|

This problem involves calculating the difference between two absolute value expressions. The given expression is |-128 - (-144)| - |-65 + 61 + (-80)|. We need to simplify each absolute value expression separately before finding the difference. Let's start with the first absolute value: |-128 - (-144)|. Subtracting a negative number is equivalent to adding its positive counterpart, so we have -128 + 144. Performing this addition, we get -128 + 144 = 16. The absolute value of 16 is |16| = 16.

Now, let's simplify the second absolute value: |-65 + 61 + (-80)|. First, we add -65 and 61, which gives us -65 + 61 = -4. Next, we add -80 to this result: -4 + (-80) = -84. The absolute value of -84 is |-84| = 84. Now that we have the absolute values of both expressions, we can find the difference. The original problem asks for |-128 - (-144)| - |-65 + 61 + (-80)|, which simplifies to 16 - 84. Subtracting 84 from 16 gives us 16 - 84 = -68. Therefore, the final answer is -68.

In summary, we approached this problem by first simplifying each absolute value expression independently. The key to simplifying the expressions was to correctly apply the rules of addition and subtraction, especially when dealing with negative numbers. Subtracting a negative number was transformed into adding its positive counterpart. After simplifying the expressions inside the absolute value bars, we took the absolute values, which are always non-negative. Finally, we performed the subtraction between the two absolute values to arrive at the final answer. This problem illustrates the importance of breaking down complex expressions into smaller, manageable parts to ensure accurate calculations.

b) [+7 - (25 - 68)] + [-14 - (9 + 7)]

This problem involves simplifying an expression with multiple parentheses and brackets. The given expression is [+7 - (25 - 68)] + [-14 - (9 + 7)]. To solve this, we need to work from the innermost parentheses outwards, following the order of operations. First, let's simplify the expression within the first set of parentheses: (25 - 68). Subtracting 68 from 25 gives us 25 - 68 = -43. Now, we substitute this back into the expression: [+7 - (-43)]. Subtracting a negative number is the same as adding its positive counterpart, so we have 7 + 43, which equals 50.

Next, let's simplify the expression within the second set of parentheses: (9 + 7). Adding 9 and 7 gives us 16. Substituting this back into the second set of brackets, we have [-14 - 16]. Subtracting 16 from -14 gives us -14 - 16 = -30. Now, we have simplified the two bracketed expressions to 50 and -30, respectively. The original expression is [+7 - (25 - 68)] + [-14 - (9 + 7)], which now simplifies to 50 + (-30). Adding 50 and -30 is the same as subtracting 30 from 50, which results in 50 - 30 = 20. Therefore, the final answer is 20.

In summary, we tackled this problem by systematically simplifying the expressions within the parentheses and brackets. The order of operations is crucial in these types of problems: we started with the innermost parentheses and worked our way outwards. Subtracting a negative number was correctly handled by adding the corresponding positive number. Each step was performed carefully, ensuring accurate arithmetic. By breaking the problem down into smaller parts, we were able to simplify the expression and arrive at the final answer of 20. This problem highlights the importance of meticulousness and a step-by-step approach when dealing with complex arithmetic expressions.