Simplifying Expressions And Absolute Values A Step By Step Guide

In the realm of mathematics, simplifying expressions and determining their absolute values are fundamental skills. This article delves into the process of simplifying mathematical expressions and subsequently finding their absolute values. We will explore various operations such as division, multiplication, subtraction, and the order of operations, while emphasizing the significance of the absolute value concept. Understanding these concepts is crucial for building a strong foundation in mathematics and its applications in various fields. We will dissect each expression step by step, ensuring a clear understanding of the underlying principles. This comprehensive guide aims to equip you with the knowledge and confidence to tackle similar mathematical challenges.

The following expressions will be simplified and their absolute values will be determined:

i) |20 ÷ (-5)| - 4 ii) |6 × (-3)| iii) |(-40) - (-45) ÷ 5 - (-2)| iv) |6 × (-5) - (-3)|

i) |20 ÷ (-5)| - 4

Let's begin by simplifying the first expression: |20 ÷ (-5)| - 4. The key here is to remember the order of operations, often remembered by the acronym PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this expression, we have division within the absolute value and then subtraction outside the absolute value. The absolute value of a number is its distance from zero on the number line, so it is always non-negative.

First, we perform the division inside the absolute value: 20 ÷ (-5). When dividing a positive number by a negative number, the result is negative. 20 divided by 5 is 4, so 20 ÷ (-5) equals -4. Now the expression becomes |-4| - 4. The absolute value of -4 is 4, because -4 is 4 units away from zero. So, we have 4 - 4. Finally, subtracting 4 from 4 gives us 0. Therefore, the simplified value of the expression |20 ÷ (-5)| - 4 is 0. This step-by-step approach ensures accuracy and clarity in solving mathematical problems. Understanding the rules of signs in division and the concept of absolute value is essential for mastering this type of calculation. The ability to break down complex expressions into simpler steps is a fundamental skill in mathematics, and this example demonstrates how to apply that skill effectively. The final result, 0, highlights the importance of carefully following the order of operations and the properties of absolute values.

ii) |6 × (-3)|

Next, we'll simplify the expression |6 × (-3)|. This expression involves multiplication within the absolute value. Similar to the previous example, we first address the operation inside the absolute value. We are multiplying 6 by -3. When multiplying a positive number by a negative number, the result is negative. 6 multiplied by 3 is 18, so 6 × (-3) equals -18. The expression now becomes |-18|. The absolute value of -18 is the distance of -18 from zero, which is 18. Thus, |6 × (-3)| simplifies to 18. This example underscores the straightforward application of multiplication rules involving negative numbers and the definition of absolute value. It's a concise demonstration of how mathematical operations within absolute value symbols are performed before the absolute value is applied. The result, 18, is a positive number, reflecting the nature of absolute value as a measure of distance. This type of problem is common in basic algebra and arithmetic, and mastering it is crucial for more advanced mathematical concepts. The clarity of this example helps to reinforce the fundamental principles at play.

iii) |(-40) - (-45) ÷ 5 - (-2)|

Now, let's tackle the expression |(-40) - (-45) ÷ 5 - (-2)|. This expression is more complex, involving subtraction, division, and negative numbers, all within the absolute value. We must adhere strictly to the order of operations (PEMDAS/BODMAS) to simplify it correctly. First, we address the division: (-45) ÷ 5. Dividing a negative number by a positive number results in a negative number. 45 divided by 5 is 9, so (-45) ÷ 5 equals -9. Now our expression looks like |(-40) - (-9) - (-2)|. Next, we handle the subtractions. Subtracting a negative number is equivalent to adding its positive counterpart. So, (-40) - (-9) becomes (-40) + 9, which equals -31. Similarly, subtracting -2 is the same as adding 2, so the expression further simplifies to |-31 - (-2)|, which is |-31 + 2|. Adding 2 to -31 gives us -29. Thus, the expression inside the absolute value is now -29. Finally, we take the absolute value of -29, which is the distance from -29 to zero, or 29. Therefore, |(-40) - (-45) ÷ 5 - (-2)| simplifies to 29. This example showcases the importance of meticulously following the order of operations and correctly handling negative numbers. The multiple steps involved highlight how seemingly complex expressions can be simplified methodically by breaking them down into smaller, manageable parts. The final result, 29, is a testament to the power of careful calculation and adherence to mathematical rules.

iv) |6 × (-5) - (-3)|

Finally, we'll simplify the expression |6 × (-5) - (-3)|. This expression involves multiplication and subtraction within the absolute value. Following the order of operations (PEMDAS/BODMAS), we first perform the multiplication: 6 × (-5). Multiplying a positive number by a negative number yields a negative result. 6 times 5 is 30, so 6 × (-5) equals -30. The expression now becomes |-30 - (-3)|. Next, we address the subtraction. Subtracting a negative number is the same as adding its positive counterpart. So, -30 - (-3) becomes -30 + 3. Adding 3 to -30 gives us -27. The expression inside the absolute value is now -27. Finally, we take the absolute value of -27, which is the distance from -27 to zero, or 27. Therefore, |6 × (-5) - (-3)| simplifies to 27. This example illustrates the straightforward application of multiplication and subtraction rules, along with the concept of absolute value. The step-by-step simplification process demonstrates how complex expressions can be made manageable by addressing operations in the correct order. The final result, 27, is a positive number, consistent with the nature of absolute value as a measure of distance. This type of problem is fundamental to algebraic manipulation and understanding the properties of numbers.

In conclusion, simplifying expressions and finding their absolute values is a critical skill in mathematics. By meticulously following the order of operations and understanding the properties of absolute values, we can effectively solve a variety of mathematical problems. The examples provided demonstrate how complex expressions can be broken down into simpler steps, leading to accurate results. Mastering these skills is essential for further studies in mathematics and its applications in various fields. The ability to manipulate numbers and expressions with confidence is a cornerstone of mathematical proficiency. This article serves as a guide to understanding and applying these principles, paving the way for more advanced mathematical explorations.