Evaluate Absolute Value Of |-xz| For X=3 And Z=-4

Jul 10, 2025 by ADMIN 50 views

Evaluate |-xz| for x=3 and z=-4 - A Comprehensive Guide

In this article, we will walk through the process of evaluating the expression |-xz| when x=3 and z=-4. This involves understanding the concept of absolute value and applying basic arithmetic operations. We'll break down each step to ensure clarity and comprehension. Whether you're a student learning algebra or someone looking to brush up on your math skills, this guide will provide a clear and thorough explanation.

Understanding Absolute Value

Before we dive into the evaluation, it’s crucial to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line. It is always non-negative. The absolute value of a number a is denoted as |a|. For example, |5| = 5, and |-5| = 5 because both 5 and -5 are 5 units away from zero. The absolute value essentially strips away the sign of the number, leaving only its magnitude. This concept is fundamental in mathematics and has various applications in fields like physics, engineering, and computer science.

The significance of absolute value extends beyond simple arithmetic. In more complex mathematical contexts, such as calculus and real analysis, absolute value is used to define distance, convergence, and continuity. For instance, in defining the limit of a function, the absolute value helps to quantify the closeness of a function's value to a certain limit. Similarly, in vector spaces, the norm (a generalization of absolute value) is used to measure the length of vectors. Understanding absolute value, therefore, lays a crucial foundation for more advanced mathematical studies. In real-world scenarios, absolute value helps in situations where direction or sign is irrelevant, and only the magnitude matters. For instance, in calculating the error in measurements, we are often concerned with the absolute difference between the measured value and the true value, regardless of whether the error is positive or negative. Similarly, in financial analysis, absolute value can be used to measure the volatility of an asset, focusing on the size of price fluctuations rather than their direction. Thus, mastering the concept of absolute value is not just an academic exercise but a practical skill applicable in diverse fields.

Step-by-Step Evaluation of |-xz|

Now, let’s proceed with evaluating the expression |-xz| given that x=3 and z=-4. We'll break this down into manageable steps:

Step 1: Substitute the Values

The first step is to substitute the given values of x and z into the expression. This means replacing x with 3 and z with -4 in the expression |-xz|. So, we have |-(3)(-4)|. This substitution is crucial because it sets the stage for the arithmetic operations that will follow. Accurate substitution is a fundamental skill in algebra, ensuring that we're working with the correct numerical values. A small error in substitution can lead to a completely incorrect final answer. Therefore, double-checking the substituted values against the original equation and given values is a good practice. This step is not just about replacing variables with numbers; it's about translating an algebraic expression into an arithmetic problem that can be solved using basic operations.

Step 2: Perform the Multiplication Inside the Absolute Value

Next, we need to perform the multiplication inside the absolute value. We have -(3)(-4). Multiplying 3 by -4 gives us -12. Thus, we now have |-(-12)|. Remember the rules of multiplication with negative numbers: a positive number multiplied by a negative number results in a negative number, and a negative number multiplied by a negative number results in a positive number. This is a critical concept in arithmetic and algebra, and a clear understanding of these rules is essential for accurate calculations. The order of operations (PEMDAS/BODMAS) dictates that we perform operations inside parentheses or grouping symbols first. In this case, the absolute value acts as a grouping symbol, so we must complete the multiplication within it before we can apply the absolute value.

Step 3: Simplify the Expression Inside the Absolute Value

Now, we simplify the expression inside the absolute value. We have |-(-12)|. The negative of a negative number is positive, so -(-12) becomes 12. Our expression is now |12|. This step highlights the importance of understanding how negative signs interact. A common mistake is to overlook the double negative, which can lead to an incorrect answer. This simplification step is not just a matter of applying a rule; it’s about understanding the underlying concept of additive inverses. Every number has an additive inverse, which, when added to the number, results in zero. In this case, -12 is the additive inverse of 12, and the negative of -12 is 12 itself. This concept is fundamental in algebra and is used extensively in solving equations and simplifying expressions.

Step 4: Evaluate the Absolute Value

Finally, we evaluate the absolute value of 12. The absolute value of a positive number is the number itself. Therefore, |12| = 12. This completes the evaluation of the expression. The absolute value essentially gives us the magnitude of the number, disregarding its sign. In this context, 12 is 12 units away from zero on the number line, so its absolute value is 12. This final step reinforces the initial concept of absolute value as the distance from zero. It’s a straightforward step, but it's crucial for arriving at the correct final answer. Understanding absolute value is not just about applying a rule; it's about grasping the underlying concept of magnitude and distance, which has broad applications in mathematics and other fields.

Final Answer

Therefore, the value of |-xz| when x=3 and z=-4 is 12. This result is obtained by systematically substituting the given values, performing the multiplication, simplifying the expression inside the absolute value, and finally, evaluating the absolute value itself. Each step is crucial and builds upon the previous one, demonstrating the importance of following the correct order of operations and understanding the underlying mathematical concepts. This example illustrates how algebraic expressions can be evaluated methodically, breaking down a complex problem into simpler, manageable steps. The process not only provides the correct answer but also reinforces fundamental mathematical principles, fostering a deeper understanding of algebra and arithmetic.

Conclusion

In summary, evaluating |-xz| for x=3 and z=-4 involves a clear understanding of substitution, multiplication, and the concept of absolute value. By following a step-by-step approach, we can accurately evaluate the expression and arrive at the final answer of 12. This exercise not only reinforces basic algebraic skills but also highlights the importance of precision and attention to detail in mathematical calculations. Mastering these fundamental concepts is essential for further studies in mathematics and its applications in various fields. The ability to break down complex problems into simpler steps and apply the correct operations is a valuable skill that extends beyond mathematics, fostering logical thinking and problem-solving abilities in all aspects of life.