Evaluate The Expression 4|·3|: A Step-by-Step Guide
Understanding Absolute Value
In the realm of mathematics, absolute value is a fundamental concept that represents the distance of a number from zero on the number line. It's crucial to grasp this concept before diving into evaluating expressions involving absolute values. The absolute value of a number is always non-negative, as distance cannot be negative. 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 instance, the absolute value of 5, denoted as |5|, is simply 5, as 5 is 5 units away from zero. Similarly, the absolute value of -5, denoted as |-5|, is also 5, because -5 is also 5 units away from zero. This concept is vital for understanding how to handle negative signs within mathematical expressions and ensures accurate calculations.
To further illustrate the significance of absolute value, consider its applications in various mathematical contexts. In geometry, absolute value is used to calculate distances between points, regardless of their relative positions on a coordinate plane. In algebra, it helps in solving equations and inequalities involving variables, ensuring that we account for both positive and negative solutions. Understanding absolute value is also essential in calculus, particularly when dealing with limits and continuity of functions. By grasping the core principle of absolute value – its representation of distance from zero – we lay a solid foundation for tackling more complex mathematical problems. The concept of absolute value is not just a theoretical abstraction; it has practical implications in real-world scenarios as well, such as in physics where it's used to measure magnitudes of quantities like velocity and acceleration, irrespective of direction.
Moreover, the absolute value function, often denoted as f(x) = |x|, is a cornerstone in mathematical analysis. Its unique properties, such as its non-differentiability at the point x = 0, make it a fascinating subject of study for mathematicians and students alike. The absolute value function serves as a building block for constructing more intricate functions and models, providing a versatile tool for mathematical exploration. In summary, a thorough understanding of absolute value is not only essential for evaluating expressions but also crucial for comprehending a wide array of mathematical concepts and applications. Its role extends beyond basic arithmetic, permeating various branches of mathematics and finding relevance in diverse scientific and engineering disciplines.
Breaking Down the Expression
To effectively evaluate the expression 4|·3|, we need to dissect it step by step, ensuring we adhere to the correct order of operations and understand the role of each component. The expression involves multiplication and the absolute value function, which is denoted by the vertical bars surrounding the number 3. The core of this expression lies in the absolute value of 3, which, as we've established, represents the distance of 3 from zero. In this case, the absolute value of 3 is simply 3, because 3 is already a positive number and its distance from zero is 3 units. This initial step is crucial, as it simplifies the expression and sets the stage for the subsequent operation.
Once we've determined the absolute value of 3, which is 3, we can move on to the multiplication operation. The expression now becomes 4 multiplied by 3. This is a straightforward arithmetic calculation. Multiplying 4 by 3 yields 12. This final result represents the evaluated value of the entire expression 4|·3|. It's important to note that the absolute value operation takes precedence over the multiplication in this expression. This is because the absolute value function acts as a grouping symbol, similar to parentheses or brackets, and operations within grouping symbols are always performed before operations outside them. In other words, we must first find the absolute value of 3 before multiplying the result by 4.
Understanding the order of operations is paramount in mathematics, as it dictates the sequence in which calculations should be performed to arrive at the correct answer. In this case, the absolute value operation takes priority, followed by multiplication. This methodical approach ensures that we accurately evaluate the expression and avoid any potential errors. The breakdown of the expression into its constituent parts – the absolute value and the multiplication – allows us to tackle the problem in a systematic manner, ultimately leading to the correct solution. This step-by-step approach is a valuable problem-solving technique in mathematics, enabling us to break down complex problems into manageable steps and arrive at the final answer with confidence.
Step-by-Step Evaluation
Let's embark on a step-by-step evaluation of the expression 4|·3| to solidify our understanding and ensure clarity. This process involves carefully applying the order of operations and understanding the concept of absolute value. Our initial step is to focus on the absolute value component of the expression, which is denoted as |·3|. The absolute value of a number, as we've discussed, represents its distance from zero on the number line. In this instance, we are dealing with the absolute value of 3. Since 3 is already a positive number, its distance from zero is simply 3. Therefore, |·3| evaluates to 3. This step is crucial, as it simplifies the expression and sets the stage for the next operation.
Having determined that |·3| equals 3, we can now substitute this value back into the original expression. This transforms the expression from 4|·3| to 4 multiplied by 3, which can be written as 4 * 3. The next step involves performing this multiplication operation. Multiplying 4 by 3 is a basic arithmetic calculation that yields 12. This calculation is straightforward and forms the final step in evaluating the expression. Thus, 4 * 3 equals 12.
Therefore, after meticulously following the steps, we arrive at the final answer: the expression 4|·3| evaluates to 12. This result represents the solution to the problem, obtained by carefully applying the concept of absolute value and the order of operations. The step-by-step evaluation process allows us to break down the problem into manageable components, ensuring that we address each aspect correctly and arrive at the accurate answer. This approach is not only effective for solving mathematical expressions but also for developing a deeper understanding of the underlying mathematical principles. By systematically working through each step, we reinforce our knowledge and enhance our problem-solving skills.
The Final Result
After a thorough and methodical evaluation, we have determined the final result of the expression 4|·3|. By carefully considering the absolute value and applying the correct order of operations, we have arrived at a definitive answer. As we've outlined in the previous sections, the initial step involved evaluating the absolute value component of the expression, which is |·3|. The absolute value of 3 is simply 3, as it represents the distance of 3 from zero on the number line. This simplification is crucial, as it allows us to proceed with the next operation.
Following the evaluation of the absolute value, we moved on to the multiplication aspect of the expression. Substituting the value of |·3| with 3, the expression transformed into 4 multiplied by 3. This multiplication operation is straightforward and yields the result 12. Therefore, 4 * 3 equals 12. This final calculation provides us with the solution to the expression 4|·3|.
Thus, the final result of the expression 4|·3| is 12. This answer is the culmination of our step-by-step evaluation, where we meticulously considered each component of the expression and applied the appropriate mathematical principles. The process involved understanding the concept of absolute value, simplifying the expression, and performing the necessary multiplication. This systematic approach not only allows us to arrive at the correct answer but also reinforces our understanding of mathematical operations and the order in which they should be performed. The final result of 12 represents the evaluated value of the expression 4|·3|, providing a clear and concise solution to the problem. This comprehensive evaluation process highlights the importance of careful and methodical problem-solving in mathematics, ensuring accuracy and fostering a deeper understanding of the underlying concepts.
