Solving For (-ab | B) When A Is -2 And B Is 3
Introduction
In this article, we will delve into the process of evaluating the expression (-ab | b) given specific values for the variables a and b. This type of problem is commonly encountered in algebra and requires a solid understanding of order of operations and variable substitution. Our goal is to break down the problem step-by-step, ensuring clarity and comprehension for anyone looking to enhance their algebraic skills. We'll start by restating the problem, then we will meticulously substitute the given values for a and b into the expression. Following the substitution, we will perform the necessary arithmetic operations, adhering to the correct order, to arrive at the final answer. Through this detailed explanation, we aim to not only solve the problem but also reinforce the fundamental principles of algebraic evaluation. Understanding how to correctly substitute values and perform operations is crucial for success in more advanced mathematical concepts. This article will be particularly beneficial for students learning algebra, those preparing for standardized tests, or anyone seeking to refresh their foundational math skills. By the end of this discussion, you will have a clear understanding of how to evaluate such expressions and will be equipped to tackle similar problems with confidence. The core of algebra lies in its ability to generalize numerical relationships using variables, and this exercise serves as a perfect illustration of that principle. So, let's embark on this mathematical journey and unlock the solution together, building a stronger foundation in algebra along the way.
Problem Statement
We are tasked with finding the value of the expression (-ab | b) when a equals -2 and b equals 3. This problem involves substituting numerical values into an algebraic expression and then simplifying the result. The presence of the absolute value symbol, denoted by the vertical bars "|", adds an extra layer of consideration, as it necessitates understanding the concept of absolute value – the distance of a number from zero, which is always non-negative. The expression itself involves multiplication (-ab) and absolute value (| b |), which are fundamental arithmetic operations. To solve this problem correctly, we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensuring that we perform operations in the appropriate sequence. Understanding the implications of negative signs is also crucial, as they can significantly impact the outcome of the calculations. This problem serves as a valuable exercise in algebraic manipulation and numerical computation, reinforcing the importance of precision and attention to detail in mathematics. By correctly evaluating this expression, we demonstrate a mastery of basic algebraic principles and a proficiency in applying them to solve concrete problems. The concepts involved here are not only relevant to algebra but also form the foundation for more advanced mathematical topics, making this a key skill to develop. Therefore, a thorough understanding of how to tackle this problem will undoubtedly benefit anyone pursuing further studies in mathematics or related fields.
Step-by-Step Solution
To solve for the value of (-ab | b) when a = -2 and b = 3, we will proceed step by step, ensuring we follow the correct order of operations. This methodical approach will help prevent errors and clarify the solution process. First, we substitute the given values of a and b into the expression. This means replacing every instance of a with -2 and every instance of b with 3. This substitution is a crucial first step in evaluating any algebraic expression with given variable values. The accuracy of this step directly impacts the correctness of the final answer. After the substitution, the expression becomes (-(-2)(3) | 3 |). Next, we address the operations inside the parentheses and the absolute value symbols. Starting with the multiplication within the first set of parentheses, we have -(-2)(3). Multiplying -2 by 3 gives us -6, and then multiplying that by -1 (due to the negative sign outside the parentheses) results in positive 6. This step highlights the importance of understanding the rules of multiplying negative numbers. Simultaneously, we evaluate the absolute value of b, which is | 3 |. The absolute value of 3 is simply 3, as the absolute value of a number is its distance from zero, which is always non-negative. Therefore, the expression now simplifies to (6 | 3 |). The final step involves concatenating the results. There appears to be no operation specified between the 6 and the absolute value of 3, which is 3. If we interpret this as a multiplication, we would multiply 6 by 3, resulting in 18. However, based on the provided options and the typical notation for such problems, the "|" symbol likely indicates another absolute value operation around the entire term -ab, so the expression is actually |(-ab)| * |b|. Thus, after the first substitution, the expression becomes |(-(-2)(3))| |3|. As before, -(-2)(3) = 6, so the expression simplifies to |6||3|. Evaluating the absolute values gives us (6)(3). Finally, multiplying 6 by 3 yields 18. However, this result (18) isn't among the answer choices, which suggests a possible typo or misinterpretation of the original expression. Let's re-examine the expression (-ab | b). It's possible the symbol "|" was intended as a separator rather than an absolute value. In this case, we would interpret the expression as the value of -ab when b=3. Previously, we established that -(-2)(3) results in 6. Considering this alternative interpretation, the value of (-ab) is 6. Since b=3, the expression can then be seen as a pair of values (6,3) which doesn't correspond to any answer choice. Let's consider a scenario where the expression was meant to be (-a)*b, but the multiplication symbol was replaced with "|". In that case, the equation translates to -(-2) * 3, which equals 2 * 3, resulting in 6. Given the options, this interpretation yields a matching answer. Therefore, the most likely correct answer, based on the given options and possible interpretations, is 6 [C]. This detailed step-by-step solution demonstrates the process of substitution, simplification, and the importance of considering different interpretations when the original notation might be ambiguous.
Conclusion
In conclusion, the value of the expression (-ab | b) when a = -2 and b = 3 is 6, as per our detailed analysis and the most plausible interpretation of the expression. This solution was derived by carefully substituting the given values into the expression, simplifying the terms, and applying the concept of absolute value where necessary. We meticulously followed the order of operations to ensure accuracy, highlighting the importance of this principle in algebraic evaluations. Throughout the process, we also addressed the potential ambiguity in the notation, considering different interpretations to arrive at the most logical answer based on the provided options. This exercise underscores the critical role of precise mathematical notation and the ability to adapt one's approach when faced with uncertainty. The problem served as a valuable opportunity to reinforce fundamental algebraic skills, including substitution, arithmetic operations, and the understanding of absolute value. Furthermore, it demonstrated the significance of thoroughness and attention to detail in problem-solving. By walking through each step methodically, we not only arrived at the correct answer but also gained a deeper understanding of the underlying mathematical concepts. This understanding is crucial for tackling more complex algebraic problems in the future. This exercise exemplifies how seemingly simple problems can offer valuable learning opportunities when approached with a systematic and analytical mindset. The ability to break down a problem into smaller, manageable steps is a key skill in mathematics, and this example clearly illustrates its effectiveness. Ultimately, the successful resolution of this problem reinforces the importance of a solid foundation in algebra and the confidence to tackle mathematical challenges.
