Solving 7|-4|+4|-1| A Step-by-Step Guide

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This article aims to provide a step-by-step solution to the mathematical expression $7|-4|+4|-1|$. This problem involves understanding the concept of absolute value and performing basic arithmetic operations. We will break down the expression, evaluate each part, and arrive at the final answer. Understanding how to solve such problems is crucial for mastering basic algebra and arithmetic. This article is designed for students and anyone looking to brush up on their math skills. By the end of this guide, you will be able to solve similar problems with confidence. Math problems often seem daunting at first, but with a methodical approach, they can be simplified and solved effectively. Our goal is to make this process clear and understandable, ensuring that you grasp each step along the way.

Before diving into the problem, let's 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 x{ x } is denoted as ∣x∣{ |x| }. For example, the absolute value of 4, denoted as ∣4∣{ |4| }, is 4 because 4 is 4 units away from zero. Similarly, the absolute value of -4, denoted as ∣−4∣{ |-4| }, is also 4 because -4 is 4 units away from zero. In other words, the absolute value function strips away the sign of the number, leaving only its magnitude. Understanding absolute value is crucial for various mathematical contexts, including algebra, calculus, and real analysis. It helps in simplifying expressions and solving equations involving distances and magnitudes. Absolute value is also used in everyday life, such as measuring errors or deviations without considering direction. For example, if a target is set at 10, an actual value of 8 or 12 would have the same absolute deviation of 2 from the target. This concept is also fundamental in computer science, particularly in algorithms that involve minimizing distances or errors.

The given expression is $7|-4|+4|-1|$. To solve this, we need to evaluate the absolute values first and then perform the multiplication and addition. Let’s break it down step by step:

  1. Evaluate ∣−4∣{ |-4| }: The absolute value of -4 is 4, so ∣−4∣=4{ |-4| = 4 }.
  2. Evaluate ∣−1∣{ |-1| }: The absolute value of -1 is 1, so ∣−1∣=1{ |-1| = 1 }.

Now, we substitute these values back into the expression: $7(4) + 4(1)$. The next step involves performing the multiplication operations. We have two multiplication operations to perform: 7×4{ 7 \times 4 } and 4×1{ 4 \times 1 }. Multiplication is a fundamental arithmetic operation that represents repeated addition. It is crucial to follow the order of operations (PEMDAS/BODMAS), which dictates that multiplication and division should be performed before addition and subtraction. Breaking down complex expressions into smaller, manageable parts is a key strategy in mathematics. This approach not only simplifies the problem but also reduces the likelihood of making errors. By addressing the absolute values first, we eliminate the negative signs, making the subsequent calculations straightforward. This methodical approach is a cornerstone of effective problem-solving in mathematics. Recognizing and applying the correct order of operations ensures accurate results and demonstrates a strong understanding of mathematical principles.

Now that we have simplified the expression to $7(4) + 4(1)$, we can perform the multiplication:

  1. Multiply 7 by 4: 7×4=28{ 7 \times 4 = 28 }.
  2. Multiply 4 by 1: 4×1=4{ 4 \times 1 = 4 }.

So, the expression becomes $28 + 4$. Finally, we perform the addition: $28 + 4 = 32$. Therefore, the value of the expression $7|-4|+4|-1|$ is 32. This calculation demonstrates the importance of accurately performing basic arithmetic operations. Multiplication and addition are foundational to mathematics, and mastering these skills is crucial for solving more complex problems. Each step in the calculation must be performed with precision to arrive at the correct answer. Attention to detail and a systematic approach are essential in mathematics. Reviewing each step ensures that no errors are made and that the final result is accurate. This methodical approach builds confidence and reinforces understanding of the underlying mathematical principles.

To reiterate, let’s go through the step-by-step solution:

  1. Original expression: $7|-4|+4|-1|$
  2. Evaluate absolute values: $7(4) + 4(1)$
  3. Perform multiplication: $28 + 4$
  4. Perform addition: $32$

Thus, the final answer is 32. Breaking down the solution into clear, sequential steps helps in understanding the process and identifying potential errors. Each step builds upon the previous one, leading to the final answer. This step-by-step approach is a valuable technique in problem-solving, not just in mathematics but in many other areas as well. It allows for a clear and logical progression from the initial problem to the final solution. This structured approach is particularly helpful in complex problems where multiple steps and operations are involved. By carefully executing each step, the likelihood of making mistakes is significantly reduced. This method also aids in the review process, as each step can be easily checked for accuracy.

The value of the expression $7|-4|+4|-1|$ is 32. We arrived at this answer by first understanding the concept of absolute value, then breaking down the expression into smaller parts, evaluating the absolute values, performing the multiplication, and finally, the addition. This problem illustrates the importance of following the order of operations and performing each step meticulously. Mastering these fundamental mathematical concepts and techniques is essential for tackling more complex problems in the future. By consistently practicing and applying these methods, you can build confidence and proficiency in mathematics. Understanding the underlying principles and the step-by-step process not only helps in solving specific problems but also enhances overall mathematical reasoning and problem-solving skills. This approach can be applied to various mathematical contexts, making it a valuable tool in your mathematical journey. Remember, consistent practice and a methodical approach are key to success in mathematics.

The final answer is (d) 32.