Solving (x+4)/(6x) = 1/x A Step-by-Step Guide

JU07/09/2025 08, 2025 by THE IDEN 46 views

$Solving Equations A Comprehensive Guide to ${\frac{x+4}{6x} = \frac{1}{x}}$$

In the realm of mathematics, solving equations is a fundamental skill. This article delves into a step-by-step solution of the equation ${\frac{x+4}{6x} = \frac{1}{x}}$ , providing a clear and detailed explanation to enhance your understanding. Whether you're a student tackling algebra problems or a math enthusiast looking to sharpen your skills, this guide will offer valuable insights and practical techniques. Let's embark on this mathematical journey together, unraveling the complexities and arriving at the correct solution. Understanding the nuances of solving such equations is crucial for various mathematical applications and real-world problem-solving scenarios. This comprehensive guide aims to equip you with the necessary tools and knowledge to confidently approach similar problems in the future. We will explore the underlying principles, potential pitfalls, and efficient strategies to ensure you grasp the concepts thoroughly. So, let's dive in and master the art of equation-solving!

Problem Overview

The given equation is:

${ \frac{x+4}{6x} = \frac{1}{x} }$

Our goal is to find the value(s) of x that satisfy this equation. This involves algebraic manipulation, careful consideration of potential restrictions, and a systematic approach to isolate the variable x. The solution process will highlight the importance of understanding fractions, algebraic operations, and the concept of undefined expressions in mathematics. By working through this problem, we will reinforce fundamental mathematical principles and develop problem-solving skills that are applicable in various contexts. The equation itself is a rational equation, which means it involves fractions with variables in the denominator. These types of equations require special attention to avoid division by zero, a concept we will explore in detail as we proceed. The journey to finding the solution will not only reveal the correct answer but also deepen our understanding of algebraic concepts and techniques.

Step-by-Step Solution

1. Identify Restrictions

Before we begin solving, it's crucial to identify any values of x that would make the denominators zero, as division by zero is undefined. In this equation, the denominator is 6x and x. Thus, x cannot be 0.

This initial step is paramount in solving rational equations. By recognizing and noting the restrictions, we prevent the inclusion of extraneous solutions that may arise from algebraic manipulations. The restriction x ≠ 0 stems from the fundamental principle that division by zero is undefined in mathematics. If we were to proceed without acknowledging this, we might arrive at a solution that, when substituted back into the original equation, would result in an invalid expression. Therefore, identifying restrictions is not just a preliminary step but a critical safeguard in ensuring the accuracy and validity of our solution. It demonstrates a thorough understanding of the mathematical principles governing rational expressions and equations. Moreover, this practice cultivates a meticulous approach to problem-solving, a skill that is invaluable in both academic and real-world scenarios.

2. Multiply Both Sides by the Least Common Denominator (LCD)

The LCD of 6x and x is 6x. Multiply both sides of the equation by 6x to eliminate the fractions:

${ 6x \cdot \frac{x+4}{6x} = 6x \cdot \frac{1}{x} }$

This step is a pivotal maneuver in simplifying the equation. By multiplying both sides by the least common denominator (LCD), we effectively clear the fractions, transforming the equation into a more manageable form. The LCD, in this case, is 6x, which is the smallest expression that is divisible by both denominators, 6x and x. This technique leverages the fundamental principle of algebraic manipulation: performing the same operation on both sides of an equation maintains its balance. By eliminating the fractions, we reduce the complexity of the equation, making it easier to isolate the variable x and ultimately find the solution. This approach is not only efficient but also reduces the likelihood of errors that can occur when dealing with fractional expressions. It exemplifies a strategic and methodical approach to problem-solving, highlighting the importance of identifying and utilizing key mathematical tools to simplify complex expressions.

3. Simplify the Equation

Simplify both sides:

${ x + 4 = 6 }$

After multiplying both sides by the LCD, the equation simplifies dramatically. The elimination of fractions leads us to a more straightforward linear equation, which is easier to solve. This simplification step is a direct result of the previous operation and showcases the power of algebraic manipulation in making complex problems more accessible. The equation x + 4 = 6 is a testament to the elegance of mathematics, where a series of logical steps can transform a seemingly intricate problem into a simple one. This step is not just about reducing the equation to its simplest form; it's about revealing the underlying structure and making the solution more apparent. It underscores the importance of precision and attention to detail in algebraic operations, ensuring that each step brings us closer to the final answer. The simplified equation is now in a form that can be easily solved, setting the stage for the final steps in our problem-solving journey.

4. Isolate x

Subtract 4 from both sides:

${ x = 6 - 4 }$

${ x = 2 }$

Isolating x is the penultimate step in solving the equation. This involves performing algebraic operations to get x by itself on one side of the equation. In this case, subtracting 4 from both sides effectively isolates x. This step highlights the core principle of solving equations: to undo operations and maintain equality. The operation performed here is the inverse of addition, which is subtraction. This strategic maneuver brings us closer to the solution by simplifying the equation further. The clarity and directness of this step demonstrate the power of algebraic manipulation in revealing the value of the unknown variable. It’s a crucial step that bridges the gap between the simplified equation and the final solution. The result, x = 2, is a potential solution, but it's essential to verify it against the initial restrictions to ensure its validity.

5. Check the Solution

We found that x = 2. Since this value does not violate our restriction (x ≠ 0), it is a valid solution.

Verifying the solution against the initial restrictions is a crucial step in the problem-solving process, especially when dealing with rational equations. We previously established that x cannot be 0 due to the denominators in the original equation. Our solution, x = 2, does not violate this restriction, confirming its validity. This step is not merely a formality; it’s a safeguard against extraneous solutions that may arise from algebraic manipulations. By checking the solution, we ensure that it not only satisfies the simplified equation but also the original equation in its entirety. This meticulous approach underscores the importance of precision and thoroughness in mathematical problem-solving. It reinforces the understanding that a solution is only valid if it adheres to all the conditions and constraints of the problem. The confirmation of x = 2 as a valid solution marks the successful completion of our problem-solving journey.

Final Answer

Therefore, the solution to the equation is:

${ x = 2 }$

Thus, the correct answer is:

C. x = 2

This final answer is the culmination of a systematic and methodical approach to solving the equation. We began by identifying restrictions, cleared fractions using the LCD, simplified the equation, isolated the variable x, and finally, verified the solution. The result, x = 2, is the value that satisfies the original equation ${\frac{x+4}{6x} = \frac{1}{x}}$ . This process underscores the importance of each step in solving mathematical problems. From identifying restrictions to verifying the solution, every action plays a crucial role in ensuring accuracy and validity. The correct answer, option C, is not just a numerical value; it’s the embodiment of a successful problem-solving journey. It demonstrates the power of algebraic manipulation, the significance of attention to detail, and the satisfaction of arriving at a correct solution through logical and methodical steps. This journey serves as a valuable learning experience, reinforcing mathematical principles and enhancing problem-solving skills.

Conclusion

In conclusion, solving the equation ${\frac{x+4}{6x} = \frac{1}{x}}$ involves several key steps, including identifying restrictions, eliminating fractions, simplifying the equation, isolating the variable, and verifying the solution. The correct answer is x = 2, which corresponds to option C. This exercise demonstrates the importance of a systematic approach to solving algebraic equations and highlights the significance of each step in the process. By understanding and applying these principles, you can confidently tackle similar problems and enhance your mathematical proficiency. The journey through this problem has not only revealed the solution but also reinforced fundamental mathematical concepts and techniques. The ability to solve equations is a cornerstone of mathematical understanding and a valuable skill in various fields. This comprehensive guide aims to empower you with the knowledge and confidence to approach and solve algebraic equations effectively. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the underlying principles.