Solving X/4 - 5/(2x) = 1/(4x) A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving the algebraic equation x/4 - 5/(2x) = 1/(4x). This equation involves fractions and variables, requiring a systematic approach to find the value(s) of x that satisfy the equation. Mastering the techniques to solve such equations is crucial for various mathematical applications and problem-solving scenarios. Throughout this exploration, we will emphasize the underlying principles and demonstrate each step meticulously, ensuring a comprehensive understanding of the solution. Solving algebraic equations like this one is a fundamental skill in mathematics, finding applications in various fields such as physics, engineering, and economics. It involves manipulating the equation to isolate the variable, thereby determining the value or values that make the equation true. This particular equation presents an interesting challenge due to the presence of fractions and the variable x in the denominators. Our goal is to systematically eliminate these fractions and simplify the equation to a form that is easily solvable. We will begin by identifying the least common multiple (LCM) of the denominators, which will allow us to clear the fractions and transform the equation into a more manageable form. Subsequently, we will apply algebraic manipulations such as combining like terms and isolating the variable to arrive at the solution. By carefully following each step, we can not only find the solution to this specific equation but also develop a deeper understanding of the techniques involved in solving algebraic equations in general. This understanding will prove invaluable in tackling more complex problems in mathematics and related disciplines. The process of solving this equation will involve several key algebraic techniques, including finding the least common multiple (LCM) of the denominators, clearing fractions, simplifying the equation, and isolating the variable. Each of these steps is essential to arriving at the correct solution and requires careful attention to detail. By working through this example, we will reinforce these fundamental skills and develop a systematic approach to solving similar equations. In addition to the step-by-step solution, we will also discuss some common pitfalls to avoid and strategies for checking our work. This will help to ensure that we not only arrive at the correct answer but also develop a robust understanding of the underlying concepts. So, let's embark on this mathematical journey and unravel the solution to the equation x/4 - 5/(2x) = 1/(4x).

Step 1: Identifying the Common Denominator

To effectively solve the equation x/4 - 5/(2x) = 1/(4x), the initial crucial step involves identifying the common denominator of all the fractions present. This is essential as it allows us to eliminate the fractions, simplifying the equation and making it easier to solve. The denominators in our equation are 4, 2x, and 4x. To find the common denominator, we need to determine the least common multiple (LCM) of these terms. The LCM is the smallest expression that is divisible by each of the denominators. Analyzing the denominators, we can see that 4 and 2 are numerical coefficients, while x is a variable. The LCM of 4 and 2 is 4, as 4 is the smallest number that both 4 and 2 divide into evenly. Additionally, we have x and 4x as denominators. The LCM of x and 4x is simply 4x, as 4x is divisible by both x and itself. Therefore, the least common denominator (LCD) for the entire equation is 4x. This means that 4x is the smallest expression that each of the denominators 4, 2x, and 4x can divide into without leaving a remainder. Understanding how to find the LCD is a fundamental skill in algebra and is used extensively when dealing with equations involving fractions. It is a critical step in simplifying complex equations and allows us to manipulate the terms more easily. Once we have identified the LCD, we can multiply both sides of the equation by it. This will eliminate the fractions and transform the equation into a more manageable form, typically a linear or quadratic equation. By carefully determining the LCD, we can avoid errors and ensure that our subsequent steps lead to the correct solution. In this case, identifying 4x as the LCD sets the stage for the next crucial step: clearing the fractions from the equation. This will be achieved by multiplying each term in the equation by 4x, which will cancel out the denominators and leave us with a simpler equation to solve. The process of finding the common denominator is not just a mechanical step; it requires a deep understanding of the underlying principles of fractions and multiples. By mastering this skill, we can approach more complex algebraic problems with confidence and efficiency.

Step 2: Clearing the Fractions

Having identified the least common denominator (LCD) as 4x in the equation x/4 - 5/(2x) = 1/(4x), the next crucial step is to clear the fractions. This is achieved by multiplying both sides of the equation by the LCD. Multiplying both sides of an equation by the same non-zero value maintains the equality, allowing us to eliminate the fractions without altering the solution. In this case, we will multiply both sides of the equation by 4x. This process involves distributing 4x to each term in the equation. Let's break it down step by step: On the left-hand side (LHS) of the equation, we have two terms: x/4 and -5/(2x). When we multiply 4x by x/4, the 4 in the denominator cancels out with the 4 in 4x, leaving us with xx*, which simplifies to x^2. Next, we multiply 4x by -5/(2x). Here, the 2x in the denominator cancels out with 4x, leaving us with 2(-5)*, which simplifies to -10. So, the LHS of the equation becomes x^2 - 10. On the right-hand side (RHS) of the equation, we have the term 1/(4x). When we multiply 4x by 1/(4x), the 4x in the denominator cancels out with the 4x, leaving us with just 1. Thus, the RHS of the equation simplifies to 1. Now, our equation looks much simpler: x^2 - 10 = 1. We have successfully cleared the fractions, transforming the original equation into a quadratic equation. This simplification is a significant step forward in solving for x. By eliminating the fractions, we have made the equation easier to manipulate and solve. This technique of clearing fractions is a fundamental tool in algebra and is applicable to a wide range of equations. It is important to note that we must multiply every term in the equation by the LCD to maintain the equality. Skipping this step for even one term can lead to an incorrect solution. The process of clearing fractions is not just about simplifying the equation; it also helps to reveal the underlying structure of the equation. In this case, we have transformed a fractional equation into a quadratic equation, which we can then solve using standard techniques. With the fractions cleared, we are now in a position to isolate the variable x and find the values that satisfy the equation. The next step will involve further algebraic manipulations to solve the quadratic equation x^2 - 10 = 1.

Step 3: Simplifying the Equation

Following the step of clearing fractions in the equation x/4 - 5/(2x) = 1/(4x), we arrived at the simplified form x^2 - 10 = 1. Now, the next logical step is to further simplify this equation to isolate the variable x. This involves performing algebraic operations on both sides of the equation to group like terms and bring us closer to solving for x. In our current equation, x^2 - 10 = 1, we want to isolate the x^2 term on one side of the equation. To do this, we can add 10 to both sides of the equation. Adding the same value to both sides of an equation maintains the equality, a fundamental principle in algebra. By adding 10 to both sides, we eliminate the -10 term on the left-hand side, leaving us with just x^2. On the right-hand side, we have 1 + 10, which simplifies to 11. So, our equation now becomes x^2 = 11. This is a significant simplification, as we have isolated the x^2 term. The equation is now in a form where we can easily solve for x by taking the square root of both sides. This step highlights the importance of algebraic manipulation in solving equations. By carefully applying operations to both sides, we can transform a complex equation into a simpler, more manageable form. The goal of simplification is to isolate the variable, making it easier to determine its value. In this case, we have successfully isolated x^2, which is a crucial step towards finding the value(s) of x. The simplified equation x^2 = 11 is a quadratic equation, but it is in a particularly simple form. It tells us that the square of x is equal to 11. To find the values of x that satisfy this equation, we need to take the square root of both sides. This will give us two possible solutions, as both the positive and negative square roots of 11 will satisfy the equation. The process of simplifying equations is not just a mechanical exercise; it requires a deep understanding of the properties of equality and the rules of algebra. By mastering these skills, we can approach a wide range of equations with confidence and efficiency. With the equation simplified to x^2 = 11, we are now ready to take the final step: solving for x by taking the square root of both sides. This will reveal the two possible values of x that satisfy the original equation.

Step 4: Solving for x

Having simplified the equation to x^2 = 11, the final step in solving for x involves taking the square root of both sides. This operation will undo the squaring on the left-hand side, allowing us to isolate x and find its value(s). When we take the square root of both sides of an equation, it's crucial to remember that there are both positive and negative solutions. This is because both a positive number and its negative counterpart, when squared, will result in the same positive number. In our case, the square root of x^2 is ±√11, where the ± symbol indicates that there are two possible solutions: a positive square root and a negative square root. Therefore, we have x = ±√11. This means that there are two values of x that satisfy the equation: x = √11 and x = -√11. These are the solutions to the original equation x/4 - 5/(2x) = 1/(4x). We can express these solutions exactly using the square root symbol, or we can approximate them as decimal values using a calculator. The square root of 11 is an irrational number, meaning it cannot be expressed as a simple fraction. Its decimal approximation is approximately 3.3166. Therefore, the approximate solutions are x ≈ 3.3166 and x ≈ -3.3166. It is always a good practice to check our solutions by substituting them back into the original equation to ensure they satisfy the equation. This helps to catch any errors made during the solving process. In this case, substituting both √11 and -√11 back into the original equation will confirm that they are indeed the solutions. Solving for x by taking the square root is a fundamental operation in algebra and is used extensively in various mathematical contexts. It is important to remember the ± sign to account for both positive and negative solutions. By carefully applying this operation, we can successfully solve equations involving squared variables. In conclusion, the solutions to the equation x/4 - 5/(2x) = 1/(4x) are x = √11 and x = -√11, or approximately x ≈ 3.3166 and x ≈ -3.3166. We have successfully solved the equation by systematically applying algebraic techniques, including finding the common denominator, clearing fractions, simplifying the equation, and taking the square root. This process demonstrates the power of algebraic manipulation in solving complex equations.

Conclusion

In summary, we have successfully navigated through the process of solving the equation x/4 - 5/(2x) = 1/(4x). This journey has underscored the importance of several key algebraic techniques and concepts. We began by identifying the least common denominator (LCD), a crucial step in clearing fractions from the equation. The LCD, in this case, was 4x, which allowed us to transform the equation into a more manageable form. Next, we multiplied both sides of the equation by the LCD, effectively eliminating the fractions. This transformed the original fractional equation into a quadratic equation: x^2 - 10 = 1. This step highlighted the power of algebraic manipulation in simplifying complex expressions. Following the clearing of fractions, we simplified the equation further by isolating the x^2 term. This involved adding 10 to both sides of the equation, resulting in x^2 = 11. This step demonstrated the importance of maintaining equality while manipulating equations. Finally, we solved for x by taking the square root of both sides of the equation. This operation yielded two solutions: x = √11 and x = -√11. We emphasized the significance of considering both positive and negative roots when solving equations involving squared variables. These solutions can also be approximated as decimal values: x ≈ 3.3166 and x ≈ -3.3166. Throughout this process, we have highlighted the importance of a systematic approach to solving algebraic equations. Each step, from identifying the LCD to taking the square root, plays a crucial role in arriving at the correct solution. Furthermore, we have emphasized the need to check our solutions by substituting them back into the original equation. This practice helps to ensure the accuracy of our work and catch any potential errors. The techniques and concepts discussed in this article are fundamental to algebra and have wide-ranging applications in mathematics and other disciplines. Mastering these skills is essential for success in more advanced mathematical studies. By carefully following each step and understanding the underlying principles, we can confidently tackle a variety of algebraic equations. The solution to the equation x/4 - 5/(2x) = 1/(4x) not only provides a specific answer but also serves as a valuable learning experience, reinforcing our understanding of algebraic problem-solving. In conclusion, the journey through this equation has been a testament to the power and elegance of algebraic methods. By systematically applying these methods, we have successfully unveiled the solutions and deepened our appreciation for the beauty of mathematics.