Simplifying 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x^2) A Step-by-Step Guide

Introduction

In this article, we will delve into the simplification of a complex algebraic expression. Specifically, we aim to simplify the expression 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2). This type of problem is common in algebra and requires a solid understanding of fractions, factoring, and algebraic manipulation. Mastering these skills is crucial for solving more advanced mathematical problems. Our goal is to break down each step of the simplification process, making it easy to follow and understand. We'll begin by identifying the key components of the expression and then strategically apply algebraic techniques to reduce it to its simplest form. This involves finding common denominators, combining fractions, and potentially factoring expressions to cancel out terms. Ultimately, this exercise not only enhances your ability to manipulate algebraic expressions but also reinforces fundamental concepts in mathematics. By the end of this article, you should feel confident in your ability to tackle similar problems, which is a valuable skill for both academic and practical applications. Understanding the nuances of algebraic manipulation allows for a clearer grasp of mathematical structures and relationships, setting a strong foundation for more advanced studies in mathematics and related fields. This simplification process also highlights the importance of precision and attention to detail, as a single error can significantly alter the final result. So, let's embark on this journey of algebraic simplification and uncover the simplified form of the given expression, gaining insights and reinforcing our mathematical acumen along the way.

Step-by-Step Simplification

The first step in simplifying the expression 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2) is to find a common denominator for all the fractions. This involves recognizing the structure of the denominators and identifying a suitable expression that each denominator can divide into. Observing the denominators, we have (x+2y), (x-2y), and (4y^2-x2). Notice that (4y^2-x2) can be factored as -(x^2-4y2), which further factors into -(x+2y)(x-2y). This factorization is a key observation as it reveals the common factors present in the denominators. The common denominator, therefore, is (x+2y)(x-2y), or equivalently, -(4y^2-x2). Using this common denominator, we can rewrite each fraction with the same denominator, allowing us to combine the numerators. This process involves multiplying the numerator and denominator of each fraction by the appropriate factor to achieve the common denominator. For the first fraction, 1/(x+2y), we multiply both the numerator and the denominator by (x-2y). For the second fraction, 1/(x-2y), we multiply both the numerator and the denominator by (x+2y). For the third fraction, 2x/(4y^2-x2), we rewrite the denominator as -(x+2y)(x-2y), and then we can adjust the sign of the fraction as needed. Once all fractions have the same denominator, we can combine the numerators and proceed with simplifying the expression. This step is crucial for bringing the fractions together and setting the stage for further simplification. The common denominator provides a unified foundation, enabling us to perform addition and subtraction operations on the numerators, which is a fundamental technique in simplifying algebraic expressions.

Rewriting with a Common Denominator

To effectively rewrite the given expression 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2) with a common denominator, we must first identify the common denominator itself. As established earlier, the common denominator is (x+2y)(x-2y), which is equivalent to -(4y^2-x2) or (x^2-4y2). This is derived from the factorization of the third denominator, (4y^2-x2), into -(x+2y)(x-2y). Once the common denominator is clear, we proceed to rewrite each fraction with this denominator. For the first fraction, 1/(x+2y), we multiply both the numerator and the denominator by (x-2y), resulting in (x-2y)/((x+2y)(x-2y)). For the second fraction, 1/(x-2y), we multiply both the numerator and the denominator by (x+2y), resulting in (x+2y)/((x+2y)(x-2y)). For the third fraction, 2x/(4y^2-x2), we rewrite the denominator as -(x+2y)(x-2y). This means we can rewrite the fraction as -2x/((x+2y)(x-2y)). Now that all three fractions have the common denominator (x+2y)(x-2y), we can combine their numerators. This step is pivotal as it brings all the fractions under a single denominator, making it easier to perform the necessary algebraic operations. The process of rewriting each fraction with a common denominator is a fundamental technique in algebra, and it allows us to combine terms and simplify expressions more efficiently. By ensuring each fraction shares the same denominator, we create a uniform structure that facilitates addition and subtraction, which is a cornerstone of algebraic manipulation. This careful preparation sets the stage for the subsequent steps in simplifying the expression, paving the way for a clear and concise solution.

Combining the Numerators

After rewriting each fraction with the common denominator (x+2y)(x-2y), we can now combine the numerators of the expression 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2). The rewritten expression looks like this: (x-2y)/((x+2y)(x-2y)) - (x+2y)/((x+2y)(x-2y)) - 2x/((x+2y)(x-2y)). Now, we combine the numerators over the common denominator, resulting in the single fraction: ((x-2y) - (x+2y) - 2x) / ((x+2y)(x-2y)). This step is crucial as it consolidates the expression into a single fraction, making it easier to simplify the numerator. The next step involves simplifying the numerator by distributing the negative signs and combining like terms. In the numerator, we have (x-2y) - (x+2y) - 2x. Distributing the negative signs gives us x - 2y - x - 2y - 2x. Combining like terms, the x terms (x - x - 2x) result in -2x, and the y terms (-2y - 2y) result in -4y. Therefore, the simplified numerator is -2x - 4y. Now, the entire expression looks like (-2x - 4y) / ((x+2y)(x-2y)). This combined fraction is a significant milestone in our simplification process. By bringing all the terms together under one denominator, we've set the stage for further simplification, which will involve factoring and canceling out common factors. This consolidation is a key technique in algebraic manipulation, making complex expressions more manageable and easier to understand. The process of combining numerators is a fundamental step in fraction arithmetic and algebraic simplification, providing a clear path toward the final simplified form of the expression.

Simplifying the Numerator

To simplify the numerator of the combined fraction (-2x - 4y) / ((x+2y)(x-2y)) from the expression 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2), we focus on factoring out the greatest common factor. Looking at the numerator, -2x - 4y, we can identify that both terms have a common factor of -2. Factoring out -2, we rewrite the numerator as -2(x + 2y). This factorization is a pivotal step as it reveals a common factor with the denominator, which will allow for cancellation and further simplification. Now, the expression looks like -2(x + 2y) / ((x+2y)(x-2y)). The presence of the (x + 2y) term in both the numerator and the denominator is a clear indication that we can simplify the expression further. This step highlights the importance of factoring in algebraic simplification. By identifying and factoring out common factors, we can reduce the complexity of the expression and reveal underlying structures that might not be immediately apparent. Factoring is a fundamental skill in algebra, and it is crucial for solving equations, simplifying expressions, and understanding mathematical relationships. The simplified numerator, -2(x + 2y), is now in a form that allows us to easily identify and cancel out common factors with the denominator, bringing us closer to the final simplified form of the original expression. This process underscores the power of algebraic manipulation in transforming complex expressions into more manageable and transparent forms, which is a core objective in mathematics. The ability to simplify algebraic expressions is not only valuable for solving mathematical problems but also for gaining a deeper understanding of the underlying concepts and principles.

Canceling Common Factors

After simplifying the numerator to -2(x + 2y), the expression now appears as -2(x + 2y) / ((x+2y)(x-2y)) which stemmed from 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2). The next crucial step in simplifying this expression is to identify and cancel common factors between the numerator and the denominator. We observe that the term (x + 2y) appears in both the numerator and the denominator. This common factor can be canceled out, provided that x ≠ -2y (since division by zero is undefined). Canceling (x + 2y) from both the numerator and the denominator, we are left with -2 / (x - 2y). This cancellation significantly simplifies the expression, removing a potentially complex term and leading us closer to the final simplified form. The act of canceling common factors is a fundamental technique in algebraic simplification. It allows us to reduce the expression to its simplest form by eliminating redundant terms. This process is based on the principle that dividing both the numerator and the denominator of a fraction by the same non-zero quantity does not change the value of the fraction. The simplified expression, -2 / (x - 2y), is now much more manageable and easier to understand than the original expression. This step highlights the power of algebraic manipulation in transforming complex expressions into simpler, equivalent forms. The ability to identify and cancel common factors is a valuable skill in algebra, and it is essential for solving equations, simplifying expressions, and working with rational functions. This simplification process not only makes the expression easier to work with but also provides a clearer insight into the underlying mathematical structure, revealing the fundamental relationships between the variables and terms involved. Canceling common factors is a key step in achieving clarity and conciseness in algebraic expressions.

Final Simplified Form

After canceling the common factor (x + 2y) in the previous step, we arrived at the simplified expression -2 / (x - 2y) derived from 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2). This fraction represents the final simplified form of the original expression. At this point, there are no more common factors to cancel, and the expression is in its most reduced state. This final form is much more concise and easier to work with compared to the initial expression. The journey from the original expression to this simplified form demonstrates the power of algebraic manipulation and the importance of techniques such as finding common denominators, combining numerators, factoring, and canceling common factors. The simplified form, -2 / (x - 2y), provides a clear and concise representation of the relationship between the variables x and y in the original expression. It allows for easier evaluation, analysis, and further algebraic operations if needed. This final simplified form is not only the solution to the simplification problem but also a testament to the elegance and efficiency of algebraic techniques. The process of simplification is a core skill in mathematics, and it is essential for solving equations, understanding mathematical relationships, and applying mathematical concepts in various fields. This result underscores the importance of algebraic manipulation in transforming complex expressions into more manageable and transparent forms, which is a core objective in mathematics. The ability to simplify algebraic expressions is not only valuable for solving mathematical problems but also for gaining a deeper understanding of the underlying concepts and principles. The simplified form, -2 / (x - 2y), is a clear and concise representation of the original expression, making it easier to work with and interpret.

Conclusion

In conclusion, the simplification of the algebraic expression 1/(x+2y) - 1/(x-2y) + 2x/(4y^2-x2) has led us to its final simplified form: -2 / (x - 2y). This process involved several key steps, including finding a common denominator, combining numerators, factoring, and canceling common factors. Each step played a crucial role in reducing the complexity of the expression and revealing its underlying structure. The initial expression, which appeared complex and cumbersome, was systematically transformed into a concise and manageable form through the application of fundamental algebraic techniques. This simplification not only makes the expression easier to work with but also provides a clearer understanding of the relationship between the variables x and y. The journey from the original expression to the simplified form highlights the power and elegance of algebraic manipulation. It demonstrates how complex mathematical expressions can be reduced to their simplest forms through careful and methodical application of algebraic principles. The ability to simplify algebraic expressions is a fundamental skill in mathematics, essential for solving equations, understanding mathematical relationships, and applying mathematical concepts in various fields. This exercise serves as a valuable learning experience, reinforcing key algebraic concepts and techniques. It underscores the importance of precision, attention to detail, and a systematic approach when working with algebraic expressions. The final simplified form, -2 / (x - 2y), is a testament to the effectiveness of algebraic simplification and provides a clear and concise representation of the original expression. This outcome not only solves the problem at hand but also enhances our mathematical acumen and problem-solving abilities, which are crucial for success in mathematics and related disciplines. The process of simplifying algebraic expressions is an integral part of mathematical education, fostering critical thinking and analytical skills.