Solving (x-1)/(x+1) + (x+2)/(x-2) = (x-3)/(x+3) + (x+4)/(x-4) A Detailed Guide
This article delves into the intricacies of solving the algebraic equation: (x-1)/(x+1) + (x+2)/(x-2) = (x-3)/(x+3) + (x+4)/(x-4). This equation, while seemingly complex, can be tackled using a combination of algebraic manipulation, simplification, and careful consideration of potential solutions and restrictions. In the realm of mathematical problem-solving, equations like this offer a fascinating challenge, requiring a blend of skill and strategic thinking to arrive at the correct solution. Our journey begins with a thorough examination of the equation's structure, identifying key components and potential pathways to simplification. Each step we take will be meticulously explained, ensuring that the underlying logic is transparent and accessible. So, let's embark on this mathematical adventure and unravel the solution to this intriguing equation.
To begin, it is paramount to recognize the domain restrictions imposed by the denominators. The denominators in the equation are (x+1), (x-2), (x+3), and (x-4). To avoid division by zero, which is undefined in mathematics, we must ensure that none of these denominators equal zero. This gives us the following restrictions: x ≠-1, x ≠2, x ≠-3, and x ≠4. These restrictions are critical because any solution we find must satisfy these conditions. Disregarding these restrictions can lead to extraneous solutions, which are values that satisfy the transformed equation but not the original one. With these restrictions firmly in mind, we can now proceed to manipulate the equation algebraically. The initial strategy often involves combining the fractions on each side of the equation. This is typically achieved by finding a common denominator for the fractions on each side. For the left-hand side, the common denominator is (x+1)(x-2), and for the right-hand side, it is (x+3)(x-4). By rewriting each fraction with its respective common denominator, we can then add the numerators and simplify. This process transforms the equation into a more manageable form, paving the way for further simplification and eventual solution.
Combining Fractions and Simplifying
Combining fractions is a crucial step in solving the equation (x-1)/(x+1) + (x+2)/(x-2) = (x-3)/(x+3) + (x+4)/(x-4). To achieve this, we'll focus on finding a common denominator for each side of the equation. The left-hand side of the equation consists of two fractions, (x-1)/(x+1) and (x+2)/(x-2). The common denominator for these fractions is the product of their individual denominators, which is (x+1)(x-2). Similarly, on the right-hand side, we have the fractions (x-3)/(x+3) and (x+4)/(x-4). Their common denominator is (x+3)(x-4). By converting each fraction to an equivalent fraction with the common denominator, we prepare the equation for the addition of the fractions on each side. This process involves multiplying the numerator and denominator of each fraction by the appropriate factors to obtain the common denominator. Once we have successfully rewritten the fractions, we can add them together. This addition leads to a single fraction on each side of the equation, which simplifies the equation's structure. The next phase involves cross-multiplication, a technique used to eliminate the fractions. This step is essential for transforming the equation into a more workable form, often a polynomial equation. Cross-multiplication involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa. This operation effectively clears the fractions, resulting in an equation that is easier to manipulate and solve.
Following the fraction combination, careful simplification of the resulting equation is necessary. This stage involves expanding the products in both the numerators and denominators, and then combining like terms. Expansion of the products, such as (x-1)(x-2) and (x+1)(x+2), requires the application of the distributive property. Each term in one binomial must be multiplied by each term in the other binomial. This process generates a series of terms that must be carefully combined. Combining like terms is a critical step in reducing the complexity of the equation. Like terms are those that have the same variable raised to the same power. For example, terms like 3x^2 and -5x^2 are like terms and can be combined to give -2x^2. Similarly, constant terms can be combined. This process of combining like terms streamlines the equation, making it more manageable and easier to analyze. After the expansion and combination of like terms, we will often be left with a polynomial equation. The degree of this polynomial equation, which is the highest power of the variable, will dictate the techniques we use to solve it. A quadratic equation, for instance, can be solved using the quadratic formula, factoring, or completing the square. Higher-degree polynomials may require more advanced techniques or numerical methods to find their solutions. Thus, simplification is a pivotal stage in the solution process, transforming a complex equation into a simpler, more tractable form that we can effectively address.
Cross-Multiplication and Rearranging
Cross-multiplication is a pivotal technique in solving equations involving fractions. In our equation, (x-1)/(x+1) + (x+2)/(x-2) = (x-3)/(x+3) + (x+4)/(x-4), after combining fractions on each side, we have two fractions equated to each other. Cross-multiplication allows us to eliminate these fractions, transforming the equation into a more manageable form. This process involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa. The resulting equation will be free of fractions but will likely involve polynomial expressions. These expressions need to be carefully expanded and simplified. Expansion often involves using the distributive property, multiplying each term in one polynomial by each term in another. This step can be lengthy and prone to errors, so meticulous attention to detail is crucial. After expanding the products, the next step is rearranging the terms. This typically involves moving all terms to one side of the equation, leaving zero on the other side. The goal is to obtain a polynomial equation in standard form, where terms are arranged in descending order of their exponents. This standard form makes it easier to identify the degree of the polynomial and apply appropriate solution techniques. The rearrangement process often involves adding or subtracting terms from both sides of the equation. It's essential to perform these operations correctly to maintain the equation's balance and avoid introducing errors. The resulting polynomial equation may be linear, quadratic, or of higher degree, each requiring different methods for finding the solutions. Thus, cross-multiplication and rearrangement are fundamental steps in simplifying and transforming the equation, setting the stage for the final solution.
Solving for x
After cross-multiplication and rearrangement, solving for x in the equation (x-1)/(x+1) + (x+2)/(x-2) = (x-3)/(x+3) + (x+4)/(x-4) involves identifying the type of polynomial equation we've obtained. The most common types are linear, quadratic, and higher-degree polynomials. The method for solving for x will depend on the degree of the polynomial. If we have a linear equation, which is a polynomial equation of degree one, the process is straightforward. We isolate x by performing algebraic operations on both sides of the equation. This usually involves adding or subtracting constants and dividing by the coefficient of x. If the equation is quadratic, meaning it's a polynomial equation of degree two, we have several options. One common method is factoring. If the quadratic expression can be factored into two linear expressions, we can set each factor equal to zero and solve for x. Another powerful technique is using the quadratic formula, which provides the solutions for any quadratic equation in the form ax^2 + bx + c = 0. Completing the square is another method that can be used to solve quadratic equations, transforming the equation into a form where we can easily take the square root of both sides. For higher-degree polynomials, such as cubics or quartics, the solution process can be more complex. There are formulas for solving some specific cases of higher-degree polynomials, but they are often cumbersome. Numerical methods, such as the Newton-Raphson method, are frequently used to approximate the solutions. These methods involve iterative processes that converge to the roots of the polynomial. Once we have found potential solutions, it's crucial to check them against the restrictions we identified at the beginning of the process. Remember, x cannot be equal to -1, 2, -3, or 4, as these values would make the denominators zero. If any potential solution violates these restrictions, it's an extraneous solution and must be discarded. Therefore, solving for x is a multi-faceted process, requiring us to choose the appropriate method based on the polynomial's degree and to verify the solutions against any initial restrictions.
Verifying Solutions and Handling Extraneous Roots
Verifying solutions is an indispensable step in the process of solving equations, particularly when dealing with rational expressions like (x-1)/(x+1) + (x+2)/(x-2) = (x-3)/(x+3) + (x+4)/(x-4). After arriving at potential solutions for x, it is crucial to substitute these values back into the original equation to ensure they hold true. This verification process safeguards against errors made during algebraic manipulation and helps identify extraneous roots. Extraneous roots are solutions that emerge from the algebraic process but do not satisfy the original equation. These roots often arise due to operations like squaring both sides of an equation or, in this case, dealing with rational expressions where certain values of x can make the denominators zero. When verifying a solution, substitute it into the original equation and simplify both sides. If the left-hand side equals the right-hand side, the solution is valid. However, if the two sides are not equal, the solution is extraneous and must be discarded. This step is not merely a formality; it is a critical check that ensures the accuracy of the solutions. In the context of our equation, we must also consider the restrictions on x that we identified at the outset. Recall that x cannot be equal to -1, 2, -3, or 4 because these values would result in division by zero, which is undefined. If any of our potential solutions coincide with these restricted values, they are extraneous roots, regardless of whether they satisfy the transformed equation. The process of verifying solutions and handling extraneous roots underscores the importance of precision and thoroughness in mathematical problem-solving. It reinforces the idea that obtaining a solution is not the final step; verifying its validity is equally essential.
In conclusion, the journey of solving the equation (x-1)/(x+1) + (x+2)/(x-2) = (x-3)/(x+3) + (x+4)/(x-4) encapsulates the core principles of algebraic problem-solving. We began by recognizing the domain restrictions imposed by the denominators, ensuring that our solutions would not lead to division by zero. The subsequent steps involved combining fractions, simplifying the expressions, cross-multiplication, and rearranging terms to form a polynomial equation. The process of solving for x required us to select appropriate techniques based on the degree of the polynomial. Whether it was a linear, quadratic, or higher-degree equation, each called for a specific approach, from simple isolation of x to the application of the quadratic formula or numerical methods. Crucially, we emphasized the necessity of verifying solutions. This step not only validates our algebraic manipulations but also safeguards against extraneous roots, which can arise from the nature of rational expressions. Extraneous roots, which satisfy the transformed equation but not the original, highlight the importance of returning to the initial problem and confirming the legitimacy of each solution. Through this comprehensive exploration, we have not only solved a specific algebraic equation but also reinforced the broader principles of mathematical rigor and problem-solving. The meticulous attention to detail, the careful consideration of restrictions, and the validation of results are all hallmarks of sound mathematical practice. This equation serves as a valuable case study in the art of algebraic manipulation and the importance of a systematic approach to mathematical challenges.
