Simplifying X/(x+1) - (2x-8)/(x^2-3x-4) A Step-by-Step Guide

In this article, we will delve into the simplification of a complex algebraic expression. Specifically, we aim to simplify the expression x/(x+1) - (2x-8)/(x^2-3x-4). This involves techniques such as factoring quadratic expressions, finding common denominators, and combining like terms. Mastering these skills is crucial for success in algebra and calculus. Our approach will be step-by-step, ensuring a clear understanding of each operation performed. The simplified form not only makes the expression easier to work with but also reveals underlying mathematical structures. Algebraic simplification is a cornerstone of mathematical manipulation, vital for solving equations and understanding functions. The expression we are about to dissect exemplifies the importance of recognizing patterns and applying algebraic rules correctly. Let's embark on this journey of simplification, enhancing our mathematical prowess along the way.

The given expression is a subtraction between two rational functions. The first term is x/(x+1), a simple rational expression. The second term is (2x-8)/(x^2-3x-4), which appears more complex due to the quadratic expression in the denominator. Before we can subtract these two terms, we need to ensure they have a common denominator. This often involves factoring the denominators to identify common factors. Factoring the quadratic expression x^2-3x-4 is the first key step. We look for two numbers that multiply to -4 and add up to -3. These numbers are -4 and 1. Therefore, x^2-3x-4 can be factored as (x-4)(x+1). Now, the expression becomes x/(x+1) - (2x-8)/((x-4)(x+1)). This reveals a common factor of (x+1) in both denominators. Recognizing such patterns is vital in simplifying algebraic expressions. The numerator of the second term, (2x-8), can also be simplified by factoring out a 2, resulting in 2(x-4). The expression now transforms into x/(x+1) - 2(x-4)/((x-4)(x+1)). By understanding the structure of the expression and identifying potential simplifications, we set the stage for the next steps in our simplification process.

As highlighted in the previous section, the key to simplifying the expression x/(x+1) - (2x-8)/(x^2-3x-4) lies in factoring. We've already identified that x^2-3x-4 factors into (x-4)(x+1) and 2x-8 can be factored as 2(x-4). Thus, the expression can be rewritten as x/(x+1) - 2(x-4)/((x-4)(x+1)). Now, we observe that the factor (x-4) appears in both the numerator and denominator of the second term. We can cancel out this common factor, provided that x ≠ 4. This simplification is a crucial step, making the expression easier to manage. After canceling the (x-4) terms, the expression becomes x/(x+1) - 2/(x+1). This simplification brings us closer to the final form, as both terms now share a common denominator. Simplifying rational expressions by factoring and canceling common factors is a fundamental technique in algebra. It allows us to reduce complex expressions into more manageable forms, revealing the underlying structure and making further calculations easier. The importance of recognizing and applying these techniques cannot be overstated, as they form the backbone of algebraic manipulation.

Now that we have the expression in the form x/(x+1) - 2/(x+1), the next step is to combine the two fractions. Since they share a common denominator, (x+1), we can simply subtract the numerators. This gives us (x - 2)/(x+1). This step is straightforward but essential in the simplification process. Combining fractions with a common denominator involves adding or subtracting the numerators while keeping the denominator the same. This operation is a fundamental concept in arithmetic and algebra, forming the basis for more complex algebraic manipulations. The resulting fraction, (x - 2)/(x+1), is the simplified form of the original expression. However, it's important to remember any restrictions on the variable x. We had the condition x ≠ 4 from an earlier simplification step, and now we also have the condition x ≠ -1 because the denominator cannot be zero. These restrictions ensure that the expression is well-defined. The simplified expression is now in its most concise form, making it easier to analyze and use in further calculations.

After performing the necessary algebraic manipulations, we've arrived at the simplified form of the expression: (x - 2)/(x + 1). This expression is significantly simpler than the original x/(x+1) - (2x-8)/(x^2-3x-4). However, it's crucial to state the restrictions on the variable x. From our simplification steps, we identified two key restrictions: x ≠ 4 and x ≠ -1. The restriction x ≠ -1 comes from the original denominators (x+1) and x^2-3x-4 = (x-4)(x+1), which cannot be zero. The restriction x ≠ 4 arises from canceling the (x-4) factor during the simplification process. If we were to substitute x = 4 into the original expression, we would encounter a division by zero, rendering the expression undefined. Therefore, it's imperative to state these restrictions alongside the simplified expression. Stating restrictions is a crucial aspect of simplifying rational expressions. It ensures that the simplified form is equivalent to the original expression for all valid values of x. Ignoring these restrictions can lead to incorrect conclusions and mathematical inconsistencies. The final, simplified expression, along with its restrictions, provides a complete and accurate representation of the original complex expression.

In conclusion, we have successfully simplified the algebraic expression x/(x+1) - (2x-8)/(x^2-3x-4) to (x - 2)/(x + 1), with the restrictions x ≠ 4 and x ≠ -1. This process involved several key algebraic techniques, including factoring quadratic expressions, finding common denominators, canceling common factors, and combining fractions. Each step was crucial in transforming the complex expression into its simplest form. Mastering these algebraic techniques is fundamental for success in mathematics, particularly in algebra and calculus. The ability to simplify expressions allows for easier manipulation, solving equations, and understanding the behavior of functions. The restrictions on the variable x highlight the importance of considering the domain of expressions. These restrictions ensure that the simplified form is mathematically equivalent to the original expression for all valid values. Through this exercise, we have not only simplified a specific expression but also reinforced essential algebraic principles. The ability to simplify expressions efficiently and accurately is a valuable skill that will serve well in further mathematical endeavors.