Simplifying (x+3)/(x-3) - (x+2)/(x-2) A Step By Step Guide
Introduction
In this article, we will delve into the process of simplifying the algebraic expression (x+3)/(x-3) - (x+2)/(x-2). This type of problem is a fundamental exercise in algebra, often encountered in high school and early college mathematics courses. Simplifying algebraic expressions is a crucial skill, as it not only makes the expression easier to understand and work with but also lays the groundwork for more advanced mathematical concepts. The process involves several key steps, including finding a common denominator, combining fractions, expanding terms, and simplifying the resulting expression. This detailed exploration will provide a comprehensive understanding of how to approach and solve such problems. The main goal of simplifying this expression is to reduce it to its simplest form, making it easier to analyze, graph, or use in further calculations. Algebraic simplification is a cornerstone of mathematical manipulation, and mastering these techniques is essential for success in higher-level mathematics. This article aims to break down each step involved in the simplification process, ensuring clarity and a thorough understanding of the underlying principles. By the end of this discussion, you should be well-equipped to tackle similar algebraic problems with confidence and precision. We will also highlight common pitfalls and strategies to avoid them, ensuring a solid grasp of the simplification process. Let’s embark on this journey to demystify the process of simplifying algebraic expressions.
Step 1: Finding a Common Denominator
The first crucial step in simplifying the expression (x+3)/(x-3) - (x+2)/(x-2) is to find a common denominator. This is essential because we cannot directly subtract fractions unless they have the same denominator. The common denominator is the least common multiple (LCM) of the denominators (x-3) and (x-2). In this case, since (x-3) and (x-2) are distinct linear factors, their least common multiple is simply their product, which is (x-3)(x-2). Finding a common denominator allows us to combine the two fractions into a single fraction, which is a crucial step in simplifying the expression. Without a common denominator, we cannot perform the subtraction operation. The process of finding the LCM is a fundamental skill in algebra, and it's not limited to just polynomial expressions; it applies to numerical fractions as well. The ability to identify and compute the LCM correctly is paramount for successful simplification. This step ensures that the fractions are in a form where they can be combined, paving the way for the subsequent steps in the simplification process. Incorrectly determining the common denominator can lead to significant errors in the final simplified expression. Thus, meticulousness in this step is critical for ensuring the accuracy of the solution. The common denominator acts as the foundation upon which we build the simplified expression, making it a pivotal element in the overall simplification process.
Step 2: Rewriting the Fractions
Now that we have identified the common denominator as (x-3)(x-2), the next step is to rewrite each fraction with this common denominator. To do this, we multiply the numerator and denominator of the first fraction, (x+3)/(x-3), by (x-2), and the numerator and denominator of the second fraction, (x+2)/(x-2), by (x-3). This process ensures that the value of each fraction remains unchanged while allowing us to combine them. Multiplying both the numerator and the denominator by the same factor is equivalent to multiplying the fraction by 1, which doesn't alter its value. Rewriting the fractions is a critical step because it transforms the expression into a form where the subtraction operation can be performed. Without this step, we would be unable to combine the fractions under a single denominator. The technique of rewriting fractions with a common denominator is a fundamental skill in algebraic manipulation, essential for adding, subtracting, and simplifying rational expressions. This step sets the stage for combining the numerators and simplifying the resulting expression. Accuracy in this step is vital, as any error in rewriting the fractions will propagate through the remaining steps and affect the final result. The process of rewriting fractions is a core concept in algebra, and mastering it is crucial for handling more complex algebraic manipulations. By ensuring that both fractions have the same denominator, we create a unified expression that can be further simplified.
Step 3: Combining the Fractions
Having rewritten the fractions with a common denominator, we can now combine them. The expression becomes [(x+3)(x-2) - (x+2)(x-3)] / (x-3)(x-2). This step involves subtracting the second numerator from the first, keeping the common denominator. Combining the fractions is a direct application of the rules of fraction arithmetic, where we subtract the numerators while preserving the common denominator. This step is crucial as it brings the expression closer to its simplified form by consolidating the two fractions into one. The resulting single fraction is easier to manipulate and simplify further. This process of combining fractions is a fundamental skill in algebra, and it's a necessary step in solving equations, simplifying expressions, and performing other mathematical operations. The accuracy of this step is paramount, as any error in subtracting the numerators will lead to an incorrect simplified expression. It is important to pay close attention to the signs and ensure that the subtraction is performed correctly. The combined fraction represents a more concise form of the original expression, setting the stage for further simplification through expansion and collection of like terms. This consolidation is a key aspect of algebraic manipulation, making the expression more manageable and easier to analyze.
Step 4: Expanding the Numerator
The next step involves expanding the numerator of the combined fraction. We need to expand both (x+3)(x-2) and (x+2)(x-3) using the distributive property (also known as the FOIL method). Expanding (x+3)(x-2) gives us x² - 2x + 3x - 6, which simplifies to x² + x - 6. Similarly, expanding (x+2)(x-3) gives us x² - 3x + 2x - 6, which simplifies to x² - x - 6. The expansion process is a critical step in simplifying the expression, as it removes the parentheses and allows us to combine like terms. Expanding the products correctly is essential for obtaining the correct simplified form. The distributive property is a fundamental concept in algebra, and its accurate application is vital for successful simplification. This step transforms the numerator into a polynomial expression, making it easier to identify and combine like terms in the subsequent steps. Accuracy in expansion is crucial, as any error here will affect the final simplified expression. The expansion process sets the stage for collecting like terms and further simplifying the numerator. This step is a key component of algebraic manipulation, allowing us to transform the expression into a more manageable form.
Step 5: Simplifying the Numerator
Now that we have expanded the numerator, we can simplify it by combining like terms. The numerator is (x² + x - 6) - (x² - x - 6). Distributing the negative sign to the second expression, we get x² + x - 6 - x² + x + 6. Combining like terms, we have x² - x² + x + x - 6 + 6, which simplifies to 2x. This step is crucial as it reduces the complexity of the numerator, making the overall expression simpler. Simplifying the numerator involves identifying and combining terms with the same variable and exponent. The process of combining like terms is a fundamental skill in algebra and is essential for simplifying polynomial expressions. Accuracy in this step is paramount, as any error in combining like terms will lead to an incorrect simplified expression. This step transforms the numerator into its simplest form, which is 2x in this case. The simplified numerator makes the overall fraction easier to understand and work with. This simplification is a key aspect of algebraic manipulation, allowing us to present the expression in its most concise form. By reducing the numerator to its simplest terms, we set the stage for the final step of simplifying the entire expression.
Step 6: Simplifying the Entire Expression
After simplifying the numerator, the expression becomes 2x / (x-3)(x-2). Now, we need to check if there are any common factors in the numerator and the denominator that can be cancelled out. In this case, there are no common factors between 2x and (x-3)(x-2), so the expression is already in its simplest form. This final step ensures that the expression is indeed in its most reduced form. Simplifying the entire expression involves checking for any further reductions that can be made, such as cancelling out common factors. The ability to identify common factors is a fundamental skill in algebra and is crucial for simplifying rational expressions. In this instance, since there are no common factors, the expression 2x / (x-3)(x-2) is the final simplified form. This step confirms that we have successfully simplified the original expression to its most basic form. The final simplified expression is easier to understand, analyze, and use in further calculations. This process of simplification is a key component of algebraic manipulation, allowing us to present expressions in their most concise and manageable form. By ensuring that no further reductions can be made, we complete the simplification process and arrive at the final solution. The simplified expression represents the original expression in a more elegant and accessible format.
Conclusion
In conclusion, simplifying the expression (x+3)/(x-3) - (x+2)/(x-2) involved several key steps: finding a common denominator, rewriting the fractions, combining the fractions, expanding the numerator, simplifying the numerator, and simplifying the entire expression. The final simplified form of the expression is 2x / (x-3)(x-2). Each step in this process is a fundamental concept in algebra, and mastering these techniques is essential for success in higher-level mathematics. Algebraic simplification is a crucial skill, allowing us to transform complex expressions into simpler, more manageable forms. The ability to simplify expressions is not only useful in academic settings but also in various real-world applications, such as engineering, physics, and computer science. The step-by-step approach outlined in this article provides a clear and comprehensive understanding of the simplification process. By breaking down the problem into smaller, more manageable steps, we can effectively tackle complex algebraic expressions. The simplification process highlights the importance of accuracy and attention to detail in algebraic manipulation. A thorough understanding of these techniques will enable you to confidently approach and solve similar algebraic problems. This exercise demonstrates the power of algebraic simplification in making complex expressions more accessible and easier to work with. The final simplified form, 2x / (x-3)(x-2), represents the original expression in its most concise and understandable form.
