Simplifying complex algebraic expressions can often feel like navigating a maze. However, with a systematic approach and a solid understanding of algebraic principles, even the most daunting expressions can be tamed. In this comprehensive guide, we'll break down the process of simplifying the expression (5/(a-3) - 4) / (2 + 1/(a-3)), providing a step-by-step solution that Ari, or anyone else, can easily follow. We will explore the underlying concepts, explain each step in detail, and offer valuable insights to ensure clarity and accuracy. Understanding how to simplify expressions like this is not just a mathematical exercise; it's a fundamental skill in algebra that paves the way for solving more complex problems in various fields, including physics, engineering, and computer science. Mastering these simplification techniques will enhance your problem-solving abilities and boost your confidence in tackling algebraic challenges. So, let's embark on this journey of simplification and unravel the mysteries of this expression together. We'll begin by identifying the common denominator, combining fractions, and then strategically simplifying the entire expression to its most basic form. This process will not only simplify the given expression but also solidify your understanding of algebraic manipulations. Remember, the key to success in algebra lies in consistent practice and a clear understanding of the fundamental rules. Let's dive in and make this expression simpler!
1. Understanding the Expression
Before diving into the simplification process, it's crucial to understand the structure of the given expression: (5/(a-3) - 4) / (2 + 1/(a-3)). This expression is a complex fraction, which means it has fractions in both the numerator and the denominator. The numerator is (5/(a-3) - 4), and the denominator is (2 + 1/(a-3)). The key to simplifying such expressions lies in breaking them down into smaller, manageable steps. First, we need to address the fractions within the numerator and the denominator separately. This involves finding a common denominator for each part and combining the terms. The expression contains the term (a-3) in the denominator of some fractions. This indicates that 'a' cannot be equal to 3, as this would make the denominator zero, resulting in an undefined expression. Keeping this in mind is crucial for understanding the domain of the expression. When simplifying complex fractions, it's essential to maintain clarity and organization. Each step should be performed carefully, ensuring that the algebraic rules are followed correctly. Errors often occur due to mishandling signs or incorrectly applying the order of operations. By understanding the expression's components and its inherent constraints (such as a ≠ 3), we set the stage for a successful simplification process. This initial understanding is the cornerstone of simplifying complex fractions and will guide us through the subsequent steps with greater confidence and accuracy. We will see how each term interacts within the expression and how to manipulate them to achieve a simplified form. This foundational step is paramount in algebraic manipulations.
2. Simplifying the Numerator
Let's focus on simplifying the numerator of the expression: 5/(a-3) - 4. To combine these terms, we need to express 4 as a fraction with the same denominator as 5/(a-3). This means rewriting 4 as 4/1 and then multiplying both the numerator and the denominator by (a-3). So, 4 becomes 4(a-3)/(a-3). Now, we can rewrite the numerator as: 5/(a-3) - 4(a-3)/(a-3). With a common denominator, we can combine the fractions: [5 - 4(a-3)] / (a-3). Next, we need to distribute the -4 across the terms inside the parentheses: 5 - 4a + 12. Combining the constant terms, we get: 17 - 4a. Therefore, the simplified numerator is (17 - 4a) / (a-3). This process demonstrates a fundamental technique in algebra: finding a common denominator to combine fractions. This skill is crucial not only for simplifying complex fractions but also for solving equations and inequalities involving fractions. When simplifying algebraic expressions, it's essential to pay close attention to the signs and the order of operations. A small mistake in these areas can lead to an incorrect result. By breaking down the simplification process into manageable steps, we minimize the risk of errors and gain a clearer understanding of the underlying algebraic principles. The simplified numerator, (17 - 4a) / (a-3), is now ready to be used in the next step of simplifying the entire expression. This methodical approach ensures accuracy and reinforces the importance of each step in the simplification process. Remember, clarity and precision are key to mastering algebraic manipulations.
3. Simplifying the Denominator
Now, let's turn our attention to simplifying the denominator of the expression: 2 + 1/(a-3). Similar to simplifying the numerator, we need to express 2 as a fraction with the same denominator as 1/(a-3). We can rewrite 2 as 2/1 and then multiply both the numerator and the denominator by (a-3). This gives us 2(a-3)/(a-3). Now, the denominator can be written as: 2(a-3)/(a-3) + 1/(a-3). With a common denominator, we can combine the fractions: [2(a-3) + 1] / (a-3). Next, we need to distribute the 2 across the terms inside the parentheses: 2a - 6 + 1. Combining the constant terms, we get: 2a - 5. Therefore, the simplified denominator is (2a - 5) / (a-3). This step reinforces the importance of finding a common denominator when adding or subtracting fractions. It's a fundamental technique that applies to a wide range of algebraic problems. By carefully following each step and paying attention to the details, we can ensure accuracy and avoid common errors. The simplified denominator, (2a - 5) / (a-3), is now ready to be used in the final simplification of the entire expression. This methodical approach not only simplifies the denominator but also strengthens our understanding of algebraic manipulations. Remember, consistent practice and attention to detail are key to mastering these skills. This step-by-step process demonstrates how to break down a complex problem into smaller, more manageable parts, making the overall task less daunting.
4. Combining Simplified Numerator and Denominator
With the numerator and denominator simplified, we can now combine them to simplify the entire expression. We have the simplified numerator as (17 - 4a) / (a-3) and the simplified denominator as (2a - 5) / (a-3). The original expression, (5/(a-3) - 4) / (2 + 1/(a-3)), now becomes: [(17 - 4a) / (a-3)] / [(2a - 5) / (a-3)]. Dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite the expression as: [(17 - 4a) / (a-3)] * [(a-3) / (2a - 5)]. Now, we can see that the (a-3) terms in the numerator and denominator cancel each other out, as long as a ≠ 3. This gives us: (17 - 4a) / (2a - 5). This is the simplified form of the original expression. This step highlights the power of simplification and how combining simplified parts can lead to a much cleaner and more manageable expression. It also demonstrates the importance of understanding the rules of fraction manipulation, such as dividing by a fraction being equivalent to multiplying by its reciprocal. The final simplified expression, (17 - 4a) / (2a - 5), is the result of a systematic and step-by-step approach. This process underscores the value of breaking down complex problems into smaller, more manageable parts and applying fundamental algebraic principles. Remember, practice and a clear understanding of the rules are essential for mastering algebraic simplification. This final simplification showcases the elegance and efficiency of algebraic techniques.
5. Final Simplified Expression and Restrictions
After simplifying the numerator and the denominator separately and then combining them, we arrived at the final simplified expression: (17 - 4a) / (2a - 5). This expression is the most reduced form of the original complex fraction. However, it's crucial to consider any restrictions on the variable 'a'. From the original expression, we know that 'a' cannot be equal to 3 because this would make the denominator (a-3) equal to zero, resulting in an undefined expression. Additionally, in the simplified expression, the denominator is (2a - 5). This means that 2a - 5 cannot be equal to zero. Solving the equation 2a - 5 = 0 gives us 2a = 5, so a = 5/2. Therefore, 'a' cannot be equal to 5/2 because this would also make the denominator zero. In summary, the simplified expression is (17 - 4a) / (2a - 5), with the restrictions that a ≠ 3 and a ≠ 5/2. These restrictions are essential to maintain the validity of the expression and avoid division by zero. Understanding these restrictions is a crucial part of the simplification process. It ensures that the simplified expression is not only mathematically correct but also applicable within the appropriate domain. The final simplified expression with its restrictions provides a complete and accurate solution to the problem. This comprehensive approach demonstrates the importance of considering all aspects of the expression, including the restrictions on the variables. Remember, a thorough understanding of algebraic principles and attention to detail are key to success in mathematical problem-solving.
6. Conclusion
In conclusion, simplifying the complex expression (5/(a-3) - 4) / (2 + 1/(a-3)) involves a series of steps that, when followed methodically, lead to a much simpler form. We began by understanding the structure of the expression, identifying the numerator and the denominator as separate entities. Then, we simplified each part individually by finding common denominators and combining terms. This involved rewriting whole numbers as fractions with the appropriate denominators and carefully applying the distributive property. Once the numerator and denominator were simplified, we combined them, recognizing that dividing by a fraction is equivalent to multiplying by its reciprocal. This allowed us to cancel out common factors and arrive at the final simplified expression: (17 - 4a) / (2a - 5). However, the simplification process doesn't end there. It's crucial to consider any restrictions on the variable 'a' that would make the denominator zero, as this would result in an undefined expression. We identified two such restrictions: a ≠ 3 and a ≠ 5/2. These restrictions are an integral part of the solution, ensuring that the simplified expression is valid within its domain. This entire process underscores the importance of a systematic approach to algebraic simplification. Breaking down complex problems into smaller, more manageable steps allows for greater clarity and accuracy. Mastering these techniques is essential for success in algebra and beyond. The ability to simplify expressions is a fundamental skill that has applications in various fields, from mathematics and physics to engineering and computer science. By understanding the underlying principles and practicing consistently, anyone can develop the confidence and competence to tackle even the most challenging algebraic expressions. Remember, the key to success lies in patience, persistence, and a commitment to understanding the fundamental concepts. Simplifying algebraic expressions is not just about finding the right answer; it's about developing a deeper understanding of mathematical relationships and honing problem-solving skills that will serve you well in many areas of life.