Simplifying 1 + 3/(2x-1) - (x-1)/(6x^2-3x) An Algebraic Guide

by ADMIN 62 views

In mathematics, simplifying expressions is a fundamental skill that allows us to rewrite complex expressions in a more manageable and understandable form. This process often involves combining like terms, factoring, and applying various algebraic identities. In this article, we will delve into the process of simplifying a specific algebraic expression, providing a step-by-step guide that will enhance your understanding of algebraic manipulation. The expression we aim to simplify is: 1 + 3/(2x-1) - (x-1)/(6x^2-3x). This expression combines integers, fractions with linear denominators, and fractions with quadratic denominators. To effectively simplify it, we'll need to employ several algebraic techniques, including finding a common denominator, combining fractions, and simplifying the resulting expression. The goal is to transform the given expression into its simplest form, which will not only make it easier to work with but also reveal its underlying structure and properties. Understanding how to simplify expressions like this is crucial for solving equations, evaluating functions, and tackling more advanced mathematical problems. By mastering these techniques, you'll be well-equipped to handle a wide range of algebraic challenges. We will start by identifying the common denominator for the fractional parts of the expression. This involves factoring the denominators to find their least common multiple (LCM). Once we have the common denominator, we can rewrite each fraction with this denominator, allowing us to combine the numerators. After combining the numerators, we will simplify the resulting expression by expanding any products and combining like terms. If possible, we will also look for opportunities to factor the numerator and denominator to further simplify the expression. Finally, we will state the simplified form of the original expression, along with any restrictions on the variable x that are necessary to avoid division by zero. This comprehensive approach will ensure that you not only understand the mechanics of simplifying this particular expression but also gain a deeper appreciation for the principles of algebraic manipulation.

1. Rewriting the Expression

The first step in simplifying this algebraic expression is to rewrite it in a more convenient form. The expression we're working with is: 1 + 3/(2x-1) - (x-1)/(6x^2-3x). To begin, we need to address the fractional part of the expression. We have two fractions: 3/(2x-1) and (x-1)/(6x^2-3x). To combine these fractions, we need to find a common denominator. Before we can do that, it's helpful to factor the denominators to see if there are any common factors. The first denominator, (2x-1), is already in its simplest form. It's a linear expression and cannot be factored further. The second denominator, (6x^2-3x), can be factored. We can factor out a 3x from both terms: 6x^2 - 3x = 3x(2x-1). Now we can rewrite the original expression incorporating the factored denominator: 1 + 3/(2x-1) - (x-1)/(3x(2x-1)). Notice that we now have a common factor of (2x-1) in both denominators. This is a crucial observation that will help us find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. In this case, the LCD is 3x(2x-1). Now that we have the LCD, we can rewrite each term in the expression with this denominator. The first term, 1, can be written as 3x(2x-1) / 3x(2x-1). This might seem like we're making things more complicated, but it's necessary to combine all the terms into a single fraction. The second term, 3/(2x-1), needs to be multiplied by 3x/3x to get the LCD in the denominator: 3/(2x-1) * (3x/3x) = 9x / 3x(2x-1). The third term, (x-1)/(3x(2x-1)), already has the LCD in the denominator, so we don't need to change it. Now we can rewrite the entire expression with the common denominator: [3x(2x-1) / 3x(2x-1)] + [9x / 3x(2x-1)] - [(x-1) / 3x(2x-1)]. This step is essential because it allows us to combine the numerators over a single denominator, which is the next step in simplifying the expression. By rewriting the expression with a common denominator, we've set the stage for combining the terms and further simplifying the expression. This careful and methodical approach is key to avoiding errors and arriving at the correct answer.

2. Combining Fractions

After rewriting the algebraic expression with a common denominator, the next crucial step is to combine the fractions. This involves adding and subtracting the numerators while keeping the common denominator. From the previous step, we have the expression: [3x(2x-1) / 3x(2x-1)] + [9x / 3x(2x-1)] - [(x-1) / 3x(2x-1)]. Now, we can combine the numerators over the common denominator, which is 3x(2x-1): [3x(2x-1) + 9x - (x-1)] / 3x(2x-1). This step consolidates the expression into a single fraction, making it easier to simplify further. The next task is to simplify the numerator. This involves expanding any products and combining like terms. Let's start by expanding the first term in the numerator, 3x(2x-1): 3x(2x-1) = 6x^2 - 3x. Now, we can rewrite the numerator with the expanded term: 6x^2 - 3x + 9x - (x-1). Next, we need to distribute the negative sign in front of the parentheses: -(x-1) = -x + 1. Now the numerator looks like this: 6x^2 - 3x + 9x - x + 1. Now we can combine the like terms in the numerator. We have terms with x^2, x, and constants. Let's combine the x terms first: -3x + 9x - x = 5x. So, the numerator becomes: 6x^2 + 5x + 1. Now we have the simplified numerator, and we can rewrite the entire expression: (6x^2 + 5x + 1) / 3x(2x-1). This fraction represents the combined form of the original expression. The next step is to see if we can further simplify this fraction by factoring the numerator and denominator. Factoring can help us identify common factors that can be canceled out, leading to a simpler expression. This step of combining fractions is a critical part of simplifying algebraic expressions. It allows us to consolidate multiple terms into a single fraction, making it easier to manipulate and simplify further. By carefully combining the numerators and simplifying the resulting expression, we've made significant progress towards our goal of simplifying the original expression.

3. Factoring and Simplifying

After combining the fractions and simplifying the numerator, the next crucial step is factoring and further simplifying the expression. We have the expression: (6x^2 + 5x + 1) / 3x(2x-1). To simplify this fraction, we need to factor both the numerator and the denominator, if possible, and then look for common factors that can be canceled out. Let's start by factoring the numerator, which is a quadratic expression: 6x^2 + 5x + 1. We're looking for two binomials that multiply to give this quadratic. We can use the factoring by grouping method. We need to find two numbers that multiply to the product of the leading coefficient (6) and the constant term (1), which is 6, and add up to the middle coefficient (5). These two numbers are 2 and 3 because 2 * 3 = 6 and 2 + 3 = 5. Now we can rewrite the middle term using these numbers: 6x^2 + 2x + 3x + 1. Next, we group the terms in pairs: (6x^2 + 2x) + (3x + 1). Now we factor out the greatest common factor (GCF) from each pair: 2x(3x + 1) + 1(3x + 1). Notice that we now have a common factor of (3x + 1) in both terms. We can factor this out: (3x + 1)(2x + 1). So, the factored form of the numerator is (3x + 1)(2x + 1). Now let's look at the denominator, which is 3x(2x-1). We've already factored this in a previous step, and it's in its simplest form. Now we can rewrite the entire expression with the factored numerator and denominator: [(3x + 1)(2x + 1)] / [3x(2x-1)]. Now we look for common factors in the numerator and denominator that can be canceled out. In this case, there are no common factors. The numerator has factors of (3x + 1) and (2x + 1), while the denominator has factors of 3x and (2x - 1). None of these factors match, so we cannot simplify the fraction further by canceling out common factors. Therefore, the simplified form of the expression is (6x^2 + 5x + 1) / 3x(2x-1) or [(3x + 1)(2x + 1)] / [3x(2x-1)]. This step of factoring and simplifying is crucial because it allows us to reduce the expression to its simplest form. By factoring both the numerator and denominator, we can identify common factors that can be canceled out, leading to a more concise and manageable expression. In this case, we were not able to cancel out any factors, but the factored form itself can be useful for further analysis or manipulation of the expression.

4. Stating the Simplified Form and Restrictions

After factoring and attempting to simplify the algebraic expression, the final step is to state the simplified form and identify any restrictions on the variable x. We've arrived at the expression: [(3x + 1)(2x + 1)] / [3x(2x-1)]. While we couldn't cancel out any common factors, this is the simplified form of the original expression. Now, we need to consider any restrictions on the variable x. Restrictions arise when there are values of x that would make the denominator of the fraction equal to zero, as division by zero is undefined in mathematics. To find these restrictions, we need to set the denominator equal to zero and solve for x: 3x(2x-1) = 0. This equation is satisfied if either 3x = 0 or (2x-1) = 0. Solving 3x = 0, we get x = 0. Solving 2x - 1 = 0, we get: 2x = 1 x = 1/2. So, the restrictions on x are x ≠ 0 and x ≠ 1/2. These values of x would make the denominator zero, so they are not allowed in the domain of the expression. Now we can state the final simplified form of the expression along with the restrictions: Simplified form: [(3x + 1)(2x + 1)] / [3x(2x-1)] Restrictions: x ≠ 0, x ≠ 1/2. This comprehensive answer includes both the simplified form of the expression and the restrictions on the variable x. Stating the restrictions is an important part of simplifying algebraic expressions because it ensures that the expression is only defined for valid values of the variable. By identifying and stating these restrictions, we provide a complete and accurate solution to the problem. In summary, we started with the expression 1 + 3/(2x-1) - (x-1)/(6x^2-3x), and through a series of algebraic manipulations, we simplified it to [(3x + 1)(2x + 1)] / [3x(2x-1)], with the restrictions x ≠ 0 and x ≠ 1/2. This process involved finding a common denominator, combining fractions, factoring the numerator and denominator, and identifying any restrictions on the variable. By following these steps, we can effectively simplify complex algebraic expressions and gain a deeper understanding of their properties. This final step of stating the simplified form and restrictions is a crucial part of the simplification process. It ensures that the answer is complete and accurate, and it provides valuable information about the domain of the expression.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics that involves a series of steps, including finding a common denominator, combining fractions, factoring, and identifying restrictions on the variable. In this article, we have demonstrated this process by simplifying the expression 1 + 3/(2x-1) - (x-1)/(6x^2-3x). We began by rewriting the expression with a common denominator, which allowed us to combine the fractions into a single term. This involved factoring the denominators to find the least common multiple and then rewriting each fraction with this common denominator. Next, we combined the numerators, which involved expanding any products and combining like terms. This step simplified the numerator and made it easier to factor in the next step. After combining the fractions, we factored both the numerator and the denominator to see if there were any common factors that could be canceled out. Factoring is a crucial step in simplifying expressions because it allows us to identify and eliminate common factors, leading to a more concise form. In this particular case, we were not able to cancel out any factors, but the factored form itself provides valuable information about the structure of the expression. Finally, we stated the simplified form of the expression and identified any restrictions on the variable x. Restrictions are important because they ensure that the expression is only defined for valid values of the variable. In this case, the restrictions were x ≠ 0 and x ≠ 1/2, as these values would make the denominator of the fraction equal to zero. The simplified form of the expression was [(3x + 1)(2x + 1)] / [3x(2x-1)]. By following these steps, we have successfully simplified the given algebraic expression. This process not only provides a simpler form of the expression but also enhances our understanding of algebraic manipulation and the properties of fractions and polynomials. Mastering these techniques is essential for success in higher-level mathematics courses and in various applications of mathematics in science, engineering, and other fields. The ability to simplify expressions efficiently and accurately is a valuable skill that can save time and reduce errors in problem-solving. Furthermore, a deep understanding of algebraic simplification can lead to a greater appreciation for the elegance and power of mathematics. By practicing these techniques and applying them to a variety of problems, you can develop your skills and confidence in algebraic manipulation.