Simplifying The Rational Expression: A Step-by-Step Guide

Introduction

In this article, we will delve into the process of simplifying the given rational expression: $\frac{1}{\frac{1}{1-2 x}}-\frac{3}{24+x}$. This problem falls under the category of mathematics, specifically dealing with algebraic expressions and their simplification. Simplifying expressions is a fundamental skill in mathematics, as it allows us to manipulate equations and formulas more easily, solve problems more efficiently, and gain a deeper understanding of the underlying relationships between variables. Let's embark on this journey of simplification, breaking down each step to ensure clarity and comprehension. This introduction sets the stage for a detailed exploration, and in the subsequent sections, we will methodically address each component of the expression, applying relevant algebraic principles to arrive at the simplest form. Understanding how to simplify such expressions is crucial for more advanced mathematical concepts, making this a valuable exercise for students and enthusiasts alike. Before we dive into the specifics, it's helpful to recall some key concepts and techniques related to fractions and algebraic manipulation, which will serve as our foundation for this simplification process. By the end of this article, you should have a firm grasp of how to tackle similar expressions and a deeper appreciation for the elegance of mathematical simplification. Now, let's get started with the first step in unraveling this expression and transforming it into its most manageable form.

Step 1: Simplify the First Term

The first term of the expression is $\frac{1}{\frac{1}{1-2x}}$. This looks like a complex fraction, but it's actually quite straightforward to simplify. Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite this term as $1 \times \frac{1-2x}{1}$, which simplifies directly to $1-2x$. This transformation is a crucial first step because it eliminates the nested fraction, making the entire expression significantly easier to work with. Understanding reciprocals and how they interact in division is fundamental to simplifying complex fractions. This step highlights the importance of recognizing and applying basic algebraic principles to more intricate expressions. By addressing the complexity of the first term in isolation, we've effectively reduced the overall complexity of the original problem. This approach of breaking down a problem into smaller, more manageable parts is a common and effective strategy in mathematics. Now that we've simplified the first term, we can move on to incorporating it back into the original expression and addressing the second term. The next step will involve combining the simplified first term with the second term, which will require finding a common denominator. But for now, we've made significant progress by simplifying the initial complex fraction into a much simpler algebraic expression. This foundation will allow us to proceed with the remaining steps more confidently and efficiently.

Step 2: Rewrite the Expression

After simplifying the first term, our expression now becomes: $(1-2x) - \frac{3}{24+x}$. This is a significant improvement from the original expression, as we've eliminated the complex fraction. Now, we need to combine these two terms. To do this, we'll need to find a common denominator. The first term, $(1-2x)$, can be thought of as a fraction with a denominator of 1. So, to combine it with the second term, which has a denominator of $(24+x)$, we'll need to rewrite the first term with the same denominator. This process involves multiplying both the numerator and the denominator of the first term by $(24+x)$. This is a standard algebraic technique used to add or subtract fractions. By rewriting the expression in this way, we're setting the stage for combining the terms into a single fraction. This step is crucial because it allows us to perform the subtraction operation. Finding a common denominator is a key skill in working with fractions, and it's essential for simplifying algebraic expressions like this one. The transformed expression will have both terms sharing the same denominator, making the subtraction straightforward. This process not only simplifies the expression but also highlights the interconnectedness of algebraic concepts. By mastering this technique, we can confidently tackle more complex problems involving rational expressions. With the expression rewritten and a common denominator in place, we're now ready to move on to the next step: combining the numerators and further simplifying the expression.

Step 3: Find a Common Denominator and Combine Fractions

To combine the terms $(1-2x)$ and $-\frac{3}{24+x}$, we need a common denominator. As discussed, the common denominator will be $(24+x)$. We rewrite the first term as $\frac{(1-2x)(24+x)}{24+x}$. Now the expression looks like this:

$\frac{(1-2x)(24+x)}{24+x} - \frac{3}{24+x}$

Now that we have a common denominator, we can combine the fractions by subtracting the numerators:

$\frac{(1-2x)(24+x) - 3}{24+x}$

This step is a critical juncture in the simplification process. We've successfully brought the two terms together under a single fraction, which is a significant step towards our goal. The numerator now contains a product and a constant, which we will need to expand and simplify further. Combining fractions using a common denominator is a fundamental algebraic operation, and it's showcased effectively in this step. The resulting expression, while not yet in its simplest form, is now more manageable and lends itself to further manipulation. The next step will involve expanding the product in the numerator and then combining like terms. This will help us to further condense the expression and identify any potential cancellations or simplifications. With the fractions now combined, we are well-positioned to complete the simplification process by focusing on the numerator and reducing it to its simplest form. The techniques we are employing are standard algebraic procedures, demonstrating the power and consistency of these methods in tackling mathematical problems.

Step 4: Expand and Simplify the Numerator

Next, we expand the product in the numerator: $(1-2x)(24+x) = 1(24) + 1(x) - 2x(24) - 2x(x) = 24 + x - 48x - 2x^2$. So the numerator becomes:

$24 + x - 48x - 2x^2 - 3$

Now, combine like terms:

$-2x^2 - 47x + 21$

So the expression now is:

$\frac{-2x^2 - 47x + 21}{24+x}$

This step is where the algebraic manipulation becomes more intricate. Expanding the product in the numerator is a key skill in simplifying expressions, and it requires careful application of the distributive property. Expanding and simplifying algebraic expressions is a cornerstone of mathematical manipulation, and this step demonstrates its practical application. The resulting quadratic expression in the numerator is a significant outcome, as it encapsulates the combined effect of the original terms. Combining like terms is a crucial follow-up to expansion, ensuring that the expression is written in its most concise form. The simplified numerator now presents a quadratic expression, which may or may not be factorable. Whether it can be factored or not will determine the next course of action in our simplification process. The current form of the expression, with a simplified numerator and a linear denominator, allows us to assess whether further simplification is possible, such as through factorization or cancellation of common factors. We've made substantial progress in reducing the complexity of the original expression, and we are now in a position to evaluate the final steps required to achieve the simplest form.

Step 5: Check for Further Simplification

Now we have the expression $\frac{-2x^2 - 47x + 21}{24+x}$. We should see if the numerator can be factored. We are looking for two numbers that multiply to $-2 \cdot 21 = -42$ and add up to $-47$. These numbers are $-1$ and $-42$. We can rewrite the quadratic as follows:

$-2x^2 - 47x + 21 = -2x^2 - 42x - 5x + 21$

Now, factor by grouping:

$-2x(x + 21) - 1(5x - 21)$ This doesn't seem to factor nicely, so we can conclude that the expression is already in its simplest form.

This step is crucial in ensuring that we have indeed reached the most simplified version of the expression. Checking for further simplification, especially through factorization, is a standard practice in algebraic manipulation. The attempt to factor the quadratic expression in the numerator is a key part of this process. Factoring, if possible, would allow us to potentially cancel common factors with the denominator, leading to a simpler expression. However, in this case, the quadratic does not factor neatly, indicating that we have reached a point where no further simplification through factorization is possible. This conclusion is valuable, as it confirms that our efforts have led us to the simplest form of the expression. The expression, as it stands, represents the most reduced form achievable through algebraic manipulation. This final check for simplification underscores the importance of rigor and thoroughness in mathematical problem-solving. By systematically evaluating the possibilities for further reduction, we ensure the accuracy and completeness of our solution. The expression we have obtained is the final simplified form, reflecting the culmination of our step-by-step simplification process.

Conclusion

In conclusion, the simplified form of the expression $\frac{1}{\frac{1}{1-2 x}}-\frac{3}{24+x}$ is $\frac{-2x^2 - 47x + 21}{24+x}$. We arrived at this solution by systematically simplifying the expression step by step. This involved dealing with the complex fraction in the first term, finding a common denominator, combining fractions, expanding and simplifying the numerator, and finally, checking for further factorization. Each step required the application of fundamental algebraic principles, demonstrating the importance of a solid foundation in these concepts. The process highlighted the significance of breaking down complex problems into smaller, more manageable parts, a strategy that is widely applicable in mathematics and beyond. The final expression represents the most reduced form of the original expression, achieved through careful and methodical simplification. This exercise underscores the elegance and power of algebraic manipulation in transforming expressions into their simplest forms. By mastering these techniques, we can confidently tackle a wide range of mathematical problems and gain a deeper appreciation for the underlying structures and relationships within algebraic expressions. Understanding how to simplify expressions is not just about finding the right answer; it's about developing a problem-solving mindset and a proficiency in mathematical reasoning. For more information on simplifying algebraic expressions, you can visit resources like Khan Academy.