Solving (8 / (4u - 8)) - 1 = 2 / (u - 2) A Step-by-Step Guide

Introduction to Rational Equations

In the world of mathematics, rational equations play a crucial role in various fields, from engineering to economics. These equations involve fractions where the numerator and/or the denominator contain variables. Solving rational equations requires a methodical approach, combining algebraic manipulation with a keen eye for potential pitfalls. One such equation is ${\frac{8}{4u-8}-1=\frac{2}{u-2}}$, which we will dissect step-by-step to illustrate the process. Understanding how to solve rational equations is not just an academic exercise; it's a fundamental skill that empowers you to tackle real-world problems with confidence. The journey through solving this equation will highlight the importance of identifying restrictions, finding common denominators, and verifying solutions. Let's embark on this mathematical adventure and unravel the mysteries of rational equations, equipping you with the knowledge and skills to master them.

This exploration begins with a clear understanding of what makes an equation rational – the presence of variables in the denominator. This seemingly simple characteristic introduces a layer of complexity, demanding careful consideration of values that might make the denominator zero, thereby rendering the expression undefined. As we delve deeper, we'll uncover strategies for simplifying these equations, transforming them into more manageable forms. The heart of our approach lies in the strategic use of algebraic techniques, such as multiplying by the least common denominator, to eliminate fractions and pave the way for a straightforward solution. However, the journey doesn't end with finding a potential solution. We must meticulously verify our answers, ensuring they don't lead to mathematical impossibilities in the original equation. This rigorous process not only validates our solution but also reinforces our understanding of the underlying principles of rational equations. So, let's dive in and transform this equation from a challenge into a triumph of mathematical understanding.

1. Identifying Restrictions

The first crucial step in solving any rational equation, including ${\frac{8}{4u-8}-1=\frac{2}{u-2}}$, is to identify the restrictions. These are values of the variable that would make the denominator of any fraction in the equation equal to zero. Why is this so important? Because division by zero is undefined in mathematics. Ignoring these restrictions can lead to extraneous solutions, which are solutions that we find algebraically but do not satisfy the original equation. For our equation, we have two denominators: ${4u-8}$ and ${u-2}$. To find the restrictions, we set each denominator equal to zero and solve for ${u}$.

For the first denominator, ${4u-8}$, we have:

${4u - 8 = 0}$

Adding 8 to both sides gives:

${4u = 8}$

Dividing both sides by 4, we find:

${u = 2}$

For the second denominator, ${u-2}$, we have:

${u - 2 = 0}$

Adding 2 to both sides gives:

${u = 2}$

Notice that both denominators yield the same restriction: ${u = 2}$. This means that ${u}$ cannot be equal to 2, or we will be dividing by zero. We must keep this in mind as we proceed to solve the equation. Identifying these restrictions at the outset is paramount, serving as a safeguard against accepting extraneous solutions later on. This initial step not only highlights the importance of mathematical rigor but also sets the stage for a more confident and accurate solution process. By acknowledging and noting these limitations, we ensure that our final answer is not just algebraically correct but also mathematically valid within the context of the original equation. Therefore, before we delve into the algebraic manipulations needed to solve, we've armed ourselves with the knowledge of what values are off-limits, guiding our journey towards a sound and reliable solution.

2. Simplifying the Equation

With the restriction ${u \neq 2}$ firmly in mind, our next step in solving the rational equation ${\frac{8}{4u-8}-1=\frac{2}{u-2}}$ is to simplify the equation. This involves algebraic manipulations to eliminate fractions and make the equation easier to solve. The first thing we can do is factor the denominator ${4u-8}$. Factoring out a 4, we get ${4(u-2)}$. Rewriting the equation, we have:

${\frac{8}{4(u-2)} - 1 = \frac{2}{u-2}}$

Now, we can simplify the first fraction by dividing both the numerator and the denominator by 4:

${\frac{2}{u-2} - 1 = \frac{2}{u-2}}$

Notice that we now have a common denominator, ${u-2}$, in two terms. This simplification has made the equation much more manageable. The process of simplification is not just about making the equation look neater; it's about revealing the underlying structure and relationships within the equation. By factoring and canceling common factors, we strip away the unnecessary complexity and bring the core problem into sharper focus. This step is crucial because it sets the stage for the next phase of the solution, where we will manipulate the equation to isolate the variable and find its value. The careful attention to detail in simplification ensures that we are working with the most straightforward form of the equation, reducing the chances of errors and making the solution process more efficient. It’s a testament to the power of algebraic manipulation, transforming a potentially daunting equation into a more approachable problem, ready to be solved with confidence.

3. Eliminating Fractions

To further simplify the equation ${\frac{2}{u-2} - 1 = \frac{2}{u-2}}$ and make it easier to solve, the next logical step is to eliminate the fractions. We can achieve this by multiplying both sides of the equation by the least common denominator (LCD). In this case, the LCD is ${u-2}$. Multiplying both sides by ${(u-2)}$, we get:

${(u-2)\left(\frac{2}{u-2} - 1\right) = (u-2)\left(\frac{2}{u-2}\right)}$

Distributing ${(u-2)}$ on the left side, we have:

${(u-2)\left(\frac{2}{u-2}\right) - (u-2)(1) = (u-2)\left(\frac{2}{u-2}\right)}$

Simplifying, we get:

${2 - (u-2) = 2}$

The fractions have now been eliminated, leaving us with a linear equation that is much simpler to solve. This step is pivotal in the process of solving rational equations because it transforms a complex equation into a familiar form that we can handle with standard algebraic techniques. The act of multiplying by the LCD is not just a mechanical step; it's a strategic move that leverages the properties of fractions and equations to our advantage. By clearing the denominators, we remove a significant hurdle and pave the way for isolating the variable. This transformation is a powerful demonstration of how algebraic manipulation can simplify seemingly complicated problems, making them accessible and solvable. The resulting linear equation is now in a form where we can easily apply the rules of algebra to find the value of the variable, bringing us closer to the solution we seek.

4. Solving for the Variable

Now that we've eliminated the fractions, we have the simplified equation ${2 - (u-2) = 2}$. Our next task is to solve for the variable ${u}$. First, let's distribute the negative sign:

${2 - u + 2 = 2}$

Combining like terms on the left side, we get:

${4 - u = 2}$

Subtracting 4 from both sides gives:

${-u = -2}$

Finally, multiplying both sides by -1, we find:

${u = 2}$

This algebraic manipulation has led us to a potential solution for our equation. The process of solving for the variable involves carefully applying the rules of algebra to isolate the variable on one side of the equation. Each step, from distributing the negative sign to combining like terms and performing inverse operations, is a deliberate action aimed at simplifying the equation and revealing the value of the unknown. The journey through these steps is a testament to the power of algebraic reasoning, transforming a complex equation into a clear and concise solution. However, the solution we've found is not the end of our quest. We must now put this potential solution to the ultimate test: verification. This crucial step will ensure that our algebraic efforts have not led us astray and that the value we've found is indeed a valid solution to the original equation. It's a reminder that in mathematics, rigor and precision are paramount, and every step must be carefully scrutinized to ensure accuracy and validity.

5. Verifying the Solution

We've arrived at a potential solution, ${u = 2}$, for the equation ${\frac{8}{4u-8}-1=\frac{2}{u-2}}$. However, before we declare victory, we must verify the solution. This is a critical step because, as we identified earlier, ${u = 2}$ is a restriction. It makes the denominators ${4u-8}$ and ${u-2}$ equal to zero, which would make the fractions undefined.

Let's substitute ${u = 2}$ into the original equation:

${\frac{8}{4(2)-8} - 1 = \frac{2}{2-2}}$

Simplifying the denominators, we get:

${\frac{8}{8-8} - 1 = \frac{2}{0}}$

${\frac{8}{0} - 1 = \frac{2}{0}}$

We can see that substituting ${u = 2}$ results in division by zero, which is undefined. Therefore, ${u = 2}$ is not a valid solution to the equation. This underscores the importance of checking solutions against the restrictions we identified at the beginning. Verification is not just a formality; it's an essential safeguard against accepting extraneous solutions, which are values that satisfy the transformed equation but not the original one. The act of verification brings the entire solution process full circle, reinforcing the importance of each step, from identifying restrictions to algebraic manipulation. It's a reminder that in mathematics, accuracy and rigor are paramount, and every solution must be subjected to scrutiny to ensure its validity. In this case, the verification process has revealed that our potential solution is, in fact, not a solution at all, highlighting the critical role of this step in solving rational equations.

Conclusion

In conclusion, we set out to solve the rational equation ${\frac{8}{4u-8}-1=\frac{2}{u-2}}$. Through a step-by-step process, we identified the restriction ${u \neq 2}$, simplified the equation, eliminated fractions, and solved for the variable, arriving at the potential solution ${u = 2}$. However, the crucial step of verifying the solution revealed that ${u = 2}$ is an extraneous solution because it results in division by zero in the original equation. Therefore, the equation has no solution. This journey through the solution process has highlighted several key aspects of solving rational equations. First, identifying restrictions is paramount. It sets the boundaries within which our solutions must lie and prevents us from accepting extraneous solutions. Second, simplifying the equation through factoring and canceling common factors makes the problem more manageable. Third, eliminating fractions by multiplying by the LCD transforms the equation into a more familiar form. Fourth, solving for the variable involves applying algebraic techniques to isolate the unknown. Finally, verifying the solution is the ultimate test, ensuring that our algebraic manipulations have led us to a valid answer. The absence of a solution in this case underscores a critical lesson in mathematics: not every equation has a solution, and it's our responsibility to rigorously verify our answers to ensure their validity. This comprehensive approach not only helps us solve rational equations but also reinforces our understanding of mathematical principles and the importance of precision in problem-solving.