Solving Rational Equations: A Step-by-Step Guide

Hey guys! Let's dive into solving a cool equation today. We're going to tackle the rational equation $\frac{4}{x^2-5 x}=\frac{3}{2 x-11}$. If we find more than one answer, we'll make sure to list them separated by commas. So, grab your pencils, and let's get started!

Understanding Rational Equations

Before we jump into solving, let's quickly recap what rational equations are all about. Rational equations are equations that contain fractions where the numerator and/or the denominator include a variable. These equations can seem a bit intimidating at first, but with a systematic approach, they become much easier to handle. The key is to eliminate the fractions by finding a common denominator and then solving the resulting equation. Remember, it's crucial to check our answers at the end because sometimes we might get solutions that don't actually work in the original equation – these are called extraneous solutions.

When dealing with rational equations, one of the primary goals is to clear the fractions. To do this, we typically look for the least common denominator (LCD) of all the fractions involved. Once we've found the LCD, we multiply both sides of the equation by it. This step is essential because it transforms the equation into a more manageable form, usually a polynomial equation, which we can then solve using standard algebraic techniques. However, it's worth noting that when we multiply both sides by an expression containing a variable, we introduce the possibility of extraneous solutions. These are solutions that satisfy the transformed equation but not the original rational equation. Therefore, meticulous checking of solutions against the original equation is an indispensable part of the solution process.

Rational equations pop up in various real-world scenarios, from calculating rates and times to modeling complex systems in physics and engineering. For instance, they can be used to determine how long it takes for two people working together to complete a task, given their individual work rates. They also play a significant role in understanding the behavior of electrical circuits, fluid dynamics, and even economic models. By mastering the techniques to solve these equations, you're not just learning a mathematical concept; you're gaining a tool that can be applied across a wide range of practical problems. So, let's keep this in mind as we proceed, and aim to not only solve the equations but also understand the broader applicability of our skills!

Step 1: Identify the Domain and Excluded Values

Okay, first things first. We need to figure out what values of x would make our denominators zero, because dividing by zero is a big no-no in math. For the first fraction, $\frac{4}{x^2-5x}$, we need to find when $x^2 - 5x = 0$. Factoring this gives us $x(x - 5) = 0$, so $x = 0$ or $x = 5$. For the second fraction, $\frac{3}{2x-11}$, we set $2x - 11 = 0$, which gives us $x = \frac{11}{2}$. So, our excluded values are 0, 5, and $\frac{11}{2}$. Remember these – they're super important!

Why are these excluded values so important, you ask? Well, these are the values that, if we plug them back into the original equation, would cause us to divide by zero. And as we all know, division by zero is undefined in mathematics. So, these values effectively create 'holes' in the domain of our equation. Think of it like a road with potholes – you can't drive through them without causing a problem! In the context of our equation, if we were to find a solution that is one of these excluded values, it wouldn't actually be a valid solution. It's what we call an extraneous solution – a solution that arises from the process of solving the equation but doesn't satisfy the original equation. This is why identifying these values at the outset is such a critical step. It sets the stage for us to verify our answers later and ensure they're legitimate solutions.

Knowing the excluded values not only helps us filter out extraneous solutions, but it also gives us a better understanding of the equation's behavior. It tells us where the equation might have discontinuities or where the graph of the related function might have vertical asymptotes. This is valuable information that can be used to sketch the graph of the function or to understand its properties in more detail. So, by taking the time to identify these values, we're not just following a mechanical step; we're gaining deeper insight into the equation itself. It’s like understanding the terrain before embarking on a journey – it helps us navigate more effectively and avoid potential pitfalls.

Step 2: Find the Least Common Denominator (LCD)

Next up, we need to find the least common denominator (LCD) of our fractions. Our denominators are $x^2 - 5x$ and $2x - 11$. We already factored $x^2 - 5x$ into $x(x - 5)$. So, the LCD is simply $x(x - 5)(2x - 11)$. Easy peasy, right?

Finding the LCD is a crucial step because it allows us to clear the fractions from our equation, making it much easier to solve. Think of the LCD as the magic ingredient that transforms our equation from a complex-looking rational equation into a more familiar polynomial equation. But why is the LCD so effective? It's because when we multiply each term in the equation by the LCD, the denominators nicely cancel out. This happens because the LCD is, by definition, divisible by each of the denominators in the equation. So, when we multiply a fraction by the LCD, the denominator divides into the LCD perfectly, leaving us with a whole number or a simpler expression.

The process of finding the LCD involves identifying all the unique factors present in the denominators and then taking the highest power of each factor. In our case, we had the factors $x$, $(x - 5)$, and $(2x - 11)$. Since each factor appears only once in the denominators, the LCD is simply the product of these factors: $x(x - 5)(2x - 11)$. In more complex scenarios, you might encounter denominators with repeated factors or higher powers, but the principle remains the same: identify the factors and their highest powers, and then multiply them together. Once you've mastered the art of finding the LCD, you'll find that rational equations become much less daunting. It's a fundamental skill that unlocks the door to solving a wide range of problems involving fractions and algebraic expressions.

Step 3: Multiply Both Sides by the LCD

Now, we're going to multiply both sides of our equation by the LCD, which is $x(x - 5)(2x - 11)$. This gives us:

$x(x - 5)(2x - 11) \cdot \frac{4}{x(x - 5)} = x(x - 5)(2x - 11) \cdot \frac{3}{2x - 11}$

On the left side, $x(x - 5)$ cancels out, and on the right side, $(2x - 11)$ cancels out. This leaves us with:

$4(2x - 11) = 3x(x - 5)$

See how the fractions magically disappeared? That's the power of the LCD!

Multiplying both sides of the equation by the LCD is a pivotal step in solving rational equations, and it's worth understanding why it works so effectively. The fundamental principle at play here is that multiplying both sides of an equation by the same non-zero quantity maintains the equality. By choosing the LCD as our multiplier, we're not just picking any quantity; we're selecting a specific expression that has the unique ability to eliminate the denominators from our equation. This transformation is what makes the equation much easier to handle.

Think of it like this: the LCD is like a common currency that allows us to trade different fractions for whole numbers. When we multiply a fraction by the LCD, we're essentially converting it into a whole expression. This conversion happens because the denominator of the fraction divides evenly into the LCD, leaving us with a whole number or a simpler polynomial. The result is a new equation that is free from fractions, which we can then solve using standard algebraic techniques. But it's crucial to remember that this transformation is valid only if we multiply every term in the equation by the LCD. Neglecting to do so would disrupt the balance of the equation and lead to incorrect solutions. So, make sure each term gets its fair share of the LCD magic!

Step 4: Simplify and Solve the Equation

Let's simplify and solve the equation we got in the last step:

$4(2x - 11) = 3x(x - 5)$

Expand both sides:

$8x - 44 = 3x^2 - 15x$

Move everything to one side to set the equation to zero:

$0 = 3x^2 - 15x - 8x + 44$

Combine like terms:

$0 = 3x^2 - 23x + 44$

Now we have a quadratic equation! We can try to factor it, but if that doesn't work, we can always use the quadratic formula. Let's try factoring:

$0 = (3x - 11)(x - 4)$

Setting each factor to zero gives us:

$3x - 11 = 0$ or $x - 4 = 0$

Solving for x, we get:

$x = \frac{11}{3}$ or $x = 4$

Alright, we've got two potential solutions: $\frac{11}{3}$ and 4.

Simplifying and solving the equation after clearing the fractions is where our algebraic skills really shine. This step often involves expanding expressions, combining like terms, and rearranging the equation into a form that we can easily solve. In our case, we arrived at a quadratic equation, which is a common outcome when dealing with rational equations. Quadratic equations are like the Swiss Army knives of algebra – they can be solved in multiple ways, depending on their specific form and characteristics. Factoring, the quadratic formula, and completing the square are all powerful tools in our arsenal.

In this particular problem, we were fortunate enough to be able to factor the quadratic equation. Factoring is often the quickest and most elegant method when it works, but it's not always possible. When factoring proves elusive, the quadratic formula stands ready to provide the solutions. The quadratic formula is a universal solver – it can handle any quadratic equation, no matter how complex. And for those who appreciate a more geometric approach, completing the square offers a way to transform the equation into a form where the solutions can be read off directly. The key is to recognize the structure of the equation and choose the most efficient method for solving it. By mastering these techniques, we gain the confidence to tackle a wide range of algebraic challenges.

Step 5: Check for Extraneous Solutions

Remember those excluded values we found in Step 1? Now's the time to check if our solutions are valid. Our excluded values were 0, 5, and $\frac{11}{2}$. Our potential solutions are $\frac{11}{3}$ and 4. Neither of our solutions matches our excluded values, so they both seem good to go!

But just to be super sure, let's plug them back into the original equation:

For $x = \frac{11}{3}$:

$\frac{4}{(\frac{11}{3})^2 - 5(\frac{11}{3})} = \frac{3}{2(\frac{11}{3}) - 11}$

$\frac{4}{\frac{121}{9} - \frac{55}{3}} = \frac{3}{\frac{22}{3} - 11}$

$\frac{4}{\frac{121 - 165}{9}} = \frac{3}{\frac{22 - 33}{3}}$

$\frac{4}{-\frac{44}{9}} = \frac{3}{-\frac{11}{3}}$

$-\frac{9}{11} = -\frac{9}{11}$ (It checks out!)

For $x = 4$:

$\frac{4}{4^2 - 5(4)} = \frac{3}{2(4) - 11}$

$\frac{4}{16 - 20} = \frac{3}{8 - 11}$

$\frac{4}{-4} = \frac{3}{-3}$

$-1 = -1$ (It checks out too!)

Both solutions are valid. Yay!

Checking for extraneous solutions is like the final quality control step in our equation-solving process. It's where we make sure that the solutions we've found not only satisfy the transformed equation but also the original equation. Why is this so important? Because, as we've seen, the process of clearing fractions by multiplying by the LCD can sometimes introduce solutions that don't actually work in the original equation. These sneaky solutions are called extraneous solutions, and they're the mathematical equivalent of imposters – they look like solutions, but they're not the real deal.

The reason extraneous solutions can arise is that multiplying by the LCD can inadvertently change the domain of the equation. Remember, we identified excluded values at the beginning – values that make the denominator zero and thus are not allowed. When we multiply by the LCD, we might be effectively canceling out a factor that, if it were zero, would make the original equation undefined. This can create a situation where a value that was previously excluded now appears to be a valid solution. That's why plugging our solutions back into the original equation is crucial. It's the ultimate test of whether a solution is genuine or an imposter. Think of it as a detective double-checking their evidence to make sure they've got the right suspect.

Step 6: State the Solution

Finally, we can confidently state our solution. The solutions to the equation $\frac{4}{x^2-5 x}=\frac{3}{2 x-11}$ are $x = \frac{11}{3}, 4$.

So, there you have it! Solving rational equations can seem tricky at first, but by following these steps carefully, you can conquer them like a math pro. Remember to always check for excluded values and extraneous solutions to ensure your answers are correct. Keep practicing, and you'll become a rational equation-solving wizard in no time!

Practice Problems

Want to test your skills? Here are a few more rational equations you can try solving:

1. $\frac{2}{x} + \frac{3}{x-1} = 1$
2. $\frac{1}{x+2} - \frac{2}{x-3} = \frac{4}{x^2-x-6}$
3. $\frac{x}{x-4} = \frac{10}{x-4} + 3$

Good luck, and happy solving! Remember, the key to mastering math is practice, practice, practice!