Solving Rational Equations: A Detailed Explanation Of `\frac{4}{x+12}-\frac{3}{x-12}=\frac{5x}{x^2-144}`

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Rational equations, such as the one presented, \frac{4}{x+12}-\frac{3}{x-12}=\frac{5x}{x^2-144}, are a fundamental topic in algebra. These equations involve fractions where the numerators and/or denominators contain variables. Solving them requires a blend of algebraic manipulation, factoring skills, and careful attention to potential extraneous solutions. This article will provide a comprehensive guide to solving this specific equation, and rational equations in general, emphasizing the underlying concepts and techniques involved.

Understanding Rational Equations

At their core, rational equations are equations that contain one or more fractions where the numerator and/or the denominator are polynomials. The key to solving these equations lies in eliminating the fractions, which transforms the equation into a more manageable form, typically a linear or quadratic equation. However, this process can introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original rational equation. Therefore, it's crucial to check all solutions obtained against the original equation to ensure their validity.

The given equation, \frac{4}{x+12}-\frac{3}{x-12}=\frac{5x}{x^2-144}, perfectly illustrates this type of problem. It involves rational expressions, and our goal is to find the values of x that satisfy the equation. The presence of variables in the denominators immediately alerts us to the possibility of extraneous solutions, particularly values of x that would make the denominators equal to zero.

Before diving into the solution, it's essential to grasp the significance of the denominator. A denominator of zero makes the fraction undefined. In our equation, the denominators are x+12, x-12, and x^2-144. Setting each of these equal to zero gives us potential values of x that must be excluded from the solution set. These values are x = -12 and x = 12. We'll need to keep these in mind as we proceed.

Now, let's break down the steps required to solve the equation, highlighting the key algebraic manipulations and considerations.

Step-by-Step Solution

1. Factor the Denominators

The first crucial step in solving rational equations is to factor all the denominators. This allows us to identify the least common denominator (LCD), which is essential for eliminating the fractions. In our equation, we have the denominator x^2-144. This expression is a difference of squares, which factors nicely as (x+12)(x-12). Our equation now looks like this:

\frac{4}{x+12}-\frac{3}{x-12}=\frac{5x}{(x+12)(x-12)}

2. Identify the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that is divisible by all the denominators in the equation. In this case, the denominators are (x+12), (x-12), and (x+12)(x-12). The LCD is simply (x+12)(x-12). Understanding how to find the LCD is crucial because it's the key to eliminating the fractions.

3. Multiply Both Sides by the LCD

This is the core step in solving rational equations. Multiplying both sides of the equation by the LCD clears the fractions. We multiply each term in the equation by (x+12)(x-12):

(x+12)(x-12) \cdot \frac{4}{x+12} - (x+12)(x-12) \cdot \frac{3}{x-12} = (x+12)(x-12) \cdot \frac{5x}{(x+12)(x-12)}

Notice how the factors cancel out:

4(x-12) - 3(x+12) = 5x

We've successfully transformed the rational equation into a linear equation. This simplification is the goal of multiplying by the LCD.

4. Simplify and Solve the Resulting Equation

Now we have a linear equation to solve. Let's distribute and combine like terms:

4x - 48 - 3x - 36 = 5x

x - 84 = 5x

Subtract x from both sides:

-84 = 4x

Divide by 4:

x = -21

5. Check for Extraneous Solutions

Remember, checking for extraneous solutions is a critical step. We found that x = -21, but we need to make sure it doesn't make any of the original denominators equal to zero. We previously identified that x cannot be -12 or 12. Since -21 is neither of these values, it's a valid solution.

Let's plug x = -21 back into the original equation to verify:

\frac{4}{-21+12}-\frac{3}{-21-12}=\frac{5(-21)}{(-21)^2-144}

\frac{4}{-9}-\frac{3}{-33}=\frac{-105}{441-144}

\frac{-4}{9} + \frac{1}{11} = \frac{-105}{297}

\frac{-44 + 9}{99} = \frac{-35}{99}

\frac{-35}{99} = \frac{-35}{99}

The solution checks out. Therefore, the solution to the equation is x = -21.

Common Mistakes and How to Avoid Them

Solving rational equations can be tricky, and several common mistakes can lead to incorrect solutions. Being aware of these pitfalls and how to avoid them is essential for success.

  • Forgetting to Check for Extraneous Solutions: This is perhaps the most common mistake. As we've emphasized, multiplying by the LCD can introduce solutions that don't work in the original equation. Always check your solutions.

  • Incorrectly Finding the LCD: A mistake in finding the LCD will derail the entire solution process. Ensure you've factored all denominators correctly and that the LCD includes all necessary factors with the highest power present in any denominator.

  • Distributing Negatives Incorrectly: When multiplying by the LCD, you'll often have to distribute negative signs. A careless mistake here can lead to an incorrect equation and, consequently, an incorrect solution. Double-check your distribution.

  • Arithmetic Errors: Simple arithmetic errors, like adding or subtracting incorrectly, can throw off your solution. Take your time and be careful with your calculations.

  • Dividing by Zero: This is a fundamental no-no in mathematics. Make sure you've identified any values of x that would make a denominator zero and exclude them from your solution set.

By being mindful of these common mistakes and carefully checking your work, you can significantly improve your accuracy in solving rational equations.

General Strategies for Solving Rational Equations

While we've focused on a specific equation, the principles we've discussed apply to a wide range of rational equations. Here's a summary of general strategies:

  1. Factor all denominators: This is the crucial first step in identifying the LCD.

  2. Identify the LCD: The LCD is the smallest expression divisible by all denominators.

  3. Multiply both sides of the equation by the LCD: This eliminates the fractions.

  4. Simplify and solve the resulting equation: This will typically be a linear or quadratic equation.

  5. Check for extraneous solutions: Plug your solutions back into the original equation to verify they are valid.

  6. State your solution(s): Clearly indicate the values of x that satisfy the equation.

By following these steps systematically, you can approach any rational equation with confidence. Practice is key to mastering this skill. Work through a variety of examples to solidify your understanding and develop your problem-solving abilities.

Conclusion

Solving the rational equation \frac{4}{x+12}-\frac{3}{x-12}=\frac{5x}{x^2-144} demonstrates the core principles involved in dealing with this type of equation. By factoring denominators, identifying the least common denominator, clearing fractions, and checking for extraneous solutions, we arrived at the solution x = -21. This process highlights the importance of careful algebraic manipulation and attention to detail. Mastering rational equations is an essential step in developing your algebraic skills and preparing for more advanced mathematical concepts. Remember to practice consistently and apply the strategies outlined in this article to tackle any rational equation you encounter. Understanding these concepts thoroughly will not only help you solve equations effectively but also build a strong foundation for further mathematical studies.