Solving 2/(x-9) = 1/(x-3) - 1/(x-4) A Step-by-Step Guide
Introduction
In this article, we will delve into the intricacies of solving the algebraic equation 2/(x-9) = 1/(x-3) - 1/(x-4). This equation falls under the category of rational equations, where the variable x appears in the denominator of one or more terms. Solving such equations requires a systematic approach to eliminate the fractions and arrive at a polynomial equation that can be solved using standard algebraic techniques. We will explore the step-by-step process, highlighting key concepts and potential pitfalls along the way. Our main keywords are solving rational equations, algebraic manipulation, finding solutions, and extraneous solutions. Understanding these concepts is crucial for anyone studying algebra, calculus, or related fields.
Before we dive into the solution, it's important to understand the fundamental principles behind solving rational equations. The primary goal is to eliminate the denominators by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. This transformation converts the rational equation into a polynomial equation, which is typically easier to solve. However, this process may introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it's essential to check all potential solutions in the original equation to identify and discard any extraneous solutions.
The equation 2/(x-9) = 1/(x-3) - 1/(x-4) presents a classic example of a rational equation that requires careful manipulation. The presence of variables in the denominators (x-9), (x-3), and (x-4) necessitates a strategic approach to eliminate these fractions. We must also be mindful of the values of x that would make any of the denominators zero, as these values are not in the domain of the equation. Specifically, x cannot be equal to 9, 3, or 4. This constraint will be crucial when we check for extraneous solutions later in the process. The process involves finding a common denominator, combining fractions, and simplifying the resulting equation. Factoring plays a key role in identifying potential common factors and simplifying the equation. Throughout the process, we will emphasize the importance of maintaining the equality and performing the same operations on both sides of the equation.
Step-by-Step Solution
1. Find a Common Denominator
The first step in solving rational equations is to find a common denominator for all the fractions in the equation. In our case, the denominators are (x-9), (x-3), and (x-4). Since these expressions do not share any common factors, the least common multiple (LCM) is simply their product: (x-9)(x-3)(x-4). This common denominator will allow us to combine the fractions on both sides of the equation. To achieve this, we need to multiply each fraction by a form of 1 that will result in the common denominator. The key here is algebraic manipulation to ensure we maintain the equality of the equation. Our goal is to transform the equation into a more manageable form by eliminating the fractions.
2. Multiply Both Sides by the Common Denominator
To eliminate the fractions, we multiply both sides of the equation by the common denominator (x-9)(x-3)(x-4). This step is crucial in solving rational equations because it transforms the equation into a polynomial equation. By multiplying both sides, we ensure that the equality is maintained. The left side of the equation becomes:
2/(x-9) * (x-9)(x-3)(x-4) = 2(x-3)(x-4)
The right side of the equation becomes:
[1/(x-3) - 1/(x-4)] * (x-9)(x-3)(x-4) = (x-9)(x-4) - (x-9)(x-3)
This step simplifies the equation significantly, removing the fractions and making it easier to work with. The next step involves expanding the products and simplifying the resulting polynomial equation. This is a critical step in finding solutions to the original equation.
3. Expand and Simplify
Now, we expand the products on both sides of the equation and simplify. This step requires careful attention to detail to avoid errors in the algebraic manipulation. Expanding the left side, we have:
2(x-3)(x-4) = 2(x^2 - 7x + 12) = 2x^2 - 14x + 24
Expanding the right side, we have:
(x-9)(x-4) - (x-9)(x-3) = (x^2 - 13x + 36) - (x^2 - 12x + 27) = -x + 9
So the equation becomes:
2x^2 - 14x + 24 = -x + 9
Next, we move all terms to one side to obtain a quadratic equation:
2x^2 - 14x + 24 + x - 9 = 0
2x^2 - 13x + 15 = 0
This quadratic equation can now be solved using factoring, completing the square, or the quadratic formula.
4. Solve the Quadratic Equation
We now have a quadratic equation: 2x^2 - 13x + 15 = 0. To solve this equation, we can attempt to factor it. We look for two numbers that multiply to (2)(15) = 30 and add up to -13. These numbers are -10 and -3. We can rewrite the middle term as -10x - 3x and factor by grouping:
2x^2 - 10x - 3x + 15 = 0
2x(x - 5) - 3(x - 5) = 0
(2x - 3)(x - 5) = 0
Setting each factor equal to zero gives us two potential solutions:
2x - 3 = 0 => x = 3/2
x - 5 = 0 => x = 5
These are our candidate solutions, but we need to check them for extraneous solutions.
5. Check for Extraneous Solutions
It's crucial to check our solutions in the original equation to ensure they are valid. This step is particularly important when solving rational equations because multiplying by the common denominator can introduce extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation. Recall that the original equation is:
2/(x-9) = 1/(x-3) - 1/(x-4)
We identified x = 3/2 and x = 5 as potential solutions. We must also remember that x cannot be 3, 4, or 9, as these values would make the denominators zero.
First, let's check x = 3/2:
2/(3/2 - 9) = 2/(-15/2) = -4/15
1/(3/2 - 3) - 1/(3/2 - 4) = 1/(-3/2) - 1/(-5/2) = -2/3 + 2/5 = (-10 + 6)/15 = -4/15
Since both sides are equal, x = 3/2 is a valid solution.
Now, let's check x = 5:
2/(5 - 9) = 2/(-4) = -1/2
1/(5 - 3) - 1/(5 - 4) = 1/2 - 1/1 = 1/2 - 1 = -1/2
Since both sides are equal, x = 5 is also a valid solution.
Conclusion
The solutions to the equation 2/(x-9) = 1/(x-3) - 1/(x-4) are x = 3/2 and x = 5. We arrived at these solutions by systematically eliminating the fractions, simplifying the resulting polynomial equation, and carefully checking for extraneous solutions. This process highlights the importance of algebraic manipulation, factoring, and solution verification when dealing with rational equations. Finding solutions to such equations often involves a multi-step approach, and each step must be executed with precision. This comprehensive guide provides a clear and detailed explanation of the steps involved, making it easier for students and enthusiasts to understand and apply these techniques to similar problems. Remember, the key to success in solving rational equations lies in understanding the underlying principles and practicing consistently.
