Solving the Linear Equation 3/x - 2/x 5/(x^2 - 1)

Introduction to Fractional Equations:

Fractional equations involve rational expressions, where the variable appears in the denominator. These types of equations can be challenging but are essential to understand for advanced mathematics and problem-solving in various fields such as engineering and physics. In this article, we will delve into a specific problem and provide a comprehensive step-by-step solution.

Problem Statement:

Given the equation: frac{3}{x} - frac{2}{x} frac{5}{x^2 - 1}

Step-by-Step Solution:

Let's break down the problem and solve it systematically.

Step 1: Simplification

First, we notice that frac{3}{x} - frac{2}{x} frac{1}{x}.

So, the equation becomes:

frac{1}{x} frac{5}{x^2 - 1}

Step 2: Cross-Multiplication

Cross-multiplying both sides, we get:

1 cdot (x^2 - 1) 5 cdot x

This simplifies to:

x^2 - 1 5x

Step 3: Rearranging the Equation

Rearranging the terms to set the equation to zero:

x^2 - 5x - 1 0

Step 4: Factorization

Next, we look to factorize the quadratic equation:

x^2 - 6x x - 6 0

This can be written as:

x(x - 6) 1(x - 6) 0

Factoring out the common term (x - 6):

(x - 6)(x 1) 0

Step 5: Solving for x

Setting each factor to zero yields:

x - 6 0 implies x 6

and

x 1 0 implies x -1

Step 6: Validating Solutions

It's important to validate the solutions to ensure they don't introduce any undefined forms in the original equation. We need to check if x -1 introduces a division by zero.

Substituting x -1 into the original equation:

frac{3}{-1} - frac{2}{-1} frac{5}{(-1)^2 - 1}

This becomes:

-3 2 frac{5}{1 - 1}

The right side is undefined because it involves division by zero.

Therefore, x -1 is not a valid solution.

Concluding, x 6 is the only valid solution.

Conclusion:

We have successfully solved the given equation and verified that the only valid solution is x 6.