Questions: Find all complex solutions for the following equation by hand. 3/(x+5) - 6/(x-5) = 6x/(25-x^2) Select the correct choice below and, if necessary, fill in the answer box to complete your answer. A. The solution set is 3. (Type an integer or a simplified fraction.) B. The solution set is the set of real numbers. C. The solution set is an empty set.

Solution Steps

The given equation is: \[ \frac{3}{x+5} - \frac{6}{x-5} = \frac{6x}{25 - x^2} \] The least common denominator (LCD) of the fractions is \( (x+5)(x-5) \).

Multiply each term by \( (x+5)(x-5) \) to eliminate the denominators: \[ (x+5)(x-5) \left( \frac{3}{x+5} - \frac{6}{x-5} \right) = (x+5)(x-5) \left( \frac{6x}{25 - x^2} \right) \] Simplifying, we get: \[ 3(x-5) - 6(x+5) = 6x \]

Expand and simplify the equation: \[ 3x - 15 - 6x - 30 = 6x \] Combine like terms: \[ -3x - 45 = 6x \] Add \(3x\) to both sides: \[ -45 = 9x \] Divide by 9: \[ x = -5 \]

We need to check if \( x = -5 \) is a valid solution by substituting it back into the original equation. However, substituting \( x = -5 \) makes the denominator zero in the original equation, which is not allowed. Therefore, \( x = -5 \) is an extraneous solution.

Revisiting the simplified equation: \[ 3(x-5) - 6(x+5) = 6x \] We missed a valid solution. Let's solve it again: \[ 3(x-5) - 6(x+5) = 6x \] Simplify: \[ 3x - 15 - 6x - 30 = 6x \] Combine like terms: \[ -3x - 45 = 6x \] Add \(3x\) to both sides: \[ -45 = 9x \] Divide by 9: \[ x = -5 \]

Since \( x = -5 \) makes the denominator zero, it is not a valid solution. Therefore, we need to check for other potential solutions.

Final Answer