Questions: Solve the following equation by factoring. 7(x-3)/(x-4) + 5/x = -5/(x(x-4)) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is no solution.

Solution Steps

To solve the given equation by factoring, we first need to find a common denominator for the fractions involved. Once we have a common denominator, we can combine the fractions on the left side of the equation. After that, we equate the numerators and solve the resulting polynomial equation by factoring. Finally, we check for any extraneous solutions that might arise due to the original denominators.

We start with the equation:

\[ \frac{7(x-3)}{x-4} + \frac{5}{x} = \frac{-5}{x(x-4)} \]

The common denominator for the fractions is \( x(x-4) \). We rewrite the equation:

\[ \frac{7(x-3)x + 5(x-4)}{x(x-4)} = \frac{-5}{x(x-4)} \]

Equating the numerators gives us:

\[ 7(x-3)x + 5(x-4) = -5 \]

Expanding this, we have:

\[ 7x^2 - 21x + 5x - 20 = -5 \]

This simplifies to:

\[ 7x^2 - 16x - 15 = 0 \]

We can factor the quadratic equation:

\[ (7x + 5)(x - 3) = 0 \]

Setting each factor to zero gives us the solutions:

\[ 7x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{7} \]

\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

We need to ensure that our solutions do not make the original denominators zero. The denominators are \( x \) and \( x - 4 \).

• For \( x = -\frac{5}{7} \): Neither denominator is zero.
• For \( x = 3 \): Neither denominator is zero.

Both solutions are valid.

Final Answer