https://file.aiquickdraw.com/mathbot/file/1726190390792yquqsaca.jpg

Let's proceed by analyzing the mathematical problem directly from the image. The problem asks to solve for \( x \) when \( f(x) = g(x) \), given the functions:

\[
f(x) = \frac{x}{x-4} - \frac{3}{x-1}
\]
and
\[
g(x) = \frac{-15x - 12}{x^2 - 5x + 4}
\]

We need to solve \( f(x) = g(x) \) by equating the two expressions.

### Steps:
1. Find a common denominator for \( f(x) \).
2. Simplify both sides.
3. Cross-multiply and solve for \( x \).

Let me now work through the steps and find the solution.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you would like, I can guide you through solving this problem manually. Would you like the steps or explanation to help you solve it on your own?

Here are some related questions to deepen your understanding:
1. How do you solve rational equations in general?
2. What are the conditions for a solution to be valid in a rational equation?
3. How can you factor the quadratic expression \(x^2 - 5x + 4\)?
4. What happens if the denominator in either function equals zero?
5. Can you always cross-multiply when solving rational equations?

**Tip:** Always check for values that make the denominator zero, as they are not valid solutions.

Solve for x when f(x) = g(x), given f(x) = \frac{x}{x-4} - \frac{3}{x-1} and g(x) = \frac{-15x - 12}{x^2 - 5x + 4}.

Learn how to solve the rational equation f(x) = g(x), where f(x) = \frac{x}{x-4} - \frac{3}{x-1} and g(x) = \frac{-15x - 12}{x^2 - 5x + 4}. This problem involves key concepts such as factoring quadratics, finding common denominators, and cross-multiplication, suitable for high school students.

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