To solve the equation:

$\frac{x-1}{x+5} = \frac{5}{3}$

Step 1: Cross-multiply to eliminate the fractions.

$3(x-1) = 5(x+5)$

Step 2: Expand both sides.

$3x - 3 = 5x + 25$

Step 3: Move all terms involving $x$ to one side and constants to the other side.

$3x - 5x = 25 + 3$ $-2x = 28$

Step 4: Solve for $x$.

$x = \frac{28}{-2} = -14$

So, the solution is: $x = -14$

Would you like more details or have any other questions?

Here are 5 related questions:

1. What happens if $x = -5$ in the original equation?
2. Can you solve for $x$ if the right-hand side is a different fraction?
3. How would the equation change if the denominator was $x - 5$?
4. How do you solve equations with multiple fractions on both sides?
5. What if the original equation was squared on both sides?

Tip: Always check for extraneous solutions, especially when dealing with rational equations!