We need to solve the equation $\frac{x-6}{x+2} = \frac{x-4}{x+1}$ for $x$.

1. Problem Description

We need to solve the equation $\frac{x-6}{x+2} = \frac{x-4}{x+1}$ for $x$.

2. Solution Steps

To solve the equation $\frac{x-6}{x+2} = \frac{x-4}{x+1}$, we can cross-multiply:

$(x-6)(x+1) = (x-4)(x+2)$

Expanding both sides, we get:

$x^2 + x - 6x - 6 = x^2 + 2x - 4x - 8$

$x^2 - 5x - 6 = x^2 - 2x - 8$

Subtracting $x^2$ from both sides gives:

$-5x - 6 = -2x - 8$

Adding $5x$ to both sides gives:

$-6 = 3x - 8$

Adding $8$ to both sides gives:

$2 = 3x$

Dividing both sides by $3$ gives:

$x = \frac{2}{3}$

3. Final Answer

$x = \frac{2}{3}$