The problem is to solve the equation $\frac{-\frac{7}{4}}{x-2} = \frac{2-x}{7}$ for $x$.

1. Problem Description

The problem is to solve the equation $\frac{-\frac{7}{4}}{x-2} = \frac{2-x}{7}$ for $x$.

2. Solution Steps

First, simplify the left-hand side (LHS) of the equation:

$\frac{-\frac{7}{4}}{x-2} = \frac{-7}{4(x-2)}$

So the equation becomes:

$\frac{-7}{4(x-2)} = \frac{2-x}{7}$

Cross-multiply:

$-7(7) = 4(x-2)(2-x)$

$-49 = 4(x-2)(2-x)$

$-49 = -4(x-2)(x-2)$

$-49 = -4(x-2)^2$

Divide both sides by $-4$:

$\frac{49}{4} = (x-2)^2$

Take the square root of both sides:

$x-2 = \pm \sqrt{\frac{49}{4}}$

$x-2 = \pm \frac{7}{2}$

Solve for $x$ in both cases:

Case 1: $x-2 = \frac{7}{2}$

$x = 2 + \frac{7}{2}$

$x = \frac{4}{2} + \frac{7}{2}$

$x = \frac{11}{2}$

Case 2: $x-2 = -\frac{7}{2}$

$x = 2 - \frac{7}{2}$

$x = \frac{4}{2} - \frac{7}{2}$

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

3. Final Answer

$x = \frac{11}{2}$ or $x = -\frac{3}{2}$