We are given the equation $x = \frac{2U - 3}{3U + 2}$ and we want to make $U$ the subject of the formula.

1. Problem Description

We are given the equation $x = \frac{2U - 3}{3U + 2}$ and we want to make $U$ the subject of the formula.

2. Solution Steps

First, multiply both sides by $3U + 2$:

$x(3U + 2) = 2U - 3$

$3Ux + 2x = 2U - 3$

Rearrange the equation to group the terms involving $U$:

$3Ux - 2U = -3 - 2x$

Factor out $U$ from the left side:

$U(3x - 2) = -2x - 3$

Divide both sides by $(3x - 2)$ to isolate $U$:

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

Multiply the numerator and denominator by -1:

$U = \frac{2x + 3}{2 - 3x} * \frac{-1}{-1} = \frac{2x + 3}{-3x + 2} = \frac{-(2x+3)}{-(3x-2)}$

$U = \frac{-(2x+3)}{-(3x-2)} = \frac{2x+3}{-(3x-2)} = \frac{-2x-3}{2-3x}$

$U = \frac{-(2x+3)}{3x-2} = \frac{-2x-3}{3x-2}$

Therefore, $U = \frac{2x + 3}{2 - 3x}$.

3. Final Answer

C. $U = \frac{2x + 3}{2 - 3x}$