Express \( \frac{3 x-2}{x^{2}-3 x+2}-\frac{3 x}{x^{2}-2 x} \) as a single fraction in its simplest form.

To combine the fractions \( \frac{3x-2}{x^{2}-3x+2} - \frac{3x}{x^{2}-2x} \), we first need to factor the denominators: 1. The first denominator \( x^{2} - 3x + 2 \) factors as: \[ x^{2}-3x+2 = (x-1)(x-2) \] 2. The second denominator \( x^{2}-2x \) can be factored out as: \[ x^{2}-2x = x(x-2) \] Now the expression can be rewritten as: \[ \frac{3x-2}{(x-1)(x-2)} - \frac{3x}{x(x-2)} \] Next, we need to find a common denominator. The least common denominator (LCD) of the two fractions is \( x(x-1)(x-2) \). We will rewrite each fraction with this common denominator: For the first fraction: \[ \frac{3x-2}{(x-1)(x-2)} \cdot \frac{x}{x} = \frac{x(3x-2)}{x(x-1)(x-2)} \] For the second fraction: \[ \frac{3x}{x(x-2)} \cdot \frac{(x-1)}{(x-1)} = \frac{3x(x-1)}{x(x-1)(x-2)} \] Now we can combine the fractions: \[ \frac{x(3x-2) - 3x(x-1)}{x(x-1)(x-2)} \] Next, we will simplify the numerator: \[ x(3x-2) - 3x(x-1) = 3x^2 - 2x - (3x^2 - 3x) = 3x^2 - 2x - 3x^2 + 3x = (3x - 2x) = x \] Thus, the expression simplifies to: \[ \frac{x}{x(x-1)(x-2)} \] We can reduce this fraction by canceling \( x \) in the numerator and denominator (assuming \( x \neq 0 \)): \[ \frac{1}{(x-1)(x-2)} \] Hence, the final result is: \[ \boxed{\frac{1}{(x-1)(x-2)}} \]