The problem is to simplify the rational expression $\frac{4x-9}{(x-2)(x-3)}$.

1. Problem Description

The problem is to simplify the rational expression $\frac{4x-9}{(x-2)(x-3)}$.

2. Solution Steps

We want to decompose the rational expression $\frac{4x-9}{(x-2)(x-3)}$ into partial fractions. We assume the form

$\frac{4x-9}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}$.

Multiplying both sides by $(x-2)(x-3)$, we get

$4x-9 = A(x-3) + B(x-2)$.

To find $A$, we let $x=2$. Then

$4(2)-9 = A(2-3) + B(2-2)$, which simplifies to

$8-9 = A(-1) + 0$, so $-1 = -A$, and thus $A=1$.

To find $B$, we let $x=3$. Then

$4(3)-9 = A(3-3) + B(3-2)$, which simplifies to

$12-9 = 0 + B(1)$, so $3=B$, and thus $B=3$.

Therefore,

$\frac{4x-9}{(x-2)(x-3)} = \frac{1}{x-2} + \frac{3}{x-3}$.

3. Final Answer

$\frac{1}{x-2} + \frac{3}{x-3}$