The problem requires simplifying the given rational expression: $\frac{4x-9}{(x-2)(x-3)}$. The instructions only say "express each of the following," so I'll assume we are to decompose the rational expression into partial fractions. We can express the given rational expression in the form: $\frac{4x-9}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}$ where $A$ and $B$ are constants.

1. Problem Description

The problem requires simplifying the given rational expression: $\frac{4x-9}{(x-2)(x-3)}$. The instructions only say "express each of the following," so I'll assume we are to decompose the rational expression into partial fractions. We can express the given rational expression in the form:

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

where $A$ and $B$ are constants.

2. Solution Steps

To find the values of $A$ and $B$, we multiply both sides of the equation by $(x-2)(x-3)$:

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

We can solve for $A$ and $B$ by substituting suitable values for $x$.

Let $x = 2$:

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

$8 - 9 = A(-1) + B(0)$

$-1 = -A$

$A = 1$

Let $x = 3$:

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

$12 - 9 = A(0) + B(1)$

$3 = B$

$B = 3$

Therefore, we have:

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

3. Final Answer

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