To add the rational expressions
[tex]\[
-\frac{9}{x-2} + \frac{2 - 3x}{x}
\][/tex]
we need to find a common denominator. The denominators are [tex]\(x - 2\)[/tex] and [tex]\(x\)[/tex]. The common denominator will be the product of these two, which is
[tex]\[
x(x - 2)
\][/tex]
We will rewrite each fraction with this common denominator.
First, rewrite [tex]\(-\frac{9}{x-2}\)[/tex]:
[tex]\[
-\frac{9}{x-2} = -\frac{9 \cdot x}{(x-2) \cdot x} = -\frac{9x}{x(x-2)}
\][/tex]
Next, rewrite [tex]\(\frac{2 - 3x}{x}\)[/tex]:
[tex]\[
\frac{2 - 3x}{x} = \frac{(2 - 3x) \cdot (x-2)}{x \cdot (x-2)} = \frac{(2 - 3x)(x-2)}{x(x-2)}
\][/tex]
Now, expand the numerator of [tex]\(\frac{(2 - 3x)(x-2)}{x(x-2)}\)[/tex]:
[tex]\[
(2 - 3x)(x - 2) = 2x - 4 - 3x^2 + 6x = -3x^2 + 8x - 4
\][/tex]
So,
[tex]\[
\frac{2 - 3x}{x} = \frac{-3x^2 + 8x - 4}{x(x-2)}
\][/tex]
Now we add the two fractions:
[tex]\[
-\frac{9x}{x(x-2)} + \frac{-3x^2 + 8x - 4}{x(x-2)} = \frac{-9x + (-3x^2 + 8x - 4)}{x(x-2)}
\][/tex]
Combine the terms in the numerator:
[tex]\[
-9x + (-3x^2 + 8x - 4) = -3x^2 + 8x - 4 - 9x = -3x^2 - x - 4
\][/tex]
Thus, the combined fraction becomes:
[tex]\[
\frac{-3x^2 - x - 4}{x(x-2)}
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
\boxed{\frac{-3x^2 - x - 4}{x(x-2)}}
\][/tex]
