To determine the simplest form of the given expression [tex]\(\frac{5(x-3)-3(2 x+4)}{9}\)[/tex], we start by simplifying the numerator.
### Step-by-Step Simplification:
1. Expand the terms in the numerator:
[tex]\[
5(x - 3) - 3(2x + 4)
\][/tex]
Distribute the coefficients:
[tex]\[
= 5x - 15 - 6x - 12
\][/tex]
2. Combine like terms:
[tex]\[
5x - 6x - 15 - 12
\][/tex]
Combine the [tex]\(x\)[/tex]-terms and the constant terms:
[tex]\[
= -x - 27
\][/tex]
3. Substitute the simplified numerator back into the fraction:
[tex]\[
\frac{-x - 27}{9}
\][/tex]
4. Ensure this fraction is simplified to its simplest form:
The expression [tex]\(\frac{-x - 27}{9}\)[/tex] can be written as [tex]\(\frac{-x}{9} - 3\)[/tex], but the form [tex]\(\frac{-x - 27}{9}\)[/tex] is already one of the choices given.
Now let's compare this to the given options to find the match:
1. Option A: [tex]\(\frac{-x - 27}{9}\)[/tex]
2. Option B: [tex]\(\frac{11x - 27}{9}\)[/tex]
3. Option C: [tex]\(-x - 3\)[/tex]
4. Option D: [tex]\(\frac{-x - 3}{9}\)[/tex]
The simplest form of [tex]\(\frac{5(x-3)-3(2 x+4)}{9}\)[/tex] matches exactly with Option A:
[tex]\[
\boxed{\mathrm{A} \,\, \frac{-x - 27}{9}}
\][/tex]
