To find the simplest form of the expression [tex]\(\frac{5(x-3) - 3(2x+4)}{9}\)[/tex], let's follow a step-by-step approach to simplify the numerator first and then divide by 9.
1. Distribute the constants inside the parentheses:
[tex]\[
5(x-3) - 3(2x+4)
\][/tex]
Distribute 5 to both terms in [tex]\( (x-3) \)[/tex]:
[tex]\[
5 \cdot x - 5 \cdot 3 = 5x - 15
\][/tex]
Distribute -3 to both terms in [tex]\( (2x+4) \)[/tex]:
[tex]\[
-3 \cdot 2x - 3 \cdot 4 = -6x - 12
\][/tex]
2. Combine the results from distribution:
[tex]\[
5x - 15 - 6x - 12
\][/tex]
3. Combine the like terms:
Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
5x - 6x = -x
\][/tex]
Combine the constant terms:
[tex]\[
-15 - 12 = -27
\][/tex]
So the simplified numerator is:
[tex]\[
-x - 27
\][/tex]
4. Express the entire fraction:
Now we can write the expression as:
[tex]\[
\frac{-x - 27}{9}
\][/tex]
5. Compare with the provided options:
- A. [tex]\(\frac{11x - 27}{9}\)[/tex]
- B. [tex]\(\frac{-x - 27}{9}\)[/tex]
- C. [tex]\(\frac{-x - 3}{9}\)[/tex]
- D. [tex]\(-x - 3\)[/tex]
Clearly, the simplified form matches option B:
[tex]\[
\boxed{\frac{-x - 27}{9}}
\][/tex]
So, the simplest form of the given expression is [tex]\(\frac{-x - 27}{9}\)[/tex] which matches option B.
