To find the difference between the two rational expressions [tex]\(\frac{g - 9}{g - 4}\)[/tex] and [tex]\(\frac{-8g - 3}{g - 4}\)[/tex], follow these steps:
1. Identify Like Denominators:
Both fractions have the same denominator [tex]\(g - 4\)[/tex], so we can directly subtract the numerators over the common denominator.
2. Subtract the Numerators:
[tex]\[
\frac{g - 9}{g - 4} - \frac{-8g - 3}{g - 4} = \frac{(g - 9) - (-8g - 3)}{g - 4}
\][/tex]
3. Simplify the Numerator:
Distribute and combine like terms inside the numerator.
[tex]\[
(g - 9) - (-8g - 3) = g - 9 + 8g + 3 = 9g - 6
\][/tex]
4. Form the New Fraction:
Place the simplified numerator over the common denominator.
[tex]\[
\frac{9g - 6}{g - 4}
\][/tex]
5. Check Simplifications:
Look to see if the fraction can be simplified further. Notice [tex]\(9g - 6\)[/tex] and [tex]\(g - 4\)[/tex] have no common factors that can be cancelled out.
Thus, the simplest form of the given expression's difference is:
[tex]\[
\frac{9g - 6}{g - 4}
\][/tex]
Comparing this with the given multiple choices, we find that the correct answer is:
[tex]\(\boxed{\frac{9g - 6}{g - 4}}\)[/tex]
