To simplify the given expressions, let's break them down step-by-step.
### Simplifying [tex]\(\frac{11}{3(x-5)} - \frac{x+1}{3x}\)[/tex]
1. Find a common denominator:
For the fractions [tex]\(\frac{11}{3(x-5)}\)[/tex] and [tex]\(\frac{x+1}{3x}\)[/tex], the common denominator is [tex]\(3x(x-5)\)[/tex].
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{11}{3(x-5)} = \frac{11 \cdot x}{3x(x-5)} = \frac{11x}{3x(x-5)}
\][/tex]
[tex]\[
\frac{x+1}{3x} = \frac{(x+1)(x-5)}{3x(x-5)} = \frac{x^2 - 5x + x - 5}{3x(x-5)} = \frac{x^2 - 4x - 5}{3x(x-5)}
\][/tex]
3. Subtract the fractions:
[tex]\[
\frac{11x}{3x(x-5)} - \frac{x^2 - 4x - 5}{3x(x-5)} = \frac{11x - (x^2 - 4x - 5)}{3x(x-5)}
\][/tex]
4. Simplify the numerator:
[tex]\[
11x - (x^2 - 4x - 5) = 11x - x^2 + 4x + 5 = -x^2 + 15x + 5
\][/tex]
5. Write the final simplified expression:
[tex]\[
\frac{-x^2 + 15x + 5}{3x(x-5)}
\][/tex]
### Simplifying [tex]\(\frac{-x^2 + 15x + 5}{3x^2 + [?]x}\)[/tex]
Given the missing term in the denominator, we assume it is zero as it is not specified:
1. Rewrite the denominator:
[tex]\[
3x^2 + [?]x \Rightarrow 3x^2 + 0x = 3x^2
\][/tex]
2. Write the fraction with the simplified denominator:
[tex]\[
\frac{-x^2 + 15x + 5}{3x^2}
\][/tex]
Therefore, the simplifications provide us with the following results:
### Final Simplified Expressions
1. The simplified form of [tex]\(\frac{11}{3(x-5)} - \frac{x+1}{3x}\)[/tex] is:
[tex]\[
\frac{-x^2 + 15x + 5}{3x(x-5)}
\][/tex]
2. The simplified form of [tex]\(\frac{-x^2 + 15x + 5}{3x^2 + [?]x}\)[/tex] is:
[tex]\[
\frac{-x^2 + 15x + 5}{3x^2}
\][/tex]
Both of these results match the expressions simplified above.
