To simplify the given expressions, let's tackle them one by one in detail.
Expression 1:
[tex]\[
\frac{11}{3(x-5)} - \frac{x+1}{3x}
\][/tex]
1. Combine the fractions: The common denominator for the fractions is [tex]\(3x(x - 5)\)[/tex].
- The first fraction, [tex]\(\frac{11}{3(x-5)}\)[/tex], needs to be multiplied by [tex]\(x/x\)[/tex] to get the common denominator:
[tex]\[
\frac{11x}{3x(x-5)}
\][/tex]
- The second fraction, [tex]\(\frac{x+1}{3x}\)[/tex], needs to be multiplied by [tex]\((x-5)/(x-5)\)[/tex] to get the common denominator:
[tex]\[
\frac{(x+1)(x-5)}{3x(x-5)} = \frac{x^2 - 4x - 5}{3x(x-5)}
\][/tex]
2. Subtract the second fraction from the first:
[tex]\[
\frac{11x}{3x(x-5)} - \frac{x^2 - 4x - 5}{3x(x-5)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{11x - (x^2 - 4x - 5)}{3x(x-5)}
\][/tex]
3. Distribute and combine like terms in the numerator:
[tex]\[
11x - x^2 + 4x + 5 = -x^2 + 15x + 5
\][/tex]
So, the simplified form of the first expression is:
[tex]\[
\frac{-x^2 + 15x + 5}{3x(x-5)}
\][/tex]
Expression 2:
[tex]\[
-x^2
\][/tex]
This expression is already simplified.
Thus, the simplified forms of the given expressions are:
[tex]\[
\frac{-x^2 + 15x + 5}{3x(x-5)} \quad \text{and} \quad -x^2
\][/tex]
