Alright, let's simplify the given rational expressions step-by-step:
We start with the expression: [tex]\[
\frac{11}{3(x-5)}-\frac{x+1}{3x}
\][/tex]
1. Find a common denominator:
The denominators in both fractions need to be the same so we can combine them. The two denominators are [tex]\(3(x-5)\)[/tex] and [tex]\(3x\)[/tex]. The least common multiple (LCM) of these denominators is: [tex]\[
3(x-5)x
\][/tex]
Rewrite each fraction with the common denominator: [tex]\[
\frac{11}{3(x-5)} = \frac{11x}{3x(x-5)}
\][/tex] [tex]\[
\frac{x+1}{3x} = \frac{(x+1)(x-5)}{3x(x-5)}
\][/tex]
2. Expand and combine the fractions:
Now that both fractions have the same common denominator, we can subtract the numerators: [tex]\[
\frac{11x}{3x(x-5)} - \frac{(x+1)(x-5)}{3x(x-5)}
\][/tex]
First, expand the second numerator: [tex]\[
(x+1)(x-5) = x^2 - 5x + x - 5 = x^2 - 4x - 5
\][/tex]
The expression now becomes: [tex]\[
\frac{11x - (x^2 - 4x - 5)}{3x(x-5)}
\][/tex]
