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]
3. Simplify the numerator:
Distribute the negative sign:
[tex]\[
11x - x^2 + 4x + 5 = -x^2 + 15x + 5
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{-x^2 + 15x + 5}{3x(x-5)}
\][/tex]
4. Separate the numerator and the denominator:
After simplification, the numerator is [tex]\(-x^2 + 15x + 5\)[/tex] and the denominator is [tex]\(3x(x-5)\)[/tex].
So, our final step-by-step solution shows that:
[tex]\[
\frac{11}{3(x-5)} - \frac{x+1}{3x} = \frac{-x^2 + 15x + 5}{3x(x-5)}
\][/tex]
The final simplified expression matches the format you asked for:
[tex]\[
\frac{-x^2 + 15x + 5}{x^2 - 5x}
\][/tex]
