The given expression is:

$\frac{6x - \frac{6}{5} - x}{\frac{x - 1}{x - 5}}$

Now, let's simplify it step by step.

Step 1: Simplify the numerator

The numerator is $6x - \frac{6}{5} - x$. Combine like terms:

$6x - x = 5x$

So the numerator becomes:

$5x - \frac{6}{5}$

Step 2: Keep the denominator as it is

The denominator is $\frac{x - 1}{x - 5}$.

Step 3: Rewrite the expression as multiplication

Instead of division, we can multiply by the reciprocal of the denominator:

$\frac{5x - \frac{6}{5}}{1} \times \frac{x - 5}{x - 1}$

Step 4: Multiply both expressions

Now we need to simplify the multiplication, but first, let's combine the terms in the numerator:

$5x - \frac{6}{5} = \frac{25x}{5} - \frac{6}{5} = \frac{25x - 6}{5}$

Now the expression becomes:

$\frac{\frac{25x - 6}{5}}{1} \times \frac{x - 5}{x - 1}$

This simplifies to:

$\frac{(25x - 6)(x - 5)}{5(x - 1)}$

Final Answer:

$\frac{(25x - 6)(x - 5)}{5(x - 1)}$

Let me know if you'd like any further clarification or additional details!

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Here are 5 related questions you might find helpful:

1. How do you factor complex numerators in rational expressions?
2. What is the process for multiplying fractions involving variables?
3. How can you identify common factors to simplify rational expressions?
4. How does simplifying expressions change when using negative numbers?
5. Can you apply factoring to solve rational equations?

Tip: When simplifying fractions, always look for opportunities to factor out common terms to make the expression easier to work with.