To simplify the given expression:

\[
\frac{6x}{(5x - 7)(x + 1)} - \frac{1}{5x - 7}
\]

### Step 1: Find a common denominator.

The first term already has the denominator \((5x - 7)(x + 1)\), while the second term has only \(5x - 7\) in the denominator. To subtract these fractions, we need to bring the second term to have the same denominator by multiplying its numerator and denominator by \(x + 1\).

\[
\frac{1}{5x - 7} \cdot \frac{x + 1}{x + 1} = \frac{x + 1}{(5x - 7)(x + 1)}
\]

Now the expression becomes:

\[
\frac{6x}{(5x - 7)(x + 1)} - \frac{x + 1}{(5x - 7)(x + 1)}
\]

### Step 2: Combine the fractions.

Now that both terms have the same denominator, we can combine the numerators:

\[
\frac{6x - (x + 1)}{(5x - 7)(x + 1)}
\]

### Step 3: Simplify the numerator.

Simplify the numerator \(6x - (x + 1)\):

\[
6x - x - 1 = 5x - 1
\]

### Final expression:

\[
\frac{5x - 1}{(5x - 7)(x + 1)}
\]

This is the simplified form of the given expression.

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Would you like further details or have any questions about this?

Here are 5 related questions to expand your understanding:

1. How do you simplify fractions with different denominators in general?
2. What are the steps to factor complex denominators?
3. How can you handle expressions with multiple variables in a similar situation?
4. What would happen if one of the terms in the expression were squared?
5. How do you find common denominators for rational expressions?

**Tip:** Always look for common factors or opportunities to factorize terms before attempting to simplify complex rational expressions!

Learn how to simplify the rational expression 6x/(5x - 7)(x + 1) - 1/(5x - 7) with a step-by-step guide. This problem involves finding common denominators, combining fractions, and simplifying algebraic terms. Ideal for students in Grades 9-11.

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