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The given expression is:

\[
\frac{\frac{3}{6 - x} + 3}{x - 7}
\]

Let's simplify it step by step.

### Step 1: Combine the terms in the numerator

We need to simplify the numerator first:

\[
\frac{3}{6 - x} + 3
\]

This can be written with a common denominator:

\[
\frac{3}{6 - x} + \frac{3(6 - x)}{6 - x} = \frac{3 + 3(6 - x)}{6 - x}
\]

Simplifying the expression inside the numerator:

\[
3 + 3(6 - x) = 3 + 18 - 3x = 21 - 3x
\]

So the numerator becomes:

\[
\frac{21 - 3x}{6 - x}
\]

### Step 2: Substitute this into the original expression

Now, substitute this back into the original expression:

\[
\frac{\frac{21 - 3x}{6 - x}}{x - 7}
\]

### Step 3: Simplify the overall expression

This expression is a fraction divided by another term. We can simplify it as follows:

\[
\frac{21 - 3x}{(6 - x)(x - 7)}
\]

This is the simplified form of the expression.

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Would you like to see more details or have any questions about the steps?

Here are some related questions:
1. How can you further simplify rational expressions?
2. What is the significance of finding common denominators?
3. How do you handle negative signs in denominators?
4. Can this expression be factored further?
5. What are the conditions for undefined expressions?

**Tip**: Always check the denominator for values that make the expression undefined (in this case, \(x = 6\) and \(x = 7\)).

Simplify the expression (3 / (6 - x) + 3) / (x - 7)

Learn how to simplify the rational expression (3 / (6 - x) + 3) / (x - 7) step by step. This problem involves algebraic concepts such as combining terms and simplifying complex fractions. Suitable for students in Grades 9-12.

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