We need to simplify the following expression:

$\frac{12}{x - 7 + x} \div \frac{3}{x - 7} + 1$

Step-by-step solution:

1. Simplify the numerator and denominator: The expression is a division of two fractions.

   First, combine the terms in the numerator and the denominator.

   $\frac{12}{x - 7 + x} = \frac{12}{2x - 7}$ So the expression becomes: $\frac{12}{2x - 7} \div \frac{3}{x - 7} + 1$

2. Division of fractions: To divide two fractions, multiply the first by the reciprocal of the second: $\frac{12}{2x - 7} \div \frac{3}{x - 7} = \frac{12}{2x - 7} \times \frac{x - 7}{3}$

3. Multiply the fractions: Multiply the numerators and the denominators: $\frac{12 \times (x - 7)}{(2x - 7) \times 3} = \frac{12(x - 7)}{3(2x - 7)}$

4. Simplify the resulting fraction: Simplify the factor of 12 in the numerator and 3 in the denominator: $\frac{12(x - 7)}{3(2x - 7)} = \frac{4(x - 7)}{2x - 7}$

5. Final expression with the added 1: Now, add 1 to this result: $\frac{4(x - 7)}{2x - 7} + 1$

This is the simplified form of the expression.

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Would you like further details on the process? Here are five related questions:

1. How do you add two fractions with different denominators?
2. Can you clarify how reciprocal multiplication works in fraction division?
3. How can you factor the terms in the expression further?
4. Why do we cancel out common factors in the numerator and denominator?
5. How do you approach simplifying complex rational expressions?

Tip: When dividing fractions, always remember to multiply by the reciprocal of the second fraction.