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The expression to simplify is:

\[
\frac{x - \frac{8}{x + 7}}{1 + \frac{1}{x + 7}}
\]

### Step-by-Step Simplification:

1. **Simplify the numerator**:
   \[
   x - \frac{8}{x + 7}
   \]
   Combine the terms into a single fraction:
   \[
   \frac{x(x + 7) - 8}{x + 7} = \frac{x^2 + 7x - 8}{x + 7}
   \]

2. **Simplify the denominator**:
   \[
   1 + \frac{1}{x + 7}
   \]
   Combine into a single fraction:
   \[
   \frac{(x + 7) + 1}{x + 7} = \frac{x + 8}{x + 7}
   \]

3. **Combine the entire expression**:
   Now, we have:
   \[
   \frac{\frac{x^2 + 7x - 8}{x + 7}}{\frac{x + 8}{x + 7}}
   \]
   Since both the numerator and denominator have a common denominator of \(x + 7\), it cancels out, leaving:
   \[
   \frac{x^2 + 7x - 8}{x + 8}
   \]

4. **Factor the numerator**:
   Factor the quadratic expression \(x^2 + 7x - 8\):
   \[
   x^2 + 7x - 8 = (x - 1)(x + 8)
   \]
   So, the simplified expression becomes:
   \[
   \frac{(x - 1)(x + 8)}{x + 8}
   \]
   For \(x \neq -8\), the \(x + 8\) terms cancel out, leaving:
   \[
   x - 1
   \]

Thus, the simplified expression is:

\[
x - 1 \quad \text{for} \quad x \neq -8
\]

Would you like further clarification or details?

### Related Questions:
1. How do you factor quadratic expressions like \(x^2 + 7x - 8\)?
2. What happens when the denominator of a fraction is zero?
3. Can you explain how to combine terms into a single fraction?
4. How do you identify and cancel out common factors in algebraic expressions?
5. Why is \(x \neq -8\) a restriction in this simplification?

### Tip:
Always check for restrictions in the denominator when simplifying algebraic fractions to avoid division by zero.

Simplify the expression: (x - 8/(x + 7)) / (1 + 1/(x + 7))

Learn how to simplify the complex rational expression (x - 8/(x + 7)) / (1 + 1/(x + 7)) with step-by-step guidance, including factoring quadratics and canceling common factors. Suitable for students in Grades 9-11.

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