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.