7. \( \frac{x^{2}-1}{x^{2}-x-2}-\frac{x-1}{x-2} \)

To simplify the expression \( \frac{x^{2}-1}{x^{2}-x-2}-\frac{x-1}{x-2} \), we can first factor the numerators and denominators. 1. **Factor the first term**: \[ x^2 - 1 = (x - 1)(x + 1) \] The denominator \( x^2 - x - 2 \) can be factored as: \[ x^2 - x - 2 = (x - 2)(x + 1) \] So the first term becomes: \[ \frac{(x-1)(x+1)}{(x-2)(x+1)} \] Provided \( x \neq -1 \) (to avoid division by zero). 2. **For the second term**, it is already simplified to: \[ \frac{x-1}{x-2} \] Now we can rewrite the entire expression: \[ \frac{(x-1)(x+1)}{(x-2)(x+1)} - \frac{x-1}{x-2} \] 3. **Find a common denominator**, which will be \((x-2)(x+1)\): \[ \frac{(x-1)(x+1) - (x-1)(x+1)}{(x-2)(x+1)} \] Since both fractions have the same numerator, we can simplify it to: \[ \frac{(x-1)(x+1) - (x-1)(x+1)}{(x-2)(x+1)} = \frac{0}{(x-2)(x+1)} = 0 \] Thus, the simplified expression is: \[ 0 \quad \text{(for all } x \neq 2, -1 \text{, where the original expression is defined).} \]