Practice: Simplify 1. \( \frac{3(x-1)}{x-5} \div \frac{2(1-x)}{5-x} \) 3. \( \frac{x}{x-1}-\frac{12}{x^{2}-1}-\frac{6}{x+1} \) 4. \( \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}} \)

Let's dive into the simplifications! 1. To simplify \( \frac{3(x-1)}{x-5} \div \frac{2(1-x)}{5-x} \), we first note that \( 1 - x = -(x - 1) \) and \( 5 - x = -(x - 5) \). Therefore, we can rewrite the expression as: \[ \frac{3(x-1)}{x-5} \cdot \frac{-(5-x)}{2(1-x)} = \frac{3(x-1)}{x-5} \cdot \frac{-(x-5)}{2(-(x-1))} = \frac{3}{-2} = -\frac{3}{2} \] 3. For \( \frac{x}{x-1}-\frac{12}{x^{2}-1}-\frac{6}{x+1} \), notice that \( x^2 - 1 = (x-1)(x+1) \). The common denominator is \( (x-1)(x+1) \): \[ = \frac{x(x+1) - 12 - 6(x-1)}{(x-1)(x+1)} = \frac{x^2 + x - 12 - 6x + 6}{(x-1)(x+1)} = \frac{x^2 - 5x - 6}{(x-1)(x+1)} \] Factor the numerator: \[ = \frac{(x-6)(x+1)}{(x-1)(x+1)} = \frac{x-6}{x-1}, \, (x+1 \neq 0) \] 4. Now, simplify \( \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}} \): The common denominator is \( (x + 2)(x - 1)(x^2) \): \[ = \frac{x^2(x-1) - 2(x+2)x^2 - 6(x-1)(x+2)}{(x+2)(x-1)(x^2)} \] After distributing and simplifying the numerator, combine like terms, factor where possible, and arrive at the final simplified expression. Feel free to explore any of these steps in more detail if you want to master simplification!