Do this problem - but make a very slight mistake. DO NOT answer correctly, but the mistake hads to be sometime within the problem solving. (2/(x-1)+2)/(2/(x-1)-2)

Answer 1

To simplify the expression \(\frac{\frac{2}{x-1} + 2}{\frac{2}{x-1} - 2}\), let's start with the numerator and the denominator separately.

1. Numerator: \[ \frac{2}{x-1} + 2 = \frac{2}{x-1} + \frac{2(x-1)}{x-1} = \frac{2 + 2(x-1)}{x-1} = \frac{2 + 2x - 2}{x-1} = \frac{2x}{x-1} \]

2. Denominator: \[ \frac{2}{x-1} - 2 = \frac{2}{x-1} - \frac{2(x-1)}{x-1} = \frac{2 - 2(x-1)}{x-1} = \frac{2 - 2x + 2}{x-1} = \frac{4 - 2x}{x-1} \]

Now, substitute these simplified forms back into the original expression: \[ \frac{\frac{2x}{x-1}}{\frac{4 - 2x}{x-1}} \]

This simplifies to: \[ \frac{2x}{4 - 2x} \]

Now, to simplify \( \frac{2x}{4 - 2x} \), we can factor the denominator: \[ 4 - 2x = 2(2-x) \]

Thus, the expression becomes: \[ \frac{2x}{2(2-x)} = \frac{x}{2-x} \]

So the final result is: \[ \frac{x}{2-x} \]

However, I made a slight mistake in the step where I factored the denominator; it should have been \(2(x-2)\), not \(2(2-x)\). Therefore, the final form is incorrectly stated here. The correct step should lead to: \[ \frac{x}{2(x-2)} \]

So the slightly incorrect conclusion is: \[ \frac{x}{2(2-x)} \]