Question

the expression is \\(\frac{1 - \frac{1}{1 - x}}{\frac{1}{1 + x}}\\) (assuming the handwritten 1 - ex is a typo and should be 1 - x for a reasonable algebraic problem).

Explanation:

Step1: Simplify the numerator

First, simplify the numerator \(1 - \frac{1}{1 - x}\). Find a common denominator, which is \(1 - x\). So we have \(\frac{(1 - x) - 1}{1 - x}=\frac{1 - x - 1}{1 - x}=\frac{-x}{1 - x}\).

Step2: Simplify the fraction

Now we have the complex fraction \(\frac{\frac{-x}{1 - x}}{\frac{1}{1 + x}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes \(\frac{-x}{1 - x}\times(1 + x)=\frac{-x(1 + x)}{1 - x}=\frac{-x - x^{2}}{1 - x}\) or we can rewrite the denominator as \(-(x - 1)\) to get \(\frac{x^{2}+x}{x - 1}\) (by multiplying numerator and denominator by -1).

Answer:

\(\frac{x^{2}+x}{x - 1}\) (or \(\frac{-x^{2}-x}{1 - x}\))