Question

Se considera expresia E(x) = $\frac{x}{x^{2} + x } - ( \frac{x}{x-1} - \frac{x}{x+1} ) : \frac{2x}{x-1}$ , unde x este numar real, x ≠ -1, x ≠ 0 si x ≠ 1. Aratati ca E(x)=0, pentru orice numar real, x ≠ -1, x ≠ 0 si x ≠ 1.

Answer 1

$\frac{x}{x(x + 1)} - ( \frac{x(x + 1)}{ {x}^{2} - 1 } - \frac{x(x - 1)}{ {x}^{2} - 1} ) \times \frac{x - 1}{2x } \\ \frac{1}{x + 1} - \frac{ {x}^{2} + x - {x}^{2} + x}{ {x}^{2} - 1 } \times \frac{x - 1}{2x} \\ \frac{1}{x + 1} - \frac{2x}{ {x}^{2} - 1 } \times \frac{x - 1}{2x } \\ \frac{1}{x + 1} - \frac{1}{x + 1} = 0$