Question

Determine all real vales of x for which the expression ${\{\frac{2}{x^2-x+1}-\frac{1}{x+1}-\frac{2x+1}{x^3+1}\}}^{1/2}$ takes real values.

Answer 1

$y=\sqrt{E}$ is real if $E\ge 0$
Now, $x^2-x+1$ is a quadratic expression whose discriminent $\Delta =-ve$
and hence its sign is same as that of $1st$ term i.e., $+ve$
Hence, we must have $\frac{-x^2+x}{x+1}>0$
Multiplying by ${}^{\prime }{-}^{\prime }$ sign and changing the sign of inequality, we must have $\frac{x(x-1)(x+1)}{{(x+1)}^2}\le 0$
The change points are $-1,0,1$ in ascending order.
Thus, the inequality is satisfied for $x\in (-\infty ,-1]\cup [1,0)$