Find the largest integral $x$ which satisfies the following inequality:
$\frac{1}{x + 1} - \frac{2}{x^{2} - x + 1} < \frac{1 - 2 x}{x^{3} + 1}$

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Solution

$\frac{1}{x + 1} - \frac{2}{x^{2} - x + 1} < \frac{1 - 2 x}{x^{3} + 1}$
$\frac{x^{2} - x - 2}{x^{3} + 1} < 0$
$\frac{(x + 1) (x - 2)}{(x + 1) (x^{2} - x + 1)} < 0 ⟹ x - 2 < 0 ⟹ x < 2$

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