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Combination

Solve the fol...

Question

Solve the following inequalities
$\frac{x + 1}{x + 5} + \frac{x}{x - 1} < \frac{2 x^{2} + 5 x}{x^{2} + 4 x - 5}$

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Solution

$\frac{x + 1}{x + 5} + \frac{x}{x - 1} < \frac{2 x^{2} + 5 x}{x^{2} + 4 x - 5}$
$\Rightarrow \frac{x^{2} - 1^{2} + x (x + 5)}{(x + 5) (x - 1)} < \frac{2 x^{2} + 5 x}{x^{2} + 4 x - 5}$
$\Rightarrow \frac{x^{2} - 1 + x^{2} + 5 x}{x^{2} + 4 x - 5} < \frac{2 x^{2} - 5 x}{x^{2} + 4 x - 5}$
$\Rightarrow 2 x^{2} + 5 x - 1 < 2 x^{2} + 5 x$
$\Rightarrow - 1 < 0$ which is true $\forall x ϵ R$ .

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