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  4. The solution set of the inequality 5x + 2 > ( (1/25) )1 /x is

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Q. The solution set of the inequality $ 5^{x + 2} > \left( \frac{1}{25} \right)^{1 /x}$ is

Linear Inequalities

Solution:

We have
$ 5^{x + 2 } > \bigg( \frac{1}{25} \bigg)^{1 /x} \Rightarrow 5^{x + 2} > 5^{- \frac{2}{x}} \Rightarrow \, x + 2 > - \frac{2}{x}$
$ [ \because \, base \, 5 > 1]$
$\Rightarrow x + 2 + \frac{2}{x} > 0 \Rightarrow \frac{x^{2} + 2x + 2}{x}> 0 \Rightarrow \frac{1}{x} > 0 $
$[\because \: x^2 + 2x + 2 > 0 \, \forall \, x \, \in R ]$
$\Rightarrow \, x > 0 \: or \: x \, \in ( 0, \infty)$