Question

The set of values of ${}^{\prime }{x}^{\prime }$ satisfying the inequation $(2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2})$ is

• A. $x\in (0,\infty )$
• B. $x\in [0,\infty )$
• C. $x\in (-\infty ,\infty )$
• D. $x\in (-\infty ,0]$

Answer 1

The correct option is A $x\in (0,\infty )$

Given,

$2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}$
$\Rightarrow 2^{x+2}(1-2-2^2)>5^{x+2}(\frac{1}{5}-1)$
$\Rightarrow \frac{2^{x+2}}{5^{x+2}}<\frac{4}{25}$
$\Rightarrow {\left(\frac{2}{5}\right)}^{x+2}<{\left(\frac{2}{5}\right)}^2$
$\Rightarrow x+2>2$
$\Rightarrow x>0$