Question

Solve the following inequalities.
$2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}$

Answer 1

$2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}$

$\Rightarrow 2^x{.2}^2-2^x{.2}^3-2^x{.2}^4>5^x{.5}^1-5^{x+2}$

$[\because a^{m+n}=a^m.a^n]$

$\Rightarrow 2^x[4-8-16]>5^x[5-25]$

$\Rightarrow 2^x[-20]>5^x[-20]$

$\Rightarrow 1>{\left(\frac{5}{2}\right)}^x$

$\Rightarrow$ Apply log on both sides

$\Rightarrow lo{g}_1>log{\left(\frac{5}{2}\right)}^x$

$\Rightarrow lo{g}_1>xlog\left(\frac{5}{2}\right)$

$\Rightarrow 0>xlog\left(\frac{5}{2}\right)$

$\Rightarrow x<0$