Question

Solve the following inequalities.
$2^{x+2}-2^{x+1}+2^{x-1}-2^{x-2}⩽9$

Answer 1

Solve the following Inequality

$1)2^{x+2}-2^{x+1}+2^{x-1}-2^{x-2}\le 9$

Sol From the formula $a^{m+n}=a^m.a^n$

$\Rightarrow 2^{x+2}=2^x{.2}^2$ Similarly from the given equation we

have

$\Rightarrow 2^x{.2}^2-2^x{.2}^1+2^x{.2}^{-1}-2^x{.2}^{-2}\le 9$

$\Rightarrow 2^x[2^2-2^1+\frac{1}{2}-\frac{1}{4}]\le 9$ $[\because a^{-m}=\frac{1}{a^m}]$

$\Rightarrow 2^x[4-2+\frac{1}{2}-\frac{1}{4}]\le 9$

$\Rightarrow 2^x[2+\frac{1}{4}]\le 9$ $[\because \frac{1}{2}-\frac{1}{4}=\frac{2-1}{4}=\frac{1}{4}]$

$\Rightarrow 2^x\left[\frac{9}{4}\right]\le 9$

$\Rightarrow 2^x\le 2^2$

$\Rightarrow x=2$ $\because a^m=a^n\Rightarrow m=n$