Question

Solve the following inequality:
$2^x+2^{|x|}⩾2\sqrt{2}$

Answer 1

$2^x+2^{\left|x\right|}\ge 2^{\frac{3}{2}}$

$whenx>0$

$2\times 2^x\ge 2^{\frac{3}{2}}$

$2^{x+1}\ge 2^{\frac{3}{2}}$

$x+1\ge \frac{3}{2}$

$x\ge \frac{1}{2},x\epsilon [\frac{1}{2},\infty )-\left(i\right)$

$whenx<0$

$2^x+2^{-x}\ge 2^{\frac{3}{2}}$

$2^x+\frac{1}{2^x}\ge 2^{\frac{3}{2}}$

$\frac{{{(2}^x)}^2+1}{2^x}\ge 2^{\frac{3}{2}}$

$let{2}^x=t(t>0)$