Question

Resolva a Inequação Exponencial:
                         
                                              $4 ^{x} + 4 > 5 . 2 ^{x}$

Answer 1

E aí mano,

use as propriedades da exponenciação:

$4^x+4>5\cdot2^x\\ (2^2)^x+4-5\cdot2^x>0\\ (2^x)^2-5\cdot2^x+4>0\\\\ 2^x=k\\\\ k^2-5k+4>0\\\\ \Delta=(-5)^2-4\cdot1\cdot4\\ \Delta=25-16\\ \Delta=9\\\\ k= \dfrac{-(-5)\pm \sqrt{9} }{2\cdot1}= \dfrac{5\pm3}{2}\begin{cases}k'= \dfrac{5-3}{2}= \dfrac{2}{2}=1\\\\ k''= \dfrac{5+3}{2}= \dfrac{8}{2}=4 \end{cases}$

$2^x>k\\2^x>1\\2^x>2^0\\ \not2^x>\not2^0\\x>0\\\\2^x>2\\2^x>2^2\\\not2^x>\not2^2\\x>2\\\\\\ \huge\boxed{\boxed{S=\{x\in\mathbb{R}~|~0<x>2}}$

Tenha ótimos estudos =))