Question

Inequação Exppnencial
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Só as q estão marcadas com x

Answer 1

$59)~3.2 ^{x+2} - 2^{2x} > 32$
$3.2^x.2^2- (2^x)^2 > 32$
$3.2^x.4-(2^x) ^2 -32\ \textgreater \ 0$
         $\large\boxed{2^x =y}$
$3.y.4-y^2-32\ \textgreater \ 0$
$12y-y^2 -32\ \textgreater \ 0$
$-y^2+12y-32\ \textgreater \ 0$

$Equa{\c~c} \~ao ~~ do~~ 2\°~grau , procurando~~ as~~rai\´zes ~e ~estudando ~~ o~~ sinal$  $temos:$
$y\ \textgreater \ 4~~~~e~~~~y\ \textless \ 8$

$substituindo~~ na~~ troca ~de~~ varia\´vel:$

$2^x\ \ \textgreater \ 4$
$\not2^x\ \textgreater \ \not2^2$
$x\ \textgreater \ 2$

$2^2\ \textless \ 8$
$\not2^x \ \textless \ \not2^3$
$x\ \textless \ 3$

$\large\boxed{S = \{ x ~e~R~|~ 2\ \textless \ x\ \textless \ 3\}~~ ou~~ S = ~ ]2,~3[}$


$60)$ $2^x + 2 ^{x+1} \leq 96$
$2^x+ 2^x.2 \leq 96$
$\boxed{2^x = y}$
$y+2y \leq 96$
$3y \leq 96$
$y= \frac{96}{3}$
$y \leq 32$

$2^x \leq 32$
$\not2^x \leq \not2^5$
$\large\boxed{x \leq 5}$