Explanation

(a)
The given inequation is
$4^{- x + 0.5} - {7 \bullet 2}^{- x} < 4,x \in R$
Let $2^{- x} = t$
$\therefore 2t^{2} - 7t < 4$
$\Rightarrow 2t^{2} - 7t - 4 < 0$
$\Rightarrow (2t + 1)(t - 4) < 0$
$\Rightarrow - \frac{1}{2} < t < 4$
$\Rightarrow 0 < t < 4\ \ \ \ (\because t = 2^{- x} > 0)$
$\Rightarrow 0 < 2^{- x} < 2^{2}$
As $2^{x}\$is an increasing function $–x < 2\$or$\ x > - 2$
$\therefore x = ( - 2,\infty)$