Exponential inequality!

Clearly, the left -hand side of the above inequality can be transformed into a quadratic with a simple substitution, $y=3^x$ Therfore,

$4y^2 - 19y + 12\leq 0$ => $(y-4)(4y-3)\leq 0$

Now, plotting the $wavy-curve$ for the above inequality we find that this inequality exists only when $y\in$ [ $3/4$ , $4$ ]

Replacing $y=3^x$ again, we get:

$3^x\leq4$ $and$ $3^x\geq3/4$

Therefore,

$x\leq log_{3}{4}$ $and$ $x\geq log_{3}{3/4}$

HENCE,

$x\in$ [ $log _{ 3 }{ 3/4 }$ , $log _{ 3}{4}$ ] :)