Question

$frac{3- 4^{x} }{2- 2^{x} } geq 1,5$

Answer 1

Пусть $y=2^x$. Тогда неравенство верно при
$frac{3-y^2}{2-y} geq frac{3}{2} Rightarrow frac{2(3-y^2)-3(2-y)}{2(2-y)} geq 0$
Тогда
$frac{-2y^2+3y}{4-2y} geq 0 Rightarrow left { {{y(3-2y) geq 0} atop {4-2y textgreater 0}} right. or left { {{y(3-2y) leq 0} atop {4-2y textless 0}} right.$

 $left { {{y(3-2y) geq 0} atop {4-2y textgreater 0}} right. Rightarrow y in (-infty,2) cap ([0,+infty)cap(-infty,frac{3}{2}] cup(-infty,0]cap[frac{3}{2},infty))$
$Rightarrow y in [0,frac{3}{2}]$
$left { {{y(3-2y) leq 0} atop {4-2y textless 0}} right. Rightarrow y in (2,+infty) cap ((-infty,0] cup [frac{3}{2},+infty)) Rightarrow y in (2,infty)$
Тогда $y in [0,frac{3}{2}] cup (2,infty) Rightarrow x in (-infty, frac{ln(1.5)}{ln(2)}] cup (1, infty)$