Question

Решите неравенство 2^x+ 3 *2^-x<=4

Answer 1

$2^{x}+3\cdot 2^{-x} \leq 4\\\\2^{x}+\frac{3}{2^{x}}-4 \leq 0\\\\t=2^{x}\ \textgreater \ 0\; ,\quad t+\frac{3}{t}-4 \leq 0\\\\ \frac{t^2-4t+3}{t} \leq 0\; ,\\\\ \frac{(t-1)(t-3)}{t} \leq 0\qquad \; \; ---(0)+++[\, 1\, ]---[\, 3\, ]+++\\\\t\in (-\infty ,0)\cup [\, 1,3\, ]\qquad \Rightarrow\; \; \; \left [ {{t\ \textless \ 0} \atop {1 \leq t \leq 3}} \right. \\\\2^{x}\ \textless \ 0\; \; net\; \; reshenij\; ,\; \; tak\; kak\; 2^{x}\ \textgreater \ 0\\\\1 \leq 2^{x} \leq 3\; \; \; \Rightarrow \; \; \; 2^0\leq 2^{x}\leq 2^{log_23}\\\\0 \leq x \leq log_23$

$x\in [\, 0\, ,\; log_23\, ]$