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Question 1

Question 2

Решите задачу:

$1)\; \; (\frac{1}{9} )^{x}-6\cdot (\frac{1}{3})^{x}\ \textgreater \ -9\\\\(\frac{1}{3})^{2x}-6\cdot (\frac{1}{3})^{x}+9\ \textgreater \ 0\\\\t=(\frac{1}{3}^{x}\ \textgreater \ 0\; ,\; \; \; t^2-6t+9\ \textgreater \ 0\; ,\; \; (t-3)^2\ \textgreater \ 0\; \; \Rightarrow \; \; \; t-3\ne 0\\\\t\ne 3\; \; \; \Rightarrow \; \; \; (\frac{1}{3})^{x}\ne 3\; ,\; \; \; 3^{-x}\ne 3\; ,\; \; \; -x\ne 1\\\\x\ne -1\\\\Otvet:\; \; x\in (-\infty ,-1)\cup (-1,+\infty )$

$2)\; \; lg^2x-lgx-2\ \textless \ 0\; ,\; \; \; ODZ:\; x\ \textgreater \ 0\\\\t=lgx\; ,\; \; \; t^2-t-2\ \textless \ 0\; ,\; \; \; t_1=-1,\; t_2=2\\\\(t+1)(t-2)\ \textless \ 0\\\\+++(-1)---(2)+++\\\\-1\ \textless \ t\ \textless \ 2\; \; \; \Rightarrow \; \; \; -1\ \textless \ lgx\ \textless \ 2\; \; \Rightarrow \; \; 10^{-1}\ \textless \ x\ \textless \ 10^2\\\\0,1\ \textless \ x\ \textless \ 100\\\\Otvet:\; \; x\in (-0,1\; ;\; 100)\; .$

$3)\; \; (\frac{1}{4})^{x}-4\cdot (\frac{1}{2})^{x}\ \textgreater \ -4\\\\(\frac{1}{2})^{2x}-4\cdot (\frac{1}{2})^{x}+4\ \textgreater \ 0\\\\t=(\frac{1}{2})^{x}\ \textgreater \ 0\; ,\; \; \; t^2-4t+4\ \textgreater \ 0\; ,\; \; (t-2)^2\ \textgreater \ 0\\\\Tak\; kak\; \; (t-2)^2 \geq 0\; ,\; \; to \; \; iz\; \; neravenstva\; \; (t-2)^2\ \textgreater \ 0\; sledyet,\\\\chto\; \; t-2\ne 0\; ,\; \; t\ne 2\\\\(\frac{1}{2})^{x}\ne 2\; \; \Rightarrow \; \; \; 2^{-x}\ne 2\; \; ,\; \; -x\ne 1\; \; ,\; \; x\ne -1\\\\Otvet:\; \; x\in (-\infty ,2)\cup (2,+\infty )\; .$