Решить логарифмические неравенства.

Question 1

Question 2

Решите задачу:

$1)\quad \frac{log_2(1-x)}{x+1} \ \textless \ 0\; ,\; ODZ:\; \; \left \{ {{1-x\ \textgreater \ 0} \atop {x+1\ne 0}} \right. \; ,\; \left \{ {{x\ \textless \ 1} \atop {x\ne -1}} \right. \\\\ a)\; \; \left \{ {{log_2(1-x)\ \textless \ 0\; ,\; 1-x\ \textgreater \ 0} \atop {x+1\ \textgreater \ 0}} \right. \; \; ili\; \; b)\; \; \left \{ {{log_2(1-x)\ \textgreater \ 0\; ,\; 1-x\ \textgreater \ 0} \atop {x+1\ \textless \ 0}} \right. \\\\a)\; \; \left \{ {{1-x\ \textless \ 1\; ,\; x\ \textless \ 1} \atop {x\ \textgreater \ -1}} \right. \; ,\; \left \{ {{x\ \textgreater \ 0\; ,\; x\ \textless \ 1} \atop {x\ \textgreater \ -1}} \right. \; \; \Rightarrow\; \; 0\ \textless \ x\ \textless \ 1$

$b)\; \; \left \{ {{1-x\ \textgreater \ 1\; ,\; x\ \textless \ 1} \atop {x\ \textless \ -1}} \right. \; ,\; \left \{ {{x\ \textless \ 0\; ,\; x\ \textless \ 1} \atop {x\ \textless \ -1}} \right. \; \; \Rightarrow \; \; \; x\ \textless \ -1$

$Otvet:\; \; x\in (-\infty \ \textless \ -1)\cup (0,1)\; .\\\\2)\quad \frac{1-log_{0,5}x}{\sqrt{6x+2}} \ \textless \ 0\; ,\; \; ODZ:\; \left \{ {{\sqrt{6x+2}\ne 0\; ,\; 6x+2\ \textgreater \ 0} \atop {x\ \textgreater \ 0}} \right. \; \left \{ {{x\ \textgreater \ -\frac{1}{3}} \atop {x\ \textgreater \ 0}} \right. \; ,\; \; x\ \textgreater \ 0\\\\Tak\; kak\; \; \sqrt{6x+2}\ \textgreater \ 0,\; to\; drob\ \textless \ 0,\; esli\; \; (1-log_{0,5}x)\ \textless \ 0\; .\\\\log_{0,5}x\ \textgreater \ 1\; ,\; \; \; log_{0,5}x\ \textgreater \ log_{0,5}0,5\; \; \Rightarrow \; \; x\ \textless \ 0,5\\\\Otvet:\; \; x\in (0\; ;\; 0,5)\; .$

$3)\; \; \; \Big |log_3(2-x)\Big |\ \textgreater \ 2\; ,\; \; \; ODZ:\; \; 2-x\ \textgreater \ 0\; \; \Rightarrow \; \; x\ \textless \ 2\\\\\star \; \; |x|\ \textgreater \ a\; \; \; \Rightarrow \; \; \; \underline {x\ \textgreater \ a\; \; ili\; \; x\ \textless \ -a}\; \; , \; \; to\; est\; \; \left [ {{x\ \textgreater \ a} \atop {x\ \textless \ -a}} \right. \; \; \star \\\\ \left [ {{log_3(2-x)\ \textgreater \ 2} \atop {log_3(2-x)\ \textless \ -2}} \right. \; ,\; \left [ {{2-x\ \textgreater \ 3^2} \atop {2-x\ \textless \ 3^{-2}}} \right. \; \left [ {{x\ \textless \ -7} \atop {x\ \textgreater \ \frac{17}{9}}} \right. \\\\ a)\; \\; \left [ {{x\ \textless \ 2} \atop {x\ \textless \ -7}} \right. \; \; ili\; \; b)\; \; \left [ {{x\ \textless \ 2} \atop {x\ \textgreater \ \frac{17}{9}}} \right.$

$a)\; \; x\ \textless \ -7\; \; \; ili\; \; \; b)\; \; \frac{17}{9}\ \textless \ x\ \textless \ 2\; \; \; \; (\frac{17}{9}=1\frac{8}{9})\\\\Otvet:\; \; x\in (-\infty ,-7)\cup (1\frac{8}{9}\, ,\, 2)\; .$