Решите неравенство 46x−6⩾76x−3\frac{4}{6^x-6} \geqslant \frac{7}{6^x-3}6x−64⩾6x−37▸Ответx∈(−∞;log63)∪(1;log610]x\in(-\infty;\log_63)\cup(1;\log_610]x∈(−∞;log63)∪(1;log610]▸Решение46x−6⩾76x−3\frac{4}{6^x-6} \geqslant \frac{7}{6^x-3}6x−64⩾6x−37 ОДЗ: {6x−6≠0(1)6x−3≠0(2)⇔{x≠1x≠log63\begin{cases} 6^x-6\neq 0 (1)\\ 6^x-3 \neq 0 (2)\\ \end{cases} \Leftrightarrow \begin{cases} x\neq 1\\ x \neq \log_6 3 \end{cases}{6x−6=0(1)6x−3=0(2)⇔{x=1x=log63 1)6x−6≠01) 6^x-6\neq 01)6x−6=0 6x≠66^x \neq 66x=6 x≠1x\neq 1x=1 2)6x−3≠02) 6^x-3 \neq 02)6x−3=0 6x≠36^x \neq 36x=3 x≠log63x \neq \log_63x=log63 Пусть 6x=t6^x=t6x=t 4t−6⩾7t−3\dfrac4{t-6} \geqslant \dfrac7{t-3}t−64⩾t−37 4t−6−7t−3⩾0\dfrac4{t-6}-\dfrac7{t-3} \geqslant 0t−64−t−37⩾0 4(t−3)−7(t−6)(t−6)(t−3)⩾0\frac{4(t-3)-7(t-6)}{(t-6)(t-3)}\geqslant 0(t−6)(t−3)4(t−3)−7(t−6)⩾0 4t−12−7t+42(t−6)(t−3)⩾0\frac{4t-12-7t+42}{(t-6)(t-3)}\geqslant 0(t−6)(t−3)4t−12−7t+42⩾0 −3t+30(t−6)(t−3)⩾0\frac{-3t+30}{(t-6)(t-3)}\geqslant 0(t−6)(t−3)−3t+30⩾0 −3(t−10)(t−6)(t−3)⩾0\frac{-3(t-10)}{(t-6)(t-3)}\geqslant 0(t−6)(t−3)−3(t−10)⩾0 (t−10)(t−6)(t−3)⩽0\frac{(t-10)}{(t-6)(t-3)}\leqslant 0(t−6)(t−3)(t−10)⩽0 1)t<31) t< 31)t<3 6x<36^x < 36x<3 x<log63x < \log_63x<log63 2)6<t⩽102) 6< t\leqslant 102)6<t⩽10 6<6x⩽106 < 6^x \leqslant 106<6x⩽10 1<x⩽log6101< x\leqslant \log_6101<x⩽log610
