0}} \right. \; \left \{ {{x\ne 2} \atop {x\ne 3\; ,\; x\ne 1,\; x>0,5}} \right. " alt="log_{(x-2)^2}\; (9^{x}-3) \leq 0\; ,\; \; \; ODZ:\; \left \{ {{(x-2)^2\ \textgreater \ 0} \atop {(x-2)^2\ne 1,9^{x}-3>0}} \right. \; \left \{ {{x\ne 2} \atop {x\ne 3\; ,\; x\ne 1,\; x>0,5}} \right. " align="absmiddle" class="latex-formula">
Метод рационализации.Запишем неравенство, равносильное заданному:
![\Big ((x-2)^2-1\Big )\cdot \Big ((9^{x}-3)-1\Big ) \leq 0\\\\(x^2-4x+3)\cdot (9^{x}-4) \leq 0\\\\x^2-4x+3=0\; \; \to \; \; x_1=1,\; x_2=3\; ,\; \; +++(1)---(3)+++\\\\ a)\; \; \left \{ {{x^2-4x+3 \geq 0} \atop {9^{x}-4 \leq 0}} \right. \; \left \{ {{x\in (-\infty ,1\, ]\cup[\, 3,+\infty )} \atop {9^{x} \leq 4}} \right. \; \left \{ {{x\in (-\infty ,1\, ]\cup [\, 3,+\infty )} \atop {x\in (-\infty ,log_32\, ]}} \right. \; \; \to \\\\x\in (-\infty ,log_32\, ]](https://tex.z-dn.net/?f=%5CBig+%28%28x-2%29%5E2-1%5CBig++%29%5Ccdot+%5CBig+%28%289%5E%7Bx%7D-3%29-1%5CBig+%29+%5Cleq+0%5C%5C%5C%5C%28x%5E2-4x%2B3%29%5Ccdot+%289%5E%7Bx%7D-4%29+%5Cleq+0%5C%5C%5C%5Cx%5E2-4x%2B3%3D0%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+x_1%3D1%2C%5C%3B+x_2%3D3%5C%3B+%2C%5C%3B+%5C%3B+%2B%2B%2B%281%29---%283%29%2B%2B%2B%5C%5C%5C%5C+a%29%5C%3B+%5C%3B+%5Cleft+%5C%7B+%7B%7Bx%5E2-4x%2B3+%5Cgeq+0%7D+%5Catop+%7B9%5E%7Bx%7D-4+%5Cleq+0%7D%7D+%5Cright.+%5C%3B++%5Cleft+%5C%7B+%7B%7Bx%5Cin+%28-%5Cinfty+%2C1%5C%2C+%5D%5Ccup%5B%5C%2C+3%2C%2B%5Cinfty+%29%7D+%5Catop+%7B9%5E%7Bx%7D+%5Cleq+4%7D%7D+%5Cright.+%5C%3B++%5Cleft+%5C%7B+%7B%7Bx%5Cin+%28-%5Cinfty+%2C1%5C%2C+%5D%5Ccup+%5B%5C%2C+3%2C%2B%5Cinfty+%29%7D+%5Catop+%7Bx%5Cin+%28-%5Cinfty+%2Clog_32%5C%2C+%5D%7D%7D+%5Cright.+%5C%3B+%5C%3B+%5Cto+%5C%5C%5C%5Cx%5Cin+%28-%5Cinfty+%2Clog_32%5C%2C+%5D "\Big ((x-2)^2-1\Big )\cdot \Big ((9^{x}-3)-1\Big ) \leq 0\\\\(x^2-4x+3)\cdot (9^{x}-4) \leq 0\\\\x^2-4x+3=0\; \; \to \; \; x_1=1,\; x_2=3\; ,\; \; +++(1)---(3)+++\\\\ a)\; \; \left \{ {{x^2-4x+3 \geq 0} \atop {9^{x}-4 \leq 0}} \right. \; \left \{ {{x\in (-\infty ,1\, ]\cup[\, 3,+\infty )} \atop {9^{x} \leq 4}} \right. \; \left \{ {{x\in (-\infty ,1\, ]\cup [\, 3,+\infty )} \atop {x\in (-\infty ,log_32\, ]}} \right. \; \; \to \\\\x\in (-\infty ,log_32\, ]")
0,5" alt="\star \; \; 9^{x} \leq 4\; \; \to \; \; x \leq log_94\; ,\; \; log_94= log_{3^2}2^2=log_32<1\; ,\; log_32>0,5" align="absmiddle" class="latex-formula">
![b)\; \; \left \{ {{x^2-4x+3 \leq 0} \atop {9^{x}-4 \geq 0}} \right. \; \left \{ {{x\in [\, 1,3\, ]} \atop {x\in [\, log_32,+\infty )}} \right. \; ,\; \; x\in [\, 1,3\, ]\\\\Otvet:\; \; x\in (0,5\, ;\, log_32\, ]\cup (1,2)\cup (2,3)\; .](https://tex.z-dn.net/?f=b%29%5C%3B+%5C%3B++%5Cleft+%5C%7B+%7B%7Bx%5E2-4x%2B3+%5Cleq+0%7D+%5Catop+%7B9%5E%7Bx%7D-4+%5Cgeq+0%7D%7D+%5Cright.+%5C%3B++%5Cleft+%5C%7B+%7B%7Bx%5Cin+%5B%5C%2C+1%2C3%5C%2C+%5D%7D+%5Catop+%7Bx%5Cin+%5B%5C%2C+log_32%2C%2B%5Cinfty+%29%7D%7D+%5Cright.+%5C%3B+%2C%5C%3B+%5C%3B+x%5Cin+%5B%5C%2C+1%2C3%5C%2C+%5D%5C%5C%5C%5COtvet%3A%5C%3B+%5C%3B+x%5Cin+%280%2C5%5C%2C+%3B%5C%2C+log_32%5C%2C+%5D%5Ccup+%281%2C2%29%5Ccup+%282%2C3%29%5C%3B+. "b)\; \; \left \{ {{x^2-4x+3 \leq 0} \atop {9^{x}-4 \geq 0}} \right. \; \left \{ {{x\in [\, 1,3\, ]} \atop {x\in [\, log_32,+\infty )}} \right. \; ,\; \; x\in [\, 1,3\, ]\\\\Otvet:\; \; x\in (0,5\, ;\, log_32\, ]\cup (1,2)\cup (2,3)\; .")