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Question 1

Question 2

Решите задачу:

$\frac{4^{x}}{4^{x}-3^{x}} \ \textless \ 4\\\\\frac{4^{x}}{4^{x}-3^{x}}-4\ \textless \ 0\\\\ \frac{4^{x}-4(4^{x}-3^{x})}{4^{x}-3^{x}}\ \textless \ 0\\\\ \frac{4^{x}-4^{x}\cdot 4+4\cdot 3^{x}}{4^{x}-3^{x}}\ \textless \ 0\\\\\frac{4^{x}(1-4)+4\cdot 3^{x}}{4^{x}-3^{x}} \ \textless \ 0\; ,\; \; ODZ:\; 4^{x}-3^{x}\ne 0\; ,\; 4^{x}\ne 3^{x}\; \to \; x\ne 0\\\\\frac{-3\cdot 4^{x}+4\cdot 3^{x}}{4^{x}-3^{x}}\ \textless \ 0\; \to \left \{ {{-3\cdot 4^{x}+4\cdot 3^{x}\ \textless \ 0} \atop {4^{x}-3^{x}\ \textgreater \ 0}} \right. \; ili\; \left \{ {{-3\cdot 4^{x}+4\cdot 3^{x}\ \textgreater \ 0} \atop {4^{x}-3^{x}\ \textless \ 0}} \right.$

$a)\; \left \{ {{4\cdot 3^{x}\ \textgreater \ 3\cdot 4^{x}} \atop {4^{x}\ \textless \ 3^{x}}} \right. \; \left \{ {{\frac{4\cdot 3^{x}}{3\cdot 4^{x}}\ \textgreater \ 1} \atop {(\frac{4}{3})^{x}\ \textless \ 1}} \right. \; \left \{ {{(\frac{3}{4})^{x}\ \textgreater \ \frac{3}{4}} \atop {(\frac{4}{3})^{x}\ \textless \ (\frac{4}{3})^0}} \right. \; \left \{ {{x\ \textless \ 1} \atop {x\ \textless \ 0}} \right. \; \; ,x\ \textless \ 0\\\\b) \left \{ {{4\cdot 3^{x}\ \textless \ 3\cdot 4^{x}} \atop {4^{x}\ \textgreater \ 3^{x}}} \right. \; ;\; \left \{ {{x\ \textgreater \ 1} \atop {x\ \textgreater \ 0}} \right. \; \; ,\; x\ \textgreater \ 1\\\\x\in (-\infty ,0)\cup (1,+\infty )$

Question 3

Шикарное решение. Благодарю. Тут безусловно лучший ответ.