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$3^x+\sqrt{3^{x+2}*7^x}\ \textless \ 3*7^x+\sqrt{21^x}\\\\ 3^x+3\sqrt{3^x*7^x}\ \textless \ 3*7^x+\sqrt{3^x*7^x}\ \ |\ \ *\ \ \frac{1}{3^x}\\\\ \frac{3^x+3\sqrt{3^x*7^x}}{3^x}\ \textless \ \frac{3*7^x+\sqrt{3^x*7^x}}{3^x}\\\\ 1+3\sqrt{(\frac{7}{3})^x}\ \textless \ 3(\frac{7}{3})^x+\sqrt{(\frac{7}{3})^x}\\\\ 1+2\sqrt{(\frac{7}{3})^x}\ \textless \ 3(\frac{7}{3})^x\\\\ \sqrt{(\frac{7}{3})^x}=y \geq 0,\ \ so\ \ (\frac{7}{3})^x=y^2\\\\ 1+2y\ \textless \ 3y^2\\\\ 3y^2-2y-1\ \textgreater \ 0\\\\ D=4+4*3=4^2\\\\ y_{1,2}=\frac{2\pm4}{2*3}=\frac{1\pm2}{3}\\\\ y_1=1\ \ \ y_2=-\frac{1}{3}\\\\$

$3*(y+\frac{1}{3})(y-1)\ \textgreater \ 0\\\\ y\in(-\infty;-\frac{1}{3})\ \cup\ (1;+\infty)\\\\ \sqrt{(\frac{7}{3})^x}\ \textless \ -\frac{1}{3}\ \ \ or\ \ \ \sqrt{(\frac{7}{3})^x}\ \textgreater \ 1\\\\ \sqrt{(\frac{7}{3})^x}\ \textgreater \ 1\\\\ \sqrt{(\frac{7}{3})^x}-1\ \textgreater \ 0\ \ |\ \ *\ (\sqrt{(\frac{7}{3})^x}+1)\\\\ (\sqrt{(\frac{7}{3})^x}-1)*(\sqrt{(\frac{7}{3})^x}+1)\ \textgreater \ 0*(\sqrt{(\frac{7}{3})^x}+1)\\\\ \[[\sqrt{(\frac{7}{3})^x}\]]^2-1^2\ \textgreater \ 0\\\\ (\frac{7}{3})^x\ \textgreater \ 1\\\\ (\frac{7}{3})^x\ \textgreater \ (\frac{7}{3})^0\\\\ x\ \textgreater \ 0\\\\ x\in(0;+\infty)$