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Question Number 84904 by john santu last updated on 17/Mar/20

$$3^{{2\mathrm{x}}^2}+3^{{\mathrm{x}}^2+2\mathrm{x}+5}⩾10.3^{4\mathrm{x}+6}$$$$$$

Commented by john santu last updated on 17/Mar/20

$$\mathrm{let}{3}^{{\mathrm{x}}^2}=\mathrm{u}\mathrm{\&}{3}^{2\mathrm{x}+3}=\mathrm{v}$$$$\Rightarrow {\mathrm{u}}^2+9\mathrm{uv}-{10\mathrm{v}}^2⩾0$$$$\Rightarrow (\mathrm{u}-\mathrm{v})(\mathrm{u}+10\mathrm{v})⩾0$$$$\Rightarrow \mathrm{u}⩾\mathrm{v},3^{{\mathrm{x}}^2}⩾3^{2\mathrm{x}+3}$$$$\Rightarrow {\mathrm{x}}^2-2\mathrm{x}-3⩾0$$$$\mathrm{x}⩽-1\vee \mathrm{x}⩾3$$

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