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Question Number 140680 by bemath last updated on 11/May/21

$$\log {}_{(\mathrm{x}+2)}({7\mathrm{x}}^2-{\mathrm{x}}^3)-\log {}_{\frac{1}{\mathrm{x}+2}}({\mathrm{x}}^2-3\mathrm{x})⩾\log {}_{\sqrt{\mathrm{x}+2}}\left(\sqrt{5-\mathrm{x}}\right)$$

Answered by liberty last updated on 11/May/21

$$\left(1\right)\mathrm{x}+2>0\wedge \mathrm{x}+2\ne 1\Rightarrow \mathrm{x}>-2\wedge \mathrm{x}\ne -1$$$$\left(2\right){7\mathrm{x}}^2-{\mathrm{x}}^3>0\Rightarrow {\mathrm{x}}^2(7-\mathrm{x})>0$$$$\Rightarrow \mathrm{x}<7\wedge \mathrm{x}\ne 0$$$$\left(3\right){\mathrm{x}}^2-3\mathrm{x}>0\Rightarrow \mathrm{x}(\mathrm{x}-3)>0$$$$\Rightarrow \mathrm{x}<0\vee \mathrm{x}>3$$$$\left(4\right)5-\mathrm{x}>0\Rightarrow \mathrm{x}<5$$$$\left(5\right)\log {}_{(\mathrm{x}+2)}({7\mathrm{x}}^2-{\mathrm{x}}^3)-\log {}_{(\mathrm{x}+2)}({\mathrm{x}}^2-3\mathrm{x})⩾\log {}_{(\mathrm{x}+2)}(5-\mathrm{x})$$$$\Rightarrow \log {}_{(\mathrm{x}+2)}({7\mathrm{x}}^2-{\mathrm{x}}^3)-\log {}_{(\mathrm{x}+2)}({\mathrm{x}}^2-3\mathrm{x})-\log (5-\mathrm{x})⩾0$$$$\Rightarrow \log {}_{(\mathrm{x}+2)}({7\mathrm{x}}^2-{\mathrm{x}}^3)-\left[\log {}_{(\mathrm{x}+2)}({\mathrm{x}}^2-3\mathrm{x})(5-\mathrm{x})\right]⩾0$$$$\Rightarrow \frac{{7\mathrm{x}}^2-{\mathrm{x}}^3-({\mathrm{x}}^2-3\mathrm{x})(5-\mathrm{x})}{\mathrm{x}+2-1}⩾0$$$$\Rightarrow \frac{{7\mathrm{x}}^2\cancel{-{\mathrm{x}}^3}-({5\mathrm{x}}^2\cancel{-{\mathrm{x}}^3}-15\mathrm{x}+{3\mathrm{x}}^2)}{\mathrm{x}+1}⩾0$$$$\Rightarrow \frac{15\mathrm{x}-{\mathrm{x}}^2}{\mathrm{x}+1}⩾0;\frac{\mathrm{x}(15-\mathrm{x})}{\mathrm{x}+1}⩾0$$$$\Rightarrow \mathrm{x}<-1\vee 0⩽\mathrm{x}⩽15$$$$\mathrm{Solution}\mathrm{set}\mathrm{is}$$$$\Rightarrow -2<\mathrm{x}<-1\vee 3<\mathrm{x}<5$$

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