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Question Number 83285 by jagoll last updated on 29/Feb/20

$$3^{(\mathrm{x}+2)(\mathrm{x}-4)}⩽7^{\mathrm{x}+2}$$$$\mathrm{find}\mathrm{solution}$$

Commented by jagoll last updated on 29/Feb/20

$$7^{{\log }_7\left(3^{(\mathrm{x}+2)(\mathrm{x}-4)}\right)}⩽7^{\mathrm{x}+2}$$$$\Rightarrow (\mathrm{x}+2)(\mathrm{x}-4){\log }_7\left(3\right)⩽\mathrm{x}+2$$$$\Rightarrow (\mathrm{x}+2)[(\mathrm{x}-4){\log }_7\left(3\right)-1]⩽0$$$$-2⩽\mathrm{x}⩽4+{\log }_3\left(7\right)$$$$$$

Answered by MJS last updated on 01/Mar/20

$$(x+2)(x-4)\ln 3⩽(x+2)\ln 7$$$$\left(1\right)x+2<0\iff x<-2$$$$(x-4)\ln 3⩾\ln 7$$$$x⩾\frac{\ln 7}{\ln 3}+4\Rightarrow \mathrm{no}\mathrm{solution}$$$$\left(2\right)x+2⩾0\iff x⩾-2\mathrm{for}x=-2:3^{(x+2)(x-4)}=7^{x+2}$$$$\mathrm{for}x>-2:$$$$(x-4)\ln 3⩽\ln 7$$$$x⩽\frac{\ln 7}{\ln 3}+4\Rightarrow -2⩽x⩽\frac{\ln 7}{\ln 3}+4$$

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