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Question Number 145829 by bramlexs22 last updated on 08/Jul/21

Answered by EDWIN88 last updated on 08/Jul/21

$$\left(1\right)\sqrt{x+\frac{15}{4}}\ne 1;x\ne -\frac{11}{4}$$$$andx>-\frac{15}{4}$$$$\left(2\right)(x^2-\frac{8}{3}x)>0\Rightarrow x(x-\frac{8}{3})>0$$$$\Rightarrow x<0\cup x>\frac{8}{3}$$$$\left(3\right)let\log {}_{\sqrt{x+\frac{15}{4}}}(x^2-\frac{8}{3}x)=t$$$$\Rightarrow t+\frac{4}{t}⩽4;\frac{t^2-4t+4}{t}⩽0$$$$\Rightarrow \frac{{(t-2)}^2}{t}⩽0wegett<0$$$$\log {}_{\sqrt{x+\frac{15}{4}}}(x^2-\frac{8}{3}x)<0$$$$\Rightarrow x^2-\frac{8}{3}x-1<0$$$$\Rightarrow 3x^2-8x-3<0$$$$\Rightarrow (3x+1)(x-3)<0$$$$\Rightarrow -\frac{1}{3}<x<3$$$$solutionsetwegetfrom$$$$\left(1\right)\cap \left(2\right)\cap \left(3\right)$$$$\Rightarrow \frac{8}{3}<x<3\cup -\frac{1}{3}<x<0$$

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