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logx+63log2x-1x+2>0Case i 0<x+63<1 i.e. log is decreasing0<x+6<3-6<x<-3logx+63log2x-1x+2>0log2x-1x+2<x+630log2x-1x+2<1x-1x+2<21x-1x+2-2<0x-1-2x-4x+2<0-x-5x+2<0Multiply both sides by -1x+5x+2>0-+++++ -5 ---- -2+++++ x-,-5-2,Take intersection with -6<x<-3we get x-6,-5Case ii x+63>1 i.e log is increasingx+6>3x>-3logx+63log2x-1x+2>0log2x-1x+2> x+630log2x-1x+2>1x-1x+2>21x-1x+2-2>0x-1-2x-4x+2>0-x-5x+2>0x+5x+2<0-+++++ -5 ---- -2+++++ x-5,-2Take intersection with x>-3x-3,-2Take union of both casesx-6,-5-3,-2Also Base of log must not be 1x+631x-3Quantity inside log should be positivelogx+63log2x-1x+2log2x-1x+2>0 and x-1x+2>0x-1x+2>20 and  x-1x+2>0x-1x+2>1 and  x-1x+2>0x-1x+2-1>0 and  x-1x+2>0x-1-x-2x+2>0 and  x-1x+2>0-3x+2>0 and  x-1x+2>01x+2<0 and    x-1x+2>0x-,-2 and x-,-21,Take intersectionx-,-2Finallly take intersection of x-,-2 and  x-6,-5-3,-2We getx-6,-5-3,-2 Answer

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