解不等式log1/2 (x^2-x-2)大于log1/2 (x-1)-1快

解不等式log1/2 (x^2-x-2)大于log1/2 (x-1)-1快
解不等式log1/2 (x^2-x-2)大于log1/2 (x-1)-1
快

解不等式log1/2 (x^2-x-2)大于log1/2 (x-1)-1快
首先由log1/2 (x^2-x-2)、log1/2 有意义
得①x^2-x-2>0
x>2或x0
x>1
初步确定范围x>2
再将原始化简
log1/2 (x^2-x-2)>log1/2 (x-1)-1
log1/2 (x^2-x-2)>log1/2 (x-1)-log1/2(1/2)
log1/2 (x^2-x-2)>log1/2 [(x-1)/(1/2)]=log1/2 [2(x-1)]=log1/2(2x-2)
由y=log1/2(x)是减函数
x^2-x-2＜2x-2
x^2-3x＜0
0＜x＜3
∴综上2＜x＜3

x^2-x-2>0===>x>2或x<-1 ①
x-1>0 ===>x>1 ②
log1/2(x^2-x-2)>log1/2 (x-1)-1
log1/2(x-2)(x+1)/(x-1)>-1
log1/2(x-2)(x+1)/(x-1)>log1/2(1/2)^(-1)
(x-2)(x+1)/(x-1)<(1/2)^(-1)
解得03
综合①②③得x>3

原不等等价于：log1/2 (x^2-x-2)+1>log1/2 (x-1)
等价于 log1/2 (x^2-x-2)+ log1/2 1/2>log1/2 (x-1)
log1/2 (x^2-x-2)*1/2 >log1/2 (x-1)
等价于 ①:x^2-x-2>0

② x-1>0
...

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原不等等价于：log1/2 (x^2-x-2)+1>log1/2 (x-1)
等价于 log1/2 (x^2-x-2)+ log1/2 1/2>log1/2 (x-1)
log1/2 (x^2-x-2)*1/2 >log1/2 (x-1)
等价于 ①:x^2-x-2>0

② x-1>0
③1/2(x^2-x-2)解得：①:x>2或x<-1
②x>1
③0＜x＜3
①②③的交集为愿不等式的解集即2＜x＜3 或空集。
综上，解集即2＜x＜3

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