题目内容
设不等式
的解集为M,
.
(1)证明:
;
(2)比较
与
的大小,并说明理由.
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626897682.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626928613.png)
(1)证明:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626944803.png)
(2)比较
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626960496.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626975508.png)
(1)证明过程详见解析;(2)|1-4ab|>2|a-b|.
试题分析:本题主要考查绝对值不等式的解法、绝对值的运算性质、作差法比较大小等基础知识,考查学生的分析问题解决问题的能力、转化能力、计算能力.第一问,先利用零点分段法将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626991447.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627006993.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627022509.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627053535.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626944803.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627022509.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627053535.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626960496.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051626975508.png)
(1)记f(x)=|x-1|-|x+2|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240516271471086.png)
由-2<-2x-1<0解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627162575.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627178749.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240516271941489.png)
(2)由(1)得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627209510.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824051627225498.png)
因为|1-4ab|2-4|a-b|2=(1-8ab+16a2b2)-4(a2-2ab+b2)
=(4a2-1)(4b2-1)>0, 9分
所以|1-4ab|2>4|a-b|2,故|1-4ab|>2|a-b|. 10分
![](http://thumb2018.1010pic.com/images/loading.gif)
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