解不等式“1/(X^2-2)<1/|X|”急急急!

来源:百度知道 编辑:UC知道 时间:2024/06/16 01:24:41
如题

分母不为零,x^2-2≠0,|x|≠0
=>x≠±√2,x≠0

当x>0时,1/(x^2-2)<1/x
1/x-1/(x^2-2)>0
(x-2)(x+1)/[x(x+√2)(x-√2)]>0
=>0<x<√2或x>2

当x<0时,1/(x^2-2)<-1/x
(x+2)(x-1)/[x(x+√2)(x-√2)]<0
=>x<-2或-√2<x<0

∴x<-2或-√2<x<0或0<x<√2或x>2

分步骤讨论下
1、当X>0时,化简得1/(X^2-2)-1/X<0
(X+1)(X-2)/[X(X^2-2)]>0
X>2或0<X<√2
2、当X<0时,化简得(X-1)(X+2)/[X(X^2-2)]<0
X<-√2
综合得X<-√2 X>2或0<X<√2