Problem: If x is such that x1​<2 and x1​>−3, then:
Answer Choices:
A. −31​<x<21​
B. −21​<x<3
C. x>21​
D. x>21​ or −31​<x<0
E. x>21​ or x<−31​
Solution:
Method I. Let y=x1​. Then (a) For y>0, when y<2,y1​>21​∴x>21​.
(b) For y<0, when y>−3,−y<3,−y1​>31​,y1​<−3 ∴x<−31​.
Method II. Let y=x1​∴xy=1; the graph is the two-branched hyperbola shown. At the point x=21​,y=2. When y<2,x is to right of 21​, that is, x>21​. At the point x=−31​,y=−3. When y>−3,x is to the left of −31​, that is, x<−31​.
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