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Question

The complete solution set of the inequality log5(x22)<log5(32|x|1) is

A
(,23)(23,)
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B
(2,23)(23,2)
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C
(2,2)(2,2)
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D
(2,2)(23,2)
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Solution

The correct option is C (2,2)(2,2)
log5(x22)<log5(32|x|1) is defined if
x22>0 and 32|x|1>0
x(,2)(2,) (1)
and x(,23)(23,) (2)

Now, log5(x22)<log5(32|x|1)
x22<32|x|1x232|x|1<02x23|x|2<0
2|x|23|x|2<0 (|x|2=x2, xR)
(2|x|+1)(|x|2)<0
12<|x|<2
But |x|0
So, 0|x|<2
x(2,2) (3)

From (1),(2) and (3), we have
x(2,2)(2,2)

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