The solution set of the inequation
`(1)/(log_(2)x) lt (1)/(log_(2)sqrt(x+2))`, is

Options

A. (0,1)
B. `(2,oo)`
C. `(0,1) cup (2,oo)`
D. None of these

Answer

C

Explanation

Two sides of the given inequation are defined for `x gt 0` and `x ne 1`.
Now, two cases arise.
CASE I When `x gt 1`
In this case, we have
`log_(2) x gt 0` and `log_(2)sqrt(x+2) gt 0`
`:. (1)/(log_(2)x) lt (1)/(log_(2)sqrt(x+2))`
`implies log_(2)sqrt(x+2)lt log_(2)x`
`rArr sqrt(x+2) lt x`
`rArr x+2 lt x^(2)`
`rArr x^(2)-x-2 gt 0`
`rArr (x-2)(x+1) gt 0`
`implies x-2 gt 0 " "[:' x gt 1 implies x+1 gt0]`
`x gt 2 rArr x in (2,oo)`
CASE II When `0 lt x lt 1`
In this case, we have
`log_(2) x lt 0` and `log_(2)sqrt(x+2) gt 0`
So, the inequation`(1)/(log_(2)x) lt (1)/(log_(2)sqrt(x+2))` holds for all `x in (0,1)`.
Hence, the solution set of the given inequation is `(0,1) cup (2,oo)`.