Solution set of the inequality `1/(2^(x) - 1) gt 1/(1-2^(x-1))` is

Options

A. ` (1, infty)`
B. ` (0, log_(2) (4//3))`
C. ` (-1, infty)`
D. ` (0, log_(2)(4//3))cup(1, infty)`

Answer

D

Explanation

Put ` 2^(x) = t." Then " t gt 0`. The given inequality becomes
` 1/(t-1) gt 2/(2 - t)`
` rArr 1/(t-1) - 2/(2 - t) gt 0`
` rArr (2-t - 2t + 2)/((t-1)(2-t)) gt 0`
` rArr (4 - 3t)/((t-1)(2-t)) gt 0`
` rArr (t-1)(t-4//3)(t-2) gt 0` .
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From above sign scheme, we get
` 1 lt t 4//3 or t gt 2`.
`rArr 1 lt 2^(x) lt 4//3 or 2^(x) gt 2`
` rArr 0 lt x lt log_(2) (4//3) or x gt 1`