The solution set of inequality
`(1)/(2^(x)-1) gt (1)/(1-2^(x-1))` is

Options

A. A. `(1,oo)`
B. B. `(o,log_(2)(4//3))`
C. C. `(-1,oo)`
D. D. `(o,log_(2)(4//3)) uu (1,oo)`

Answer

D

Explanation

`(d)` Put `2^(x)=t` then `t gt 0`. The given inequality becomes
`(1)/(t-1) gt (2)/(2-t)`
`implies (1)/(t-1)-(2)/(2-t) gt 0`
`implies (2-t-2t+2)/((t-1)(2-t)) gt 0`
`implies(4-3t)/((t-1)(2-t)) gt 0`
Sign scheme is as shown below.

From sign scheme,
`1 lt t lt 4//3` or `t gt 2`.
`implies 1 lt 2^(x) lt 4//3` or `2^(x) gt 2`
`implies 0 lt x lt log_(2)(4//3)uu(1,oo)`.