If ` log_(1//2)(4-x)gelog_(1//2)2-log_(1//2)(x-1)`,then x belongs to

Options

A. (1,2]
B. [1,3]
C. [3,4)
D. [2,3]

Answer

A, C

Explanation

We have `log_(1//2)(4-x)ge log_(1//2)2-log_(1//2)(x-1)`
It is defined if `4-x gt 0` and `x- 1 gt 0` or `1 lt x lt 4` …..(i)
`rArr log_(1//2)(4-x)(x-1)ge log_(1//2)2`
`rArr (4-x)(x-1)ge 2`
`rArr x^(2)-5x + 6 ge 0`
`rArr (x-3)(x-2)ge 0`
`rArr x ge 3` or `x le 2` ......(ii)
From (i) and (ii), we get `x in (1,2] uu [3,4)`