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The number of integers satisfying log((1...

The number of integers satisfying `log_((1)/(x))((2(x-2))/((x+1)(x-5)))ge 1` is

A

0

B

1

C

2

D

3

Text Solution

Verified by Experts

The correct Answer is:
A
Case I: `1//x gt 1` or `0 lt x lt 1`
`therefore log_((1)/(x))((2(x-2))/((x+1)(x-5)))ge 1`
`rArr (2(x-2))/((x+1)(x-5))ge(1)/(x)`
`rArr (2(x-2))/((x+1)(x-5))-(1)/(x)ge0`
`rArr (2x(x-2)-(x+1)(x-5))/(x(x+1)(x-5))ge0`
`rArr (x^(2)+5)/(x(x+1)(x-5))ge 0`
`rArr x in (-1,0) uu (5, oo)`
Hence, no solution in this case.
Case II : `0 lt (1)/(x)lt 1` or `x gt 1`
`therefore 0 lt x lt 5`
Also `(2(x-2))/((x+1)(x-5))gt 0`
`therefore x in (1,2)`
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