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Find the solution of the inequality 2log...

Find the solution of the inequality `2log_(1/4)(x+5)>9/4log_(1/(3sqrt(3)))(9)+log_(sqrt(x+5))(2)`

A

`(-5,-4)`

B

`(-3,-1)`

C

`(-4,-1)`

D

`(-5,-2)`

Text Solution

Verified by Experts

The correct Answer is:
A, B
We have
`2log_((1)/(4))(x+5)gt(9)/(4)log_((1)/(3sqrt(3)))(9)+log_(sqrt(x+5))(2)`
`rArr -log_(2)(x+5)gt(9)/(4)(-(4)/(3))+(2)/(log_(2)(x+5))`
`rArr 3gt log_(2)(x+5)+(2)/(log_(2)(x+5))`
Let `log_(2)(x+5)=y`
`rArr 3 gt +(2)/(y)`
`rArr (y^(2)+2)/(y)-3lt 0`
`rArr (y^(2)-3y+2)/(y)ly 0`
`rArr ((y-2)(y-1))/(y)lt 0`
Using sign scheme method, we get
`y in (-oo,0)uu (1,2)`
`therefore log_(2)(x+5)in (-oo,0)uu(1,2)`
`therefore (x+5)in (2^(-oo),2^(0))uu(2^(1),2^(2))`
`therefore (x+5)in (0,1)uu(2,4)`
`therefore x in (-5,-4)uu (-3,-1)`
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