Solution set of the inequality `(log)_(0. 8)((log)_6(x^2+x)/(x+4))<0` is `(-4,-3)` (b) `(-3,4)uu(8,oo)` `(-3,oo)` (d) `(-4,-3)uu(8,oo)`

Options

A. `(-4, -3)`
B. `(-3, 4) cup(8, infty)`
C. `(-3, infty)`
D. ` (-4, -3) cup(8, infty)`

Answer

D

Explanation

` log_(0.8)(log_(6).(x^(2)+x)/(x+4)) lt 0`
` rArr log_(6). (x^(2)+x)/(x+4) gt 1 `
` rArr (x^(2)+x)/(x+4) gt 6`
` rArr (x^(2)+x)/(x+4) - 6 gt 0`
` rArr (x^(2) - 5x - 24)/((x+4)) gt 0`
` rArr ((x-8)(x+3))/((x+4)) gt 0`
From the sign scheme of ` ((x-8)(x+3))/((x+4))`,
` x in (-4, -3) cup (8, infty)`.