The true solution set of inequality `(log)_((x+1))(x^2-4)>1` is equal to (a) `(2, ∞)` (b) `(2, (1+sqrt(21))/2)` (c) `((1-sqrt(21))/2,(1+sqrt(21))/2)` (d) `(∞, (1+sqrt(21))/2)`

Options

A. `(2, infty)`
B. `(2,(1+sqrt21)/2)`
C. `((1-sqrt21)/2,(1+sqrt21)/2)`
D. `((1+sqrt21)/2, infty)`

Answer

D

Explanation

` log_((x+1))(x^(2 )- 4)gt1`
We must have ` x^(2) - 4 gt 0, x + 1 gt 0" abd "x + 1 ne 1`
` :. X in (2, infty)` …(i)
Case I:
` x+ 1 gt 1 rArr x gt 0 `
` :. X ^(2) -4 gt x + 1`
` :. X ^(2) - x - 5 gt 0`
` :. X in ((1+sqrt21)/2, infty)` ...(ii)
Case II:
` (x+1) in (0, 1)`
` rArr x in (-1, 0)`
Clearly, this is not possible as ` x in (2, infty)`.