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The set of real values of x for which lo...

The set of real values of `x` for which `log_(2x + 3) x^(2)ltlog_(2x+3)(2x + 3)` is `(a,b) uu (b,c) uu (c,d)` then

A

`2a + b = c + d + 1`

B

`2a = 3b`

C

`c + d = 3`

D

`b + d = 2`

Text Solution

Verified by Experts

The correct Answer is:
B, C, D
Case - `1`
If `2x + 3 gt 1 rArr x gt -1 …..(1)`
then `x^(2) lt 2x + 3`
`x^(2) - 2x - 3 lt 0`
`x in (-1, 3) …(2)`
`rArr (1) cap (2) = (-1, 3)`
Case `-2`
`0 lt 2x + 3 lt 1 rArr (-3)/(2) lt x lt 1 …..(3)`
then `x^(2) gt 2x + 3`
`rArr x lt -1 cup x gt 3 ......(4)`
`rArr (3) cap (4) (-3)/(2) lt x lt -1`
but by definition of log `x^(2) gt 0`
`rArr x lt 0 cup gt 0, x ne 0`
so final
`((-3)/(2), -1) cup (-1, 0) cup(0, 3)`
`rArr a = (-3)/(2), b = -1, c = 0, d = 3`
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