Find the solution of given inequality for real x : `2(2x + 3) < (x - 2)`

Explanation

Let `(1 + x)^(2) (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)...`
` = 1 + P_(1)x + P_(2)x^(2) + P_(3) x^(3) + P_(4) x^(4) + ...+ P_(r-1) x^(r-1) + P_(r) x^(r) +....`...(i)
Replacing x by `(x)/(2)` , we get
` (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)(1 + (x)/(2^(4)))^(2)...`
`=[1 + P_(1) ((x)/(2)) + P_(2) ((x)/(2))^(2) + P_(3) ((x)/(2))^(3)+ ...]`
On multiplying both sides by ` (1 + x)^(2)` , we get
`(1+x)^(2) (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)...`
`= (1 + x)^(2) [1 + P_(1) ((x)/(2)) + P_(2) ((x)/(2))^(2) + P_(3) ((x)/(2))^(3)+ ...]`...(ii)
From Eqs .(i) and (ii) , we get
` 1 + P_(1) x + P_(2) x^(2) + P_(3) x^(3) + P_(4)x^(4) + ...+ P_(r-1) x^(r-1) + P_(r) x^(r) + ...`
`= (1 + x)^(2) [1 + P_(1) ((x)/(2)) + P_(2) ((x)/(2))^(2) + P_(3) ((x)/(2))^(3)+ ...]`
On equating coefficient of ` x^(r)` , we get
`P_(r) = P_(r)((1)/(2^(r))) + 2P_(r-1) ((1)/(2^(r-1))) + P_(r-2) ((1)/(2^(r-2)))`
` rArr P_(r) (1-(1)/(2^(r))) = (1)/(2^(r-2)) (P_(r-1) + P_(r-2))`
` rArr P_(r) (2^(2))/((2^(r-1))) = (P_(r-1) + P_(r-2))`
Now , ` P_(0) = 1 ,P_(1)= 2 + 1 + (1)/(2) + (1)/(2^(2)) + ...= 4`
`P_(2) = (2^(2) (P_(1) + P_(0)))/(2^(2)-1)=(20)/(3)`
` P_(3)=(2^(2) (P_(2) + P_(1)))/(2^(3)-1)=(128)/(21)`
`P_(4) = (2^(2) (P_(3) + P_(2)))/(2^(4) -1) = (4((128)/(21)+ (20)/(3)))/(15) = (1072)/(315)` .