Which of the following is a solution of `(logx)(5/2-1/x) gt (5/2-1/x)` ?

Options

A. `(2/5,1/2)`
B. `(1,2)`
C. ` (2/5, 1)`
D. none of these

Answer

A , B

Explanation

Given inequality is defined if ` x gt 2//5, x ne 1`
Case I: If ` x gt 1`
` rArr 5/2 - 1/x gt x `
` or x + 1/x lt 5/2`
` or 2(x^(2) + 1) lt 5x`
` or 2x^(2) - 5x + 2 lt 0 `
` or 2 x^(2) - 4x - x + 2 lt 0`
`or (x-2)(2x-1) lt 0`
` rArr x in (1, 2)` ....(i)
Case II: ` 2/5 lt x lt 1, " then " (x-2)(2x-1) gt 0`
` rArr x in (2/5,1/2)` ...(ii)
From Eqs. (i) and (ii), we get ` x in (2/5,1/2) cup (1, 2) `.