Solve: `(log)_((log)_2(x/x))(x^2-10 x+22)>0`

Explanation

` log_(log_(2)(x/2))(x^(2)-10x+22) gt 0` (i)
We must have
`(i)x^(2)-10x+22 gt 0`
`rArr x in (-infty, 5 - sqrt3) cup(5 + sqrt3, infty)` (ii)
`(ii) x/2 gt 0`
` rArr x gt 0` (iii)
Case (i):
` 0 lt log_(2)(x/2) lt 1`
` rArr 1 lt x/2 lt 2`
` or 2 lt x lt 4` (iv)
Therefore, from Eq. (i), `x^(2)-10x+22 lt 1`
` or x^(2) - 10x+21 lt 0`
` rArr 3 lt xlt 7` (v)
From Eqs. (ii), (iii),(iv), and(v), the common solution is
` 3 lt x lt 5 - sqrt3`
Case (ii) :
` log_(2)(x/2) gt 1`
` or x/2 gt 2 or x gt 4` (vi)
Therefore, from Eqs. (i) , ` x^(2)-10x+22 gt 1`
` or x^(2) - 10x + 21 gt 0`
` or x lt 3 or x gt 7`(vii)
From Eqs.(ii), (iii),(vi), and (vii), the common solutions is ` x in (7, infty)`
Hence, ` x in (3, 5 - sqrt3) cup(7, infty)`.