Data Sufficiency

This if from OG
If x,y, and z are positive numbers, is x > y > z ?

1) xz > yz
2) yx > yz

x,y and z are positive integers(numbers) and we have to determine if x > y and y > z
From 1) : as z is positive the inequality can be divided by z on both sides, thus x > y
Not sufficient.
From 2) : again as y is positive x > z
Not sufficient.
1&2 both :
one way to combine - x > y and x > z, we are still not able to find whether y > z. Not sufficient. OA for this is E and OE is similar to above.
My doubt is regarding statements 1&2 combined.
I think we can also divide the inequalities here.
So (1)/(2) = xz/yx > yz/yz => z / y > 1 => as z and y are integers z > y
This is sufficient to determine that x,z > y, at least we know z > y and y is not > z thus it is sufficient to answer the question.
Combining inequalities is even done in OG explanation. Is the division method wrong to follow?
Please share your thoughts.
Thanks.