Could someone explain with steps for this? I am confused when using Inequalities and Absolute values together

If n < 0 and Square of n < 1/100, then (1/n) must be ?

A. less than -10
B. between -1 and (-1/10)
C. between (-1/10) and 0
D. between 0 and (1/10)
E. greater than 10

Here are the steps that I did:

Given: n < 0 and Square of n < 1/100

(n*n) < (1/100)
Taking Square root gives:
|n| < (1/10)
Since n is negative, the above equation becomes
n > (-1/10)
Taking reciprocal gives,
1/n > (-10) [another question: When taking a reciprocal, would the inequality change?)
....
OA : [spoiler]A[/spoiler]