For a definition of county, try here: County - Wikipedia, the free encyclopedia
shake your head = move your head from side to side as a way of saying 'no', or to show sadness, disagreement, disapproval, etc.
Student or Learner
Th mechanical teacher had been giving her test after test in Geography and she had been doing worse and worse until her mother had shaken her head sorrowfully and sent for the County Inspector.
Is she really `shaken her head' ? or (It is used in figurative sense or something)
Also, please explain the meaning of `County Inspector'.
Katz covered most of it.
As for inspector: (n.):
"In the United States, an inspector is an officer in the police who is next in rank to a superintendent or police chief."
"In Britain, an inspector is an officer in the police who is higher in rank than a sergeant and lower in rank than a superintendent."
This is from an Asimov sci-fi story set in the year 2155 AD. The "County Inspector" is an inspector of the mechanical teachers used to instruct the children - nothing to do with the police on this occasion.