I would advise thinking of it logically. If 1/something is between 1/2 and 1/5, and we're only dealing with integers (as we're told in this question), then clearly you're dealing with 1/3 or 1/4. Thus, k-1 has to be 3 or 4. Sometimes that kind of simplistic, non-math thinking is far more efficient on test day than dealing with complex algebra rules.
The algebra is as follows:
Split it into 2 inequalities: 1/5 < 1/(k-1) and 1/(k-1) < 1/2
Take the reciprocal of everything. Like multiplying by a negative, taking the reciprocal of an inequality means you must flip the inequality sign, so you get:
5 > k-1 and k-1 > 2
6 > k and k > 3
Putting it together, we get, of course, 3 < k < 6, or k = {4, 5} since it must be an integer.
Now you get to ask yourself: Would you rather do that algebra, and risk a silly mistake somewhere along the line, or see through the natural counting logic of the problem (1/2, 1/3, 1/4, 1/5) and get an answer faster and more confidently on test day?
