How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$?

Case 1: m>0. Then, m<=8 so there are 8 values.

 

Case 2: m<0. Then, there are also 8 values, the negatives of the values in Case 1.

 

So, there 16 values total.