How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$?
Since $|m| > 0$, we can clear fractions from the inequalities, arriving at $8 \geq |m|$.  This is satisfied for $-8 \leq m \leq 8$.  There are 17 integers in this range, but 0 is not allowed, so our final answer is $\boxed{16}$.
16
Level 4
Algebra
Problem: How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the inequality $\frac{1}{|m|} \geq \frac{1}{8}$, we first manipulate the inequality to isolate $|m|$. We can rewrite the inequality as $|m| \leq 8$. This means that the absolute value of $m$ must be less than or equal to 8.

Next, we consider the implications of the absolute value. The condition $|m| \leq 8$ translates to two inequalities: $-8 \leq m \leq 8$. However, since we are looking for integers $m$ that are not equal to zero, we need to exclude $m = 0$ from our count.

The integers that satisfy $-8 \leq m \leq 8$ are: $-8, -7, -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7, 8$. 

Counting these integers, we find there are 8 negative integers ($-1$ to $-8$) and 8 positive integers ($1$ to $8$). 

Thus, the total number of integers $m \neq 0$ that satisfy the inequality is $8 + 8 = 16$. 

Therefore, the final answer is $16$.<|END_OF_LATENT|> The final answer is $16$.