Final answer:
The inequality \(\frac{3}{m} \le -\frac{2}{11}\) is true for 16 negative integers, specifically for the integers -16 through -1 inclusive.
Step-by-step explanation:
To determine for how many integers m the inequality \(\frac{3}{m} \le -\frac{2}{11}\) is true, we need to find the set of integer values of m that satisfy the inequality.
Since we are dealing with an inequality that involves a variable in the denominator, we need to consider the sign of m to maintain the direction of the inequality. For the fraction \(\frac{3}{m}\) to be negative, m must be a negative integer. Multiplying both sides of the inequality by m (which is negative), the inequality sign reverses, giving us 3 \ge -\frac{2}{11}m. Multiplying both sides by -11 (which is also negative), we reverse the inequality sign again, which results in -33 \le 2m.
To find the final solution, we divide both sides by 2, yielding -16.5 \le m. Since m must be an integer, we take m \ge -16 as our solution. However, we must remember that m is negative, so we only consider integers less than zero. Thus, the possible integer values for m are: -1, -2, -3, ..., -16. Counting these, we find there are 16 integers that satisfy the inequality.