173k views
3 votes
For how many integers $m$ is this inequality true? \[\dfrac{3}{m} \le -\dfrac2{11}\]

User Palma
by
8.0k points

1 Answer

3 votes

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.

User Emre Efendi
by
6.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories