Question

Which of the following are solutions to the inequality below? Select all that apply.

–41 ≥ –48/q

Answer 1

To solve the inequality \(-41 \ge -\frac{48}{q}\), we can start by multiplying both sides by \(-1\). Remember that when multiplying or dividing by a negative number, we must reverse the inequality sign:

\[
41 \le \frac{48}{q}
\]

Next, we can rearrange the inequality:

\[
\frac{48}{q} \ge 41
\]

To eliminate the fraction, we can multiply both sides by \(q\). However, we must consider the sign of \(q\) because it affects the inequality.

### Case 1: \(q > 0\)

If \(q > 0\), we can multiply both sides by \(q\) without changing the direction of the inequality:

\[
48 \ge 41q
\]

Dividing both sides by 41 yields:

\[
q \le \frac{48}{41}
\]

### Case 2: \(q < 0\)

If \(q < 0\), then when we multiply both sides by \(q\), we reverse the inequality:

\[
48 \le 41q
\]

Dividing both sides by 41 yields:

\[
q \ge \frac{48}{41}
\]

Given that \(\frac{48}{41} \approx 1.17\), this means for \(q < 0\), \(q\) would be greater than 1.17, which contradicts \(q < 0\). Therefore, there are no valid solutions in this case.

### Conclusion

The only valid range for \(q\) is:

\[
q \le \frac{48}{41}
\]

This means \(q\) can take any value less than or equal to approximately 1.17, as long as \(q > 0\).

Therefore, the solutions to the inequality \( -41 \ge -\frac{48}{q} \) are all \(q\) values in the range \( (0, \frac{48}{41}] \).