To solve the inequality \( \frac{54}{k} > 10 \), we can first isolate \( k \).
- Start by multiplying both sides by \( k \) (note: we need to consider the sign of \( k \)):
- If \( k > 0 \), we have \( 54 > 10k \).
- If \( k < 0 \), the inequality sign would flip (which would lead to a contradiction since \( 54 \) is positive).
So, we can work under the assumption that \( k > 0 \): \[ 54 > 10k \]
- Now, divide both sides by 10: \[ \frac{54}{10} > k \] \[ 5.4 > k \] or \[ k < 5.4 \]
Thus, the solutions to the inequality \( \frac{54}{k} > 10 \) are all positive values of \( k \) that are less than 5.4.
In interval notation, the solution set is: \[ (0, 5.4) \]
Multiplying both sides by a negative \( k \) would give us \( k < 0 \) which is not valid in this case since it would imply a negative left side and would contradict with the positive 54 on the left side initially.
Therefore, any \( k \) within \( (0, 5.4) \) is a valid solution.