Step 1 :First, we can write the system of inequalities based on the problem. The total hours Michael can work is at most 17, so \(x + y \leq 17\). He must earn a minimum of $280, so \(19x + 14y \geq 280\).
Step 2 :Next, we can graph these inequalities. The line \(x + y = 17\) is the boundary for the hours he can work, and the line \(19x + 14y = 280\) is the boundary for the money he must earn.
Step 3 :The feasible region is the area that satisfies both inequalities. We can find one possible solution by picking a point in this region. For example, if Michael works 10 hours lifeguarding (\(x = 10\)) and 7 hours clearing tables (\(y = 7\)), he will work a total of 17 hours and earn $310, which is above the minimum requirement. The final answer is \(\boxed{x = 10, y = 7}\).