The correct real-world problem that can be modeled using the inequality \( h + 150 \geq 350 \) is:
An on-the-job trainee is required to work at least 350 hours. So far, Jerry has worked 150 hours. How many more hours must he work in order to fulfill the requirement?
This problem correctly reflects the nature of the inequality because it indicates that Jerry's total hours worked (150 hours plus the additional hours \( h \)) must be at least (≥) 350 hours in order to meet the requirement.