There are many examples. A few are listed below.
Work Step by Step
The reading on the speedometer of a car on the freeway is something that you could model with $ x \geq 0$, for the car will never read a negative velocity, but the velocity could read 0. You could also model the reading on a scale with $ x \geq 0$, for a scale will never read a negative number but could read 0 if nothing is on it. You can model the price of food at a store with $x\gt 0$, for no matter how cheap the food is, it is not going to be free. You could also use $x\gt 0$ to describe the time after a race starts, for once the race starts, time will always be greater than 0.