Points, Vectors, and Functions
Say you want a piece of a pizza, but you don't want the crust. The crust gets stuck in between your teeth, and you have a dentist appointment right after lunch.
The pizza has a 6 inch radius with the crust, but the crust is 1/2 inch thick. When you order your slice, you ask for a slice with only the inner 5 1/2 inches in radius. You receive an odd look from the guy behind the counter, but you are given your pizza sans crust.
Like crustless pizza, there are certain situations that are easier to describe with polar inequalities than with rectangular inequalities. Sometimes giving bounds for r and θ is easier than giving bounds for x and y.
Of course we can also go the other way around, starting with inequalities and ending up with a picture.