At a Glance  Simple Polar Inequalities
Say we want a piece of a pizza, but we don't want the crust. The crust gets stuck in between our teeth, and we have a dentist appointment right after lunch.
The pizza has a 6 inch radius with the crust, but the crust is a half inch thick. When we order our slice, we ask for a slice with only the inner 5.5 inch radius. Naturally, we receive an odd look from the guy behind the counter, but 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.
Example 1
Write inequalities for r and θ describing the shaded region. 
Example 2
Write inequalities for r and θ describing the shaded region.

Example 3
Sketch the region described by the inequalities π ≤ θ ≤ 2π and 1 ≤ r ≤ 2. 
Example 4
Sketch the region described by the inequality r > 2. 
Exercise 1
Write inequalities for r and θ describing the given region.
Exercise 2
Write inequalities for r and θ describing the given region.
Exercise 3
Write inequalities for r and θ describing the given region.
Exercise 4
Sketch the region described by the given inequalities.
Exercise 5
Sketch the region described by the given inequalities.
 r < 4
Exercise 6
Sketch the region described by the given inequalities.
 and 2 ≤ r ≤ 5