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.

## Practice:

Write inequalities for *r* and θ describing the shaded region. PICTURE polar ineqs 1 | |

If we take any point in the shaded region, θ must lie between 0 and $\frac{π}{2}$. PICTURE polar ineqs 2 The appropriate bounds for θ are 0≤θ≤ \frac{π}{2}. Any point in the shaded region must have *r* between 0 and 1. PICTURE polar ineqs 3 The appropriate bounds for *r* are 0≤ r ≤ 1. | |

Write inequalities for *r* and θ describing the shaded region. PICTURE polar ineqs 4
| |

Since the region wraps all the way around the origin, we have
0≤ θ≤ 2π
Actually, any value of θ is fine, but it's tidier to give bounds.
The value of *r* goes from 3 to 4 inside the region, therefore
3≤ r≤ 4. PICTURE polar ineqs 5
| |

Sketch the region described by the inequalities
π ≤ θ ≤ 2 π
1 ≤ r ≤ 2.
| |

If θ is between π and 2π then the region in question is contained in the third and fourth quadrants.
PICTURE polar ineqs 10
Since *r* must be between 1 and 2, we shade in the portion of these quadrants which is at least 1 but not more than 2 from the origin:
PICTURE polar ineqs 11
| |

Sketch the region described by the inequality
r>2. | |

Since no bounds are given for θ, the region will wrap all the way around the origin.
The condition r > 2 describes the region outside a circle of radius 2, not including the boundary of that circle.
PICTURE graph. Dashed line at circle *r* = 2, shade outside. | |

Write inequalities for *r* and θ describing the given region.

- PICTURE polar ineqs 6 without pink

Answer

- We have $\frac{π}{4}≤ θ ≤ \frac{3π}{4}$:PICTURE polar ineqs 6 with pink
Within those values of θ the value of
*r* can be anything nonnegative, the appropriate bound is
0≤ r.
It would be possible to describe this region in a way that allowed *r* to be negative, but there's no need to
go to the extra trouble.

Write inequalities for *r* and θ describing the given region.

Answer

- The value of
*r* can be anything greater than 2, therefore
2≤ r.
Since the region wraps all the way around, bounds for θ are optional. We could say
0≤ θ≤ 2π
or we could decline to give any bounds for θ.

Write inequalities for *r* and θ describing the given region.

Answer

- Counting the rays and the rings, we see that\frac{7π}{6}≤θ≤\frac{4π}{3}
and
2≤ r≤ 3.
PICTURE polar ineqs 9
{enumerate}

Sketch the region described by the given inequalities.

Answer

- Since no bounds are given for
*r*, this region will extend outward infinitely. The conditions on θ include all of the third quadrant.
PICTURE: graph region

Sketch the region described by the given inequalities.

Answer

- The condition r < 4 describes the inside of a circle of radius 4, not including the boundary of the circle. Since no bounds are given for θ
the region wraps all the way around the origin.
PICTURE: graph region r < 4.

Sketch the region described by the given inequalities.

- \frac{11π}{6}≤ θ ≤ 2π and 2≤ r ≤ 5

Answer

- The conditions on θ describe an infinite pizza-slice-like wedge lying below the positive
*x*-axis:
PICTURE: graph infinite pizza wedge described by $\frac{11π}{6}≤ θ ≤ 2π$
The conditions on *r* say we want to take a pizza-crust-shaped portion of that wedge:
PICTURE: graph region