How Curves Divide the Plane into Regions

Try this

In many problems $x$ and $y$ are related not by an equation $F (x, y) = 0$ but by an inequality $F (x, y) > 0$ or $F (x, y) < 0$ . Such inequalities describe regions of the plane.

Choose “Line”, $F = 2 x - y + 1$ . Drag $Q$ to each side of the line and pin a point on each. Are the two markers the same? What is $F$ when $Q$ is exactly on the line?
Wander around on one side of the line. The value of $F$ changes, but does its sign ever change?
Choose “Circle”. What is the sign inside the circle, and outside?
Choose “Hyperbola”. Into how many pieces does it cut the plane? How are the signs of the three pieces arranged?
Choose “Two lines”, $F = (x - y) (x + y - 2)$ . It cuts the plane into four pieces. How does the sign change when you move into a neighboring piece? And when you go through the crossing point into the opposite piece?

Put these observations together and you have the basic fact of this section:

A curve $F (x, y) = 0$ cuts the plane into a number of pieces, and $F$ has the same sign throughout each piece.

The reason (for a continuous $F$ , such as a polynomial): within one piece you can walk from one point to another without touching the curve. If $F$ changed sign along the way, you would have to pass a point where $F = 0$ , and that point is on the curve: a contradiction. So to find the sign on a region, just test one point inside it (the origin is a popular choice). This is the test-point method.

Predict first

Are the points $(2, 3)$ and $(- 1, - 1)$ on the same side or on opposite sides of the line $2 x - y + 1 = 0$ ?

Same sideOpposite sidesOne of them is on the line

Choose an answer first; then you can see the result.

Why

For a line, “same side, same sign” can be computed directly, without relying on continuity.

Why F has one sign on each side of a line (F = 2x − y + 1)

Step 1 / 5

Take any point
Take a point not on the line, $Q (1, 1)$ : $F (Q) = 2 > 0$ .

$F (1, 1) = 2 - 1 + 1 = 2$
Find the point of the line straight above it
Subtract
Same side, same sign
Check

The region described by a system of inequalities is the common part of the regions of the individual inequalities. For example, $x^{2} + y^{2} \leq 9$ together with $2 x - y + 1 \geq 0$ is the intersection of a disc and a half-plane; the next stop looks for extreme values in regions just like this one.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The sign at one point is the sign of the whole region

The parabola $y^{2} - 4 x = 0$ . Test $(1, 0)$ inside the opening: $F = - 4 < 0$ ; test $(- 2, 0)$ outside: $F = 8 > 0$ . Each one speaks for a whole piece.

Practice

Exercise 1Basic

Of the points $(0, 0)$ , $(3, 1)$ , $(- 2, 4)$ , $(1, - 1)$ , how many are on the same side of the line $3 x + 2 y - 4 = 0$ as the point $(2, 2)$ ?

Exercise 2Basic

The point $(a, 1)$ is inside the circle $x^{2} + y^{2} - 4 x = 0$ . How many integers $a$ satisfy this?

Exercise 3Advanced

How many lattice points (points with integer coordinates) lie in the region given by $x^{2} + y^{2} \leq 4$ and $y \geq ∣ x ∣$ ?

Exercise 4Challenge

The points $A (1, 2)$ and $B (3, - 1)$ are on opposite sides of the line $y = k x$ . The possible values of $k$ form an open interval. Find its length.

Try this

Why

Why F has one sign on each side of a line (F = 2x − y + 1)

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

“Above the line means $>0$”

Testing with the origin when the origin is on the curve