Don't Decide
Which Side

Test a point

Week 7 · Day 1 CG Quarter 1 · competencies 9–10 45 minutes

Last week: solve x² − 4 > 0

x < −2  or  x > 2

One variable. A line was enough.

Same, or different?

y > x² − 4

x² − y < 4

Take a vote. Write both counts on the board. Do not settle it.

Two variables now.
The answer cannot live on a line — it lives on the whole plane.

Step 1 — the boundary. Replace the sign with =

y = x² − 4

Vertex	(0, −4)
Crosses x at	−2 and 2
Sign is >	broken curve

Broken for < and >. Solid for ≤ and ≥.

The curve cuts the plane in two.
Which side is the answer?

Someone will say “above”. Maybe. Let us not decide — let us test.

Test (0, 0) in  y > x² − 4

0 > 0² − 4

0 > −4  TRUE

True → shade the side the point is on. We did not reason about the sign. We asked one point.

Now the other one:  x² − y < 4

What is the boundary?	x² − y = 4, so y = x² − 4
Is that a new curve?	No — the same one
Broken or solid?	Broken — the sign is <
Test (0, 0)	0² − 0 = 0, and 0 < 4 TRUE
So which side?	The same side

They are the same inequality

Inequality	Test at (0, 0)	Result	Region
y > x² − 4	0 > −4	TRUE	the side with the origin
x² − y < 4	0 < 4	TRUE	the side with the origin

One says greater and one says less, and it makes no difference at all.

The sign does not decide the side.
The test point decides the side.

Test (0, −10) in both. Show your substitution.

y > x² − 4	−10 > −4  FALSE
x² − y < 4	10 < 4  FALSE

Circulate with one question

“Where is your substitution written?”

1Draw the BOUNDARY. Replace the sign with = and graph it. Broken for < and >. Solid for ≤ and ≥.

2TEST A POINT not on the boundary. True → shade that side. False → shade the other side.

Don't decide which side. Test a point.

“Any point not on the boundary.”

The origin is easiest and it usually works. Tomorrow I will give you one where it does not — and I will not tell you which one.

Exit ticket  —  y ≥ x² − 9

1. Where does the boundary cross the x-axis?   x = ______ and ______
2. Is the boundary broken or solid? ____________
3. Test (0, 0). Show your substitution. ______ TRUE / FALSE
4. Which side is shaded — the side with (0, 0), or the other side?

Quarter sheet. Name at the top. Read item 3 before item 4 — a blank substitution with a correct item 4 is a coin toss that landed well.

Don't decide which side.
Test a point.