Test a Point

The solution of an inequality in two variables is a region — and the sign does not tell you which side.

MATHEMATICS 10 MATATAG · Key Stage 3
CG Quarter 1 · competencies 9–10
Term 1 · Week 7

The routine — both steps, every time

1Draw the BOUNDARY — replace the sign with = and graph it.

2TEST A POINT not on the boundary. TRUE → shade that side. FALSE → shade the other side.

< >

Broken — the boundary
points are not solutions

≤ ≥

Solid — the boundary
points are solutions

One boundary, two sides — the test decides

y > x² − 4

0 > −4 TRUE → shade this side

y < x² − 4

0 < −4 FALSE → shade the other side

Same curve, same test point, opposite answers. Nothing but the substitution separates them.

These two are the same inequality

y > x² − 4

at (0, 0): 0 > −4 TRUE

x² − y < 4

at (0, 0): 0 < 4 TRUE

One says greater and one says less, and it makes no difference at all — moving y across the sign is what flipped it. The sign depends on how the inequality happens to be written. The region does not.

Is a point in, out, or on?

Substitute. For y ≥ x² − 4:

Point	Substitution	Where	Solution?
(0, 0)	0 ≥ −4	IN	yes
(2, 0)	0 ≥ 0	ON	yes — ≥ includes it
(1, −5)	−5 ≥ −3	OUT	no

ON is not a third region. It is the edge — and whether it counts depends only on whether the boundary is broken or solid.

The origin is not always available

y > −x y < x² y ≥ 3x y ≤ −2x

Every one of these boundaries passes through (0, 0). Substituting the origin gives 0 on both sides — a false statement, no warning, and a confidently shaded wrong half.
If both sides come out 0, the origin is on the boundary. Pick another point.

Don’t decide which side. Test a point.

MATATAG Mathematics CG (DepEd, August 2023), p. 65
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