SME
Grade 9 Math · Problem type
Graph y (sign) mx + b
Use this for “Graph y ≥ 2x − 3 and test a point”, or with <, ≤ or >.
Chart
y ≥ 2x − 3
Shaded above: every point where y is at least 2x − 3.
Solid: the line itself is included
Slope m = 2: up 2 for every 1 across
Intercept b = −3: it crosses the y-axis at (0, −3)
Test (1, 0): 2 × 1 − 3 = −1, and 0 ≥ −1 is true
It is a solution: it lies in the shaded part.
Height of the line at x₀: −1
Is the test true?: true
This is an example. Type your own values to start a new problem.
y
m
x
+
(
b
)
x₀
Test point x
example
y₀
Test point y
example
yₗᵢₙₑ
Height of the line at x₀
worked out (not typed)
h
Is the test true?
worked out (not typed)
Formulas
yₗᵢₙₑ
=
m
×
x₀
+
b
−1 = 2 × 1 + (−3)
test:
y₀
(sign
s
)
yₗᵢₙₑ
gives
h
test: 0 (sign ≥) −1 gives true
Assumptions
•
The boundary is y = mx + b: dashed for < and > (left out), solid for ≤ and ≥ (included).
•
Shade above the line for > and ≥, below it for < and ≤.
•
A test point is a solution when its y is in the shaded part at its x.
Step-by-step
WE KNOW
m = 2, b = −3, s: ≥, x₀ = 1, y₀ = 0
FIND
height of the line at x₀ (yₗᵢₙₑ), is the test true? (h)
Step 1 · Find height of the line at x₀ (yₗᵢₙₑ)
Use yₗᵢₙₑ = m × x₀ + b
The
boundary’
s
height
at
the
test
point’
s
x.
yₗᵢₙₑ = 2 × 1 − 3
yₗᵢₙₑ = −1
Step 2 · Find is the test true? (h)
Use test: y₀ (sign s) yₗᵢₙₑ gives h
Compare
the
test
point’
s
y
with
the
line’
s
:
true
means
it
is
in
the
shaded
part.
0 ≥ −1 is true
h
=
true
Check
Put the numbers back in. Both sides should match.
−1 = 2 × 1 − 3 ✓
0 ≥ −1, so the test point is a solution ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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