SME
Grade 9 Math · Problem type
System of inequalities in standard form
Use this for “Graph x − 2y < 2 and 2x + y ≤ 4 and test (−6, 2).”
Chart
Where both sides are shaded, both inequalities are true: those points are the solutions.
A dashed line is left out; a solid line is included.
The boundaries cross at (2, 0)
Test (−6, 2)
1/2 × (−6) − 1 = −4, and 2 > −4 is true
−2 × (−6) + 4 = 16, and 2 ≤ 16 is true
It is a solution.
This is an example. Type your own values to start a new problem.
a
x
+
(
b
)
y
c
d
x
+
e
y
f
m₁
First slope
worked out (not typed)
b₁
First y-intercept
worked out (not typed)
m₂
Second slope
worked out (not typed)
b₂
Second y-intercept
worked out (not typed)
x₀
Test point x
example
y₀
Test point y
example
h1
First inequality true?
worked out (not typed)
h2
Second inequality true?
worked out (not typed)
Formulas
test
(
x₀
,
y₀
)
in
ax
+
by
(sign
s
)
c
:
h1
test (−6, 2) in 1x + (−2)y (sign <) 2: true
test
(
x₀
,
y₀
)
in
dx
+
ey
(sign
t
)
f
:
h2
test (−6, 2) in 2x + 1y (sign ≤) 4: true
m₁
=
−
a
b
1
2
=
−
1
−2
b₁
=
c
b
−1
=
2
−2
m₂
=
−
d
e
−2
=
−
2
1
b₂
=
f
e
4
=
4
1
Assumptions
•
Solve each for y: divide by the y term, flipping the sign when it is negative.
•
Dashed lines (< or >) are left out; solid ones (≤ or ≥) are included.
•
The solutions are where the two shadings overlap.
Step-by-step
WE KNOW
a = 1, b = −2, s: <, c = 2, d = 2, e = 1, t: ≤, f = 4, x₀ = −6, y₀ = 2
FIND
first inequality true? (h1), second inequality true? (h2), first slope (m₁), first y-intercept (b₁), second slope (m₂), second y-intercept (b₂)
Step 1 · Find first inequality true? (h1)
Use test (x₀, y₀) in ax + by (sign s) c: h1
Put the test point into the first inequality: is it true?
1(−6) − 2(2) = −10
−10 < 2 is true
h1 = true
Step 2 · Find second inequality true? (h2)
Use test (x₀, y₀) in dx + ey (sign t) f: h2
And into the second: the point is a solution when both are true.
2(−6) + 1(2) = −10
−10 ≤ 4 is true
h2 = true
Step 3 · Find first slope (m₁)
Use m₁ = −a ÷ b
Take
ax
from
both
sides,
then
divide
by
b
:
the
slope
is
−
a
b
.
m₁
=
−
1
−2
m₁
=
1
2
Step 4 · Find first y-intercept (b₁)
Use b₁ = c ÷ b
b
is
negative:
dividing
by
it
flips
<.
b₁
=
2
−2
b₁
=
−1
→
y
>
1/2x
−
1
Step 5 · Find second slope (m₂)
Use m₂ = −d ÷ e
The
same
for
the
second:
the
slope
is
−
d
e
.
m₂
=
−
2
1
m₂
=
−2
Step 6 · Find second y-intercept (b₂)
Use b₂ = f ÷ e
Divide
by
e
;
e
is
positive,
so
the
sign
stays.
b₂
=
4
1
b₂
=
4
→
y
≤
−2x
+
4
Check
Put the numbers back in. Both sides should match.
−10 < 2, so the test point makes the first inequality true ✓
−10 ≤ 4, so the test point makes the second inequality true ✓
1
2
=
−
1
−2
✓
−1
=
2
−2
✓
−2
=
−
2
1
✓
4
=
4
1
✓
Topic check
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