SME
Grade 9 Math · Algebra 1
Solving linear and compound inequalities
Use this for “Solve 3x − 4 ≥ 5x + 6 and graph the solution,” including when the sign flips.
Diagram
x <
x ≤
x >
x ≥
x < −5: every number less than −5.
The open circle leaves out −5.
Test −6: −6 < −5 is true.
This is an example. Type your own values to start a new problem.
a
x
+
(
b
)
c
x
+
d
x
k
t
Test number
example
h
Is the test true?
worked out (not typed)
Formulas
k
=
d
−
b
a
−
c
−5
=
6 − (−4)
3 − 5
sign
f
from
sign
s
,
flipped
when
a
−
c
is
negative
sign < from sign >, flipped when 3 − 5 is negative
test
t
in
ax
+
b
(sign
s
)
cx
+
d
:
h
test −6 in 3x + (−4) (sign >) 5x + 6: true
Assumptions
•
Dividing or multiplying both sides by a negative number reverses the sign.
•
< and > leave the bound out (an open circle); ≤ and ≥ put it in (a closed circle).
•
A test number from the shaded side makes the first inequality true.
Step-by-step
WE KNOW
d = 6, a = 3, b = −4, c = 5, s: >, t = −6
FIND
bound (k), sign of the answer (f), is the test true? (h)
Step 1 · Find bound (k)
Use k = (d − b) ÷ (a − c)
Take
cx
and
b
from
both
sides,
as
in
an
equation,
then
divide
by
the
x
left.
k
=
6 − (−4)
3 − 5
Take 5x from both sides: −2x − 4 > 6
Add 4 to both sides: −2x > 10
Divide both sides by −2, a negative, so > flips to <: x < −5
k
=
−5
Step 2 · Find sign of the answer (f)
Use sign f from sign s, flipped when a − c is negative
a
−
c
is
negative:
dividing
by
it
flips
>
to
<.
f
:
<
Step 3 · Find is the test true? (h)
Use test t in ax + b (sign s) cx + d: h
Put the test number into both sides of the first inequality: is it true?
3(−6) − 4 = −22
5(−6) + 6 = −24
−22 > −24 is true
h
=
true
Check
Put the numbers back in. Both sides should match.
−5
=
6 − (−4)
3 − 5
✓
3 < 5, so > flips to < ✓
−22 > −24, so the test number is a solution ✓
Topic check
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More problem types
Compound inequality with and
Use this for “Solve −5 < 3x + 4 ≤ 13” (the x term positive).
Compound inequality with or
Use this for “Solve 2x + 3 < −1 or 3x − 2 ≥ 7.”
Graph y (sign) mx + b
Use this for “Graph y ≥ 2x − 3 and test a point”, or with <, ≤ or >.
Whole-number answers
Use this for “$12 to join plus $4 a visit, and at most $50. How many visits?”