SME
Grade 7 Math · Problem type
Two-step inequalities
Use this for “Solve −2x + 1 ≤ 7 and graph it” and “What is the least whole number x with 4x > 19?”
Diagram
<
≤
>
≥
−2x + 1 ≤ 7
Take 1 from both sides
−2x ≤ 6
Divide both sides by −2: the sign flips
x ≥ −3: every number greater than or equal to −3.
The closed circle takes in −3.
Test 1: −2 × 1 + 1 = −1, and −1 ≤ 7 is true.
This is an example. Type your own values to start a new problem.
p
x
+
q
r
b
Bound
worked out (not typed)
t
Test number
example
h
Is the test true?
worked out (not typed)
Formulas
p
×
b
+
q
=
r
(times
x
×
bound
+
added
=
right
side)
−2 × (−3) + 1 = 7
x
=
t
in
px
+
q
s
r
:
h
(test
number
put
in
for
x:
is
the
test
true?)
x = 1 in −2x + 1 ≤ 7: true
Assumptions
•
Adding or taking the same number from both sides keeps an inequality true.
•
Dividing both sides by a negative number flips the sign (< becomes >).
•
A test number is a solution when it makes the written inequality true.
•
The least whole number above a bound is the next whole number up.
Step-by-step
WE KNOW
p = −2 (times x), q = 1 (added), r = 7 (right side), s: ≤, t = 1 (test number)
FIND
bound (b), is the test true? (h)
Step 1 · Find bound (b)
Use p × b + q = r (times x × bound + added = right side)
Take
q
from
both
sides,
then
divide
both
sides
by
p
.
−2x + 1 ≤ 7
Take 1 from both sides: −2x ≤ 6
Divide both sides by −2, a negative, so ≤ flips to ≥: x ≥ −3
The least whole number with x ≥ −3 is −3
b
=
−3
Step 2 · Find is the test true? (h)
Use x = t in px + q s r: h (test number put in for x: is the test true?)
Put the test number in for x: is the inequality true?
Put x = 1 in −2x + 1 ≤ 7
−2 × 1 = −2
(−2) + 1 = −1
−1 ≤ 7 is true
h
=
true
(x
=
1
is
a
solution)
Check
Put the numbers back in. Both sides should match.
−2 × (−3) + 1 = 7 ✓
−2 × 1 + 1 = −1, −1 ≤ 7 ✓
Topic check
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Two-step equations and inequalities
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