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3-4
Solving Two-Step and
Multi-Step Inequalities
Warm Up
Lesson Presentation
Lesson Quiz
3-4
Solving Two-Step and
Multi-Step Inequalities
Warm Up
Solve each equation.
1. 2x – 5 = –17 –6
2.
14
Solve each inequality and graph the
solutions.
3. 5 < t + 9 t > –4
4.
a ≤ –8
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Solving Two-Step and
Multi-Step Inequalities
Sunshine State Standards
MA.912.A.3.5 Symbollically represent and
solve multi-step and real-world applications
that involve linear…inequalities.
Also MA.912.A.3.4, MA.912.A.10.2.
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Solving Two-Step and
Multi-Step Inequalities
Objective
Solve inequalities that contain more than one
operation.
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Solving Two-Step and
Multi-Step Inequalities
Inequalities that contain more than one
operation require more than one step to solve.
Use inverse operations to undo the operations
in the inequality one at a time.
Solving Two-Step and
Multi-Step Inequalities
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Additional Example 1A: Solving Multi-Step
Inequalities
Solve the inequality and graph the solutions.
45 + 2b > 61
45 + 2b > 61
–45
–45
Since 45 is added to 2b,
subtract 45 from both sides
to undo the addition.
2b > 16
b>8
0
2
4
6
Since b is multiplied by 2, divide
both sides by 2 to undo the
multiplication.
8 10 12 14 16 18 20
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 1B: Solving Multi-Step
Inequalities
Solve the inequality and graph the solutions.
8 – 3y ≥ 29
8 – 3y ≥ 29
–8
–8
Since 8 is added to –3y, subtract
8 from both sides to undo the
addition.
–3y ≥ 21
Since y is multiplied by –3,
divide both sides by –3 to
undo the multiplication.
Change ≥ to ≤.
y ≤ –7
–7
–10 –8 –6 –4 –2
0
2
4
6
8 10
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 1a
Solve the inequality and graph the solutions.
–12 ≥ 3x + 6
–12 ≥ 3x + 6
–6
–6
Since 6 is added to 3x, subtract 6
from both sides to undo the
addition.
–18 ≥ 3x
Since x is multiplied by 3, divide
both sides by 3 to undo the
multiplication.
–6 ≥ x
–10 –8 –6 –4 –2
0
2
4
6
8 10
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 1b
Solve the inequality and graph the solutions.
Since x is divided by –2, multiply
both sides by –2 to undo the
division. Change > to <.
x + 5 < –6
–5 –5
Since 5 is added to x, subtract 5
from both sides to undo the
addition.
x < –11
–11
–20
–16
–12
–8
–4
0
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 1c
Solve the inequality and graph the solutions.
1 – 2n ≥ 21
–1
–1
–2n ≥ 20
Since 1 – 2n is divided by 3,
multiply both sides by 3 to
undo the division.
Since 1 is added to −2n, subtract
1 from both sides to undo the
addition.
Since n is multiplied by −2, divide
both sides by −2 to undo the
multiplication. Change ≥ to ≤.
n ≤ –10
–10
–20
–16
–12
–8
–4
0
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Solving Two-Step and
Multi-Step Inequalities
To solve more complicated inequalities, you
may first need to simplify the expressions on
one or both sides by using the order of
operations, combining like terms, or using the
Distributive Property.
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 2A: Simplifying Before Solving
Inequalities
Solve the inequality and graph the solutions.
2 – (–10) > –4t
12 > –4t
Combine like terms.
Since t is multiplied by –4, divide
both sides by –4 to undo the
multiplication. Change > to <.
–3 < t (or t > –3)
–3
–10 –8 –6 –4 –2
0
2
4
6
8 10
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 2B: Simplifying Before Solving
Inequalities
Solve the inequality and graph the solutions.
–4(2 – x) ≤ 8
−4(2 – x) ≤ 8
−4(2) − 4(−x) ≤
8
–8 + 4x ≤ 8
+8
+8
4x ≤ 16
Distribute –4 on the left side.
Since –8 is added to 4x, add 8 to
both sides.
Since x is multiplied by 4, divide
both sides by 4 to undo the
multiplication.
x≤4
–10 –8 –6 –4 –2
0
2
4
6
8 10
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 2C: Simplifying Before Solving
Inequalities
Solve the inequality and graph the solutions.
Multiply both sides by 6, the LCD of
the fractions.
Distribute 6 on the left side.
4f + 3 > 2
–3 –3
4f
> –1
Since 3 is added to 4f, subtract 3
from both sides to undo the
addition.
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 2C Continued
4f > –1
Since f is multiplied by 4, divide both
sides by 4 to undo the
multiplication.
0
Solving Two-Step and
Multi-Step Inequalities
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Check It Out! Example 2a
Solve the inequality and graph the solutions.
2m + 5 > 52
2m + 5 > 25
–5>–5
2m
> 20
m > 10
0
2
4
6
Simplify 52.
Since 5 is added to 2m, subtract 5
from both sides to undo the
addition.
Since m is multiplied by 2, divide
both sides by 2 to undo the
multiplication.
8 10 12 14 16 18 20
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 2b
Solve the inequality and graph the solutions.
3 + 2(x + 4) > 3
Distribute 2 on the left side.
3 + 2(x + 4) > 3
3 + 2x + 8 > 3
Combine like terms.
Since 11 is added to 2x, subtract
11 from both sides to undo the
addition.
2x + 11 > 3
– 11 – 11
2x
> –8
Since x is multiplied by 2, divide
both sides by 2 to undo the
multiplication.
x > –4
–10 –8 –6 –4 –2
0
2
4
6
8 10
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 2c
Solve the inequality and graph the solutions.
Multiply both sides by 8, the LCD
of the fractions.
Distribute 8 on the right side.
5 < 3x – 2
+2
+2
7 < 3x
Since 2 is subtracted from 3x,
add 2 to both sides to undo
the subtraction.
Solving Two-Step and
Multi-Step Inequalities
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Check It Out! Example 2c Continued
Solve the inequality and graph the solutions.
7 < 3x
Since x is multiplied by 3, divide both
sides by 3 to undo the multiplication.
0
2
4
6
8
10
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 3: Application
To rent a certain vehicle, Rent-A-Ride charges $55.00
per day with unlimited miles. The cost of renting a
similar vehicle at We Got Wheels is $38.00 per day plus
$0.20 per mile. For what number of miles in the cost at
Rent-A-Ride less than the cost at We Got Wheels?
Let m represent the number of miles. The cost for
Rent-A-Ride should be less than that of We Got
Wheels.
Cost at
Rent-ARide
must be
less
than
55
<
daily
cost at
We Got
Wheels
38
plus
+
$0.20
per mile
0.20
times
# of
miles.

m
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 3 Continued
55 < 38 + 0.20m
Since 38 is added to 0.20m, subtract
55 < 38 + 0.20m
38 from both sides to undo the
addition.
–38 –38
17 < 0.20m
Since m is multiplied by 0.20, divide
both sides by 0.20 to undo the
multiplication.
85 < m
Rent-A-Ride costs less when the number of miles is
more than 85.
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Solving Two-Step and
Multi-Step Inequalities
Additional Example 3 Continued
Check
Check the endpoint, 85.
Check a number greater
than 85.
55 = 38 + 0.20m
55 < 38 + 0.20m
55
38 + 0.20(85)
55 < 38 + 0.20(90)
55
55
38 + 17
55 
55 < 38 + 18
55 < 56 
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 3
The average of Jim’s two test scores must
be at least 90 to make an A in the class.
Jim got a 95 on his first test. What grades
can Jim get on his second test to make an
A in the class?
Let x represent the test score needed. The
average score is the sum of each score divided
by 2.
First
test
score
(95
plus
+
second
test
score
x)
divided
by

number
of scores
2
is greater
than or
equal to
≥
total
score
90
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 3 Continued
Since 95 + x is divided by 2, multiply
both sides by 2 to undo the division.
95 + x ≥ 180
–95
–95
Since 95 is added to x, subtract 95 from
both sides to undo the addition.
x ≥ 85
The score on the second test must be 85 or higher.
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Solving Two-Step and
Multi-Step Inequalities
Check It Out! Example 3 Continued
Check
Check the end point,
85.
Check a number greater
than 85.
90
90
90
90

90.5 ≥ 90 
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1. 13 – 2x ≥ 21 x ≤ –4
2. –11 + 2 < 3p
p > –3
3. 23 < –2(3 – t)
t>7
4.
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quiz: Part II
5. A video store has two movie rental plans. Plan
A includes a $25 membership fee plus $1.25 for
each movie rental. Plan B costs $40 for
unlimited movie rentals. For what number of
movie rentals is plan B less than plan A?
more than 12 movies
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quiz for Student Response Systems
1. Identify the solution for the given
inequality.
26 + 2a ≥ 12
A. a ≤ –7
B. a ≥ –7
C. a < –7
D. a < –7
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quiz for Student Response Systems
2. Identify the solution for the given
inequality.
15 + 5s < 10
A.
s < -1
B.
s > -1
C.
s<1
D.
s≤1
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quiz for Student Response Systems
3. Identify the solution for the given
inequality.
34 > 3(3x)
A.
x > 75
B.
x > 87
C.
x>9
D.
x > 25
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quiz for Student Response Systems
4. Identify the solution for the given
inequality.
A.
C.
B.
D.
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Solving Two-Step and
Multi-Step Inequalities
Lesson Quiz for Student Response Systems
5. A cell phone provider offers two plans.
Plan A includes a fee of $25 per month
and charges $0.25 for every call. Plan B
includes a fee of $40 per month and
offers unlimited calls. For what number
of phone calls is the cost of plan B less
than the cost of plan A?
A. more than 100 calls
C. more than 60 calls
B. more than 120 calls
D. more than 160 calls