Download Holt McDougal Algebra 1 2-4

Solving Two-Step and
2-4 Multi-Step Inequalities
Objective
Solve inequalities that contain more than one
operation.
Holt McDougal Algebra 1
Solving Two-Step and
2-4 Multi-Step Inequalities
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
Holt McDougal Algebra 1
Solving Two-Step and
2-4 Multi-Step Inequalities
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
Holt McDougal Algebra 1
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4
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Solving Two-Step and
2-4 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
Holt McDougal Algebra 1
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4
6
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Solving Two-Step and
2-4 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
Holt McDougal Algebra 1
–8
–4
0
Solving Two-Step and
2-4 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
Holt McDougal Algebra 1
–16
–12
–8
–4
0
Solving Two-Step and
2-4 Multi-Step Inequalities
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
Holt McDougal Algebra 1
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Solving Two-Step and
2-4 Multi-Step Inequalities
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
Holt McDougal Algebra 1
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Solving Two-Step and
2-4 Multi-Step Inequalities
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
Holt McDougal Algebra 1
Since 3 is added to 4f, subtract 3
from both sides to undo the
addition.
Solving Two-Step and
2-4 Multi-Step Inequalities
Example 2C Continued
4f > –1
Since f is multiplied by 4, divide both
sides by 4 to undo the
multiplication.
0
Holt McDougal Algebra 1
Solving Two-Step and
2-4 Multi-Step Inequalities
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
Holt McDougal Algebra 1
Solving Two-Step and
2-4 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
Holt McDougal Algebra 1
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2
4
6
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Solving Two-Step and
2-4 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
Holt McDougal Algebra 1
Since 2 is subtracted from 3x,
add 2 to both sides to undo
the subtraction.
Solving Two-Step and
2-4 Multi-Step Inequalities
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
Holt McDougal Algebra 1
6
8
10
Solving Two-Step and
2-4 Multi-Step Inequalities
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 is 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
<
Holt McDougal Algebra 1
daily
cost at
We Got
Wheels
38
plus
+
$0.20
per mile
0.20
times
# of
miles.

m
Solving Two-Step and
2-4 Multi-Step Inequalities
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.
Holt McDougal Algebra 1
Solving Two-Step and
2-4 Multi-Step Inequalities
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 
Holt McDougal Algebra 1
Solving Two-Step and
2-4 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
+
Holt McDougal Algebra 1
x)
divided
by

number
of scores
2
is greater
than or
equal to
≥
total
score
90
Solving Two-Step and
2-4 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.
Holt McDougal Algebra 1
Solving Two-Step and
2-4 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

Holt McDougal Algebra 1
90.5 ≥ 90 