11-5 Solving Two-Step Inequalities
Warm Up
Problem of the Day
Lesson Presentation
Course 3
11-5 Solving Two-Step Inequalities
Warm Up
Solve.
1. 6x + 36 = 2x
x = –9
2. 4x – 13 = 15 + 5x x = –28
3. 5(x – 3) = 2x + 3
x=6
4. 7 + x = 3
x = – 11
16
8
Course 3
16
11-5 Solving Two-Step Inequalities
Problem of the Day
Find an integer x that makes the following
two inequalities true:
4 < x2 < 16 and x < 2.5
x = –3
Course 3
11-5 Solving Two-Step Inequalities
Learn to solve two-step inequalities and
graph the solutions of an inequality on a
number line.
Course 3
11-5 Solving Two-Step Inequalities
Solving a multistep inequality uses the
same inverse operations as solving a
multistep equation. Multiplying or
dividing the inequality by a negative
number reverses the inequality
symbol.
Course 3
11-5 Solving Two-Step Inequalities
Additional Example 1A: Solving Two-Step
Inequalities
Solve and graph.
4x + 1 > 13
4x + 1 > 13
–1 –1
4x
Course 3
> 12
4x> 12
4
4
x>3
Subtract 1 from both sides.
Divide both sides by 4.
1
2
3
4
5
6
7
11-5 Solving Two-Step Inequalities
Remember!
If both sides of an inequality are
multiplied or divided by a negative
number, the inequality symbol must
be reversed.
Course 3
11-5 Solving Two-Step Inequalities
Additional Example 1B: Solving Two-Step
Inequalities
Solve and graph.
–9x + 7  25
–9x + 7  25
–7
–9x
–7
Subtract 7 from both sides.
 18
–9x  18
–9
–9
x  –2
Course 3
Divide each side by –9;
change  to .
-6
-5
-4
-3
-2
-1
0
11-5 Solving Two-Step Inequalities
Check It Out: Example 1A
Solve and graph.
5x + 2 > 12
5x + 2 > 12
–2 –2
5x
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> 10
5x> 10
5
5
x>2
Subtract 2 from both sides.
Divide both sides by 5.
1
2
3
4
5
6
7
11-5 Solving Two-Step Inequalities
Check It Out: Example 1B
–4x + 2  18
–4x + 2  18
–2
–4x
–2
Subtract 2 from both sides.
 16
–4x  16
–4
–4
x  –4
Course 3
Divide each side by –4;
change  to .
-6
-5
-4
-3
-2
-1
0
11-5 Solving Two-Step Inequalities
Additional Example 2: Solving Inequalities That
Contain Fractions
Solve 2x + 3  9 and graph the solution.
5
4 10
2x + 3  9
5
4 10
9
2x 3
Multiply by LCD, 20.
20 5 + 4  20 10
3
9
2x
20 5
+ 20 4  20 10 Distributive Property.
8x + 15  18
(
()
)
()
()
()
– 15 – 15
8x
 3
Course 3
Subtract 15 from both
sides.
11-5 Solving Two-Step Inequalities
Additional Example 2 Continued
8x  3
8x  3
8
8
Divide both sides by 8.
x 3
8
0
Course 3
3
8
1
11-5 Solving Two-Step Inequalities
Check It Out: Example 2
Solve 3x + 1  5
5
4 10
3x + 1  5
5
4 10
5
3x 1
Multiply by LCD, 20.
20 5 + 4  20 10
1
5
3x
20 5
+ 20 4  20 10 Distributive Property.
12x + 5  10
(
()
)
()
()
()
–5 –5
12x
 5
Course 3
Subtract 5 from both
sides.
11-5 Solving Two-Step Inequalities
Check It Out: Example 2 Continued
12x  5
12x  5
12
12
x 5
12
0
Course 3
Divide both sides by 12.
5
12
11-5 Solving Two-Step Inequalities
Additional Example 3: School Application
A school’s Spanish club is selling bumper
stickers. They bought 100 bumper stickers
for $55, and they have to give the company
15 cents for every sticker sold. If they plan to
sell each bumper sticker for $1.25, how many
do they have to sell to make a profit?
Let R represent the revenue and C represent
the cost. In order for the Spanish club to
make a profit, the revenue must be greater
than the cost.
R>C
Course 3
11-5 Solving Two-Step Inequalities
Additional Example 3 Continued
The revenue from selling x bumper stickers at
$1.25 each is 1.25x. The cost of selling x bumper
stickers is the fixed cost plus the unit cost times
the number of bumper stickers sold, or 55 +
0.15x. Substitute the expressions for R and C.
1.25x > 55 + 0.15x
Course 3
Let x represent the number of
bumper stickers sold. Fixed
cost is $55. Unit cost is 15
cents.
11-5 Solving Two-Step Inequalities
Additional Example 3 Continued
1.25x > 55 + 0.15x
– 0.15x
– 0.15x
1.10x > 55
1.10x
55
>
1.10 1.10
Subtract 0.15x from both
sides.
Divide both sides by 1.10.
x > 50
The Spanish club must sell more than 50
bumper stickers to make a profit.
Course 3
11-5 Solving Two-Step Inequalities
Check It Out: Example 3
A school’s French club is selling bumper
stickers. They bought 200 bumper stickers
for $45, and they have to give the company
25 cents for every sticker sold. If they plan to
sell each bumper sticker for $2.50, how many
do they have to sell to make a profit?
Let R represent the revenue and C represent
the cost. In order for the French club to make
a profit, the revenue must be greater than the
cost.
R>C
Course 3
11-5 Solving Two-Step Inequalities
Check It Out: Example 3 Continued
The revenue from selling x bumper stickers at
$2.50 each is 2.5x. The cost of selling x bumper
stickers is the fixed cost plus the unit cost times
the number of bumper stickers sold, or 45 +
0.25x. Substitute the expressions for R and C.
2.5x > 45 + 0.25x
Course 3
Let x represent the number of
bumper stickers sold. Fixed
cost is $45. Unit cost is 25
cents.
11-5 Solving Two-Step Inequalities
Check It Out: Example 3 Continued
2.5x > 45 + 0.25x
– 0.25x
– 0.25x
2.25x > 45
2.25x
45
>
2.25 2.25
Subtract 0.25x from both
sides.
Divide both sides by 2.25.
x > 20
The French club must sell more than 20 bumper
stickers to make a profit.
Course 3
11-5 Solving
Insert Lesson
Two-Step
Title
Inequalities
Here
Lesson Quiz: Part I
Solve and graph.
1. 4x – 6 > 10
x>4
2. 7x + 9 < 3x – 15
x < –6
3. w – 3w < 32
w > –16
4. 2 w + 1  1
4
3
3
w
8
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2
1
2
3
4
5
6
7
-10 -9
-8
-7
-6
-5 -4
-18 -17 -16
-15 -14 -13 -12
0
3
8
11-5 Solving Two-Step Inequalities
Lesson Quiz: Part II
5. Antonio has budgeted an average of $45
a month for entertainment. For the first five
months of the year he has spent $48, $39,
$60, $48, and $33. How much can Antonio
spend in the sixth month without exceeding
his average budget?
no more than $42
Course 3