3-8 Solving Two-Step Inequalities
Solving
Two-Step
Inequalities
Evaluating
Algebraic
Expressions
Objective:
Solve two-step linear inequalities
and interpret the solution.
3-8 Solving Two-Step Inequalities
Additional Example 1A: Solving Two-Step
Inequalities
Evaluating
Solve
and graph.
Algebraic Expressions
4x + 1 > 13
4x + 1 > 13
–1 –1
4x
> 12
4x> 12
4
4
x>3
Since 1 is added to 4x,
subtract 1 from both sides.
Since x is multiplied by 4,
divide both sides by 4.
1
2
3
4
5
6
7
3-8 Solving Two-Step Inequalities
Additional Example 1A Continued
Evaluating
Check
Algebraic Expressions
According to the graph 4 should be a solution
and 2 should not be a solution.
x>3
?
4>3
Substitute
4 for x.
So 4 is a solution.
x>3
?
2>3 
Substitute
2 for x.
So 2 is not a solution.
3-8 Solving Two-Step Inequalities
Evaluating Algebraic Expressions
Remember!
If both sides of an inequality are
multiplied or divided by a negative
number, the inequality symbol must
be reversed.
3-8 Solving Two-Step Inequalities
Additional Example 1B: Solving Two-Step
Inequalities
Evaluating
Solve
and graph.
Algebraic Expressions
–9x + 7  25
–9x + 7  25
–7
–9x
–7
Subtract 7 from both sides.
 18
–9x  18
–9
–9
x  –2
Divide each side by –9;
change  to .
-6
-5
-4
-3
-2
-1
0
3-8 Solving Two-Step Inequalities
Check It Out! Example 1A
SolveEvaluating
and graph.
Algebraic Expressions
5x + 2 > 12
5x + 2 > 12
–2 –2
5x
> 10
5x> 10
5
5
x>2
Subtract 2 from both sides.
Divide both sides by 5.
1
2
3
4
5
6
7
3-8 Solving Two-Step Inequalities
Check It Out! Example 1A Continued
Evaluating
Check
Algebraic Expressions
According to the graph 4 should be a solution
and 1 should not be a solution.
x>2
?
4>2
Substitute
4 for x.
So 4 is a solution.
x>2
?
1>2 
Substitute
1 for x.
So 1 is not a solution.
3-8 Solving Two-Step Inequalities
Check It Out! Example 1B
Evaluating Algebraic Expressions
–4x + 2  18
–4x + 2  18
–2
–4x
–2
Subtract 2 from both sides.
 16
–4x  16
–4
–4
x  –4
Divide each side by –4;
change  to .
-6
-5
-4
-3
-2
-1
0
3-8 Solving Two-Step Inequalities
Additional Example 2: Solving Inequalities That
Contain Fractions
Evaluating
Algebraic Expressions
2x 3
9
Solve

and graph the solution.
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
5
+
(
)
()
()
()
()
– 15 – 15
8x
 3
Since 15 is added to 8x,
subtract 15 from both
sides.
3-8 Solving Two-Step Inequalities
Additional Example 2 Continued
8xEvaluating
3
8x  3
8
8
x 3
8
0
Algebraic Expressions
Since x multiplied by 8, divide both
sides by 8.
3
8
1
3-8 Solving Two-Step Inequalities
Check It Out! Example 2
Evaluating
Expressions
3x
1 Algebraic
5
Solve
+ 
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
Since 5 is added to 12x,
subtract 5 from both
sides.
3-8 Solving Two-Step Inequalities
Check It Out! Example 2 Continued
Evaluating
Algebraic Expressions
12x  5
12x  5
12
12
x 5
12
0
Since x is multiplied by 12,
divide both sides by 12.
5
12
3-8 Solving Two-Step Inequalities
Additional Example 3: School Application
AEvaluating
school’s Spanish
club is selling
bumper
Algebraic
Expressions
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?
In order for the Spanish club to make a profit,
the revenue must be greater than the cost.
1.25x > 55 + 0.15x
3-8 Solving Two-Step Inequalities
Additional Example 3 Continued
Evaluating Algebraic Expressions
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.
3-8 Solving Two-Step Inequalities
Check It Out! Example 3
A Evaluating
school’s French
club is selling
bumper
Algebraic
Expressions
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?
In order for the French club to make a profit,
the revenue must be greater than the cost.
2.5x > 45 + 0.25x
3-8 Solving Two-Step Inequalities
Check It Out! Example 3 Continued
Evaluating Algebraic Expressions
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.
3-8 Solving Two-Step Inequalities
Lesson Quiz: Part I
Solve and graph.
Evaluating Algebraic Expressions
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
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
3-8 Solving Two-Step Inequalities
Lesson Quiz: Part II
Algebraic
Expressions
5.Evaluating
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