Section 3.5
Solving Inequalities with
Variables on Both Sides
California
Standards
4.0 Students simplify expressions before
solving linear equations and inequalities in one variable,
such as 3(2x – 5) + 4(x – 2) = 12.
5.0 Students solve multi-step problems,
including word problems, involving linear equations and
linear inequalities in one variable and provide
justification for each step.
Some inequalities have variable terms on both
sides of the inequality symbol.You can solve
these inequalities like you solved equations with
variables on both sides.
Use the properties of inequality to “collect” all the
variable terms on one side and all the constant
terms on the other side.
Don’t call me after midnight
1. D= Distributive property
2. C= combine like term
3. M = move variable to one side
4. A = addition/subtraction
5. M = Multiplication/division
Solve the Inequality then GRAPH
the solution
Steps:
-Distribute
2x + 8 < 6x
-Combine Like Terms
-Move Variable
2x + 8 < 6x
-2x
-2x
8 < 4x
4
-The opposite of
adding 2x is
subtracting 2x.
-Undo multiplication…
4
-…so divide by four
2 < x
-4
-2
0
-Graph
2
4
Now You Try One…
Solve the Inequality then GRAPH the solution
Steps:
-Distribute
3x - 12 < 6x
-Combine Like Terms
-Move Variable
3x - 12 < 6x
-3x
-3x
-12 < 3x
3
-The opposite of
adding 3x is
subtracting 3x.
-Undo multiplication…
3
-…so divide by three
-4 < x
-4
-2
0
-Graph
2
4
Lets try one more
Solve the Inequality then GRAPH the solution
5t + 1 < –2t – 6
Steps:
-Distribute
5t + 1 < –2t – 6
+2t
+2t
7t + 1 <
-6
-1
-1
7t
<
-7
-Combine Like Terms
-Move Variable
-The opposite of
Negative 2t is
Adding 2t.
t < -1
–5 –4 –3 –2 –1
0
1
2
Graph
3
4
5
Now You Try
Solve and Graph
1.
2x > 4x – 6
2. 7y + 1 < y – 5
3. -3r < 10 – r
Match the Following
1.
Inequality
A.
mathematical statement that two
expressions are equivalent
2.
Equation
B.
value of a variable that makes a
statement true
3.
Inverse Operations
C.
terms that contain the same variable
raised to the same power
4.
Like Terms
D.
5.
Solution of an
Equation
a mathematical statement that
compares two expressions by using
one of the following signs: <, >, <, >,
or ≠
E.
operation that “undo” each other
Let’s try this one together
Solve the Inequality then GRAPH the solution
Steps:
-Distribute
6x < 4(x + 1)
6x < 4x + 4
-Combine Like Terms
-Move Variable
6x <4x +4
-4x -4x
2x <
4
2
2
-The opposite of
adding 4x is
Subtracting 4x.
-Undo multiplication…
-…so divide by two
x < 2
-4
-2
0
-Graph
2
4
Now You Try
Solve the Inequality then GRAPH the solution
Steps:
-Distribute
2(6 – x) < 4x
12 - 2x < 4x
-Combine Like Terms
-Move Variable
12 – 2x <4x
+2x +2x
12
< 6x
6
6
-The opposite of
subtracting 2x is
adding 2x.
-Undo multiplication…
-…so divide by six
2 < x
-4
-2
0
-Graph
2
4
Lets try one more
Solve the Inequality then GRAPH the solution
x+5>x+3
x+5>x+3
-x
-x
0 +5>0+3
-5
-5
0
-2
0 > -2
All real numbers
Steps:
-Distribute
-Combine Like Terms
-Move Variable
-The opposite of
adding x is
subtracting x.
-Does it make a true
statement?
Lets try another one
Solve the Inequality then GRAPH the solution
2x + 6 < 5 + 2x
Steps:
-Distribute
2x + 6 < 5 + 2x
-2x
-2x
0 +6<5+0
-5 -5
0
0
-Combine Like Terms
1 < 0
-Does it make a true
statement?
No Solutions
-Move Variable
-The opposite of
adding 2x is
subtracting 2x.
Now You Try
Solve and Graph
1. 4x > 3(7 – x)
2. 2(x – 2) < -2(1 – x)
3. 4(y + 1) < 4y + 2
Lesson Quiz
Solve each inequality and graph the solutions.
1. t < 5t + 24
t > –6
2. 5x – 9 ≤ 4.1x – 81
x ≤ –80
3. 4b + 4(1 – b) > b – 9 b < 13
Solve each inequality.
5. 2y – 2 ≥ 2(y + 7)
ø
6. 2(–6r – 5) < –3(4r + 2)
all real numbers