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Solving Inequalities by
Multiplication and Division
A solution of an inequality is a value for the variable that
makes the inequality true.
How many solutions are there to an inequality problem?
To solve an inequality isolate the variable on one side of
the inequality symbol. Follow the basic rule:
Whatever you do to one
side of the inequality sign,
you must also do to the
other side of the inequality
sign.
Inductive Reasoning is making conclusions on patterns
you observe.
Look for the pattern that results
from dividing each side of an
inequality by an integer.
6 3
6 2
2 < 4
3< 6
6 < 12
6 1
The inequality symbol reverses
6   1  6 >  12
direction when you divide by a
6   2  3 >  6
negative number.
6   3  2 >  4
•
-12 -10
-8
•
-6
An inequality
symbol points to
the smaller
number.
12  3
12  2
12  1
12   1
12   2
12   3
Multiplication and Division Properties of Inequalities
If you multiply or divide each side of an inequality by a
positive number, the direction of the inequality symbol
is unchanged.
3x  12
3x  12
3
3
x  4
If you multiply or divide each side of an inequality by a
negative number, the direction of the inequality symbol
is reversed.
 3x  12
 3x 
/ 12
3  3
x4
Example 1 Solve
m
 6. Then graph the solution.
3
Write the inequality.
Use inverse property.
When you multiply or
divide by a negative,
reverse inequality symbol.
Graph.
m
 6
3
 3 m /
 6 3
3
>
m  18
•
Solve. Then graph the solution.
Ex. 2
m
 2
2
Ex. 3
42   6w
Ex. 4
1
1
3 m2
4
2
Ex. 5
 2.8d  9.8
Solve. Then graph the solution.
Ex. 2 m  2
2
2 m  2 2
2
m  4
Ex. 3
•
42   6w
42 
/  6w
6 > 6
7  w
w  7
O
–7
–4
Check
the
endpoint!
Solve. Then graph the solution.
1
1
3 m2
4
2 2
5 4 
 4  13
  m  
 13  4
2  13 
10
m
13
Ex. 4
O
10
13
Ex. 5  2.8d  9.8
 2.8d / 9.8
 2.8  2.8
>
d  3.5
O
–3.5
Solve. Then graph the solution.
3
Ex.
7

2
y  22
Ex. 6  11.78  1.9f
5
Write an inequality and then solve.
Ex. 8 Negative three eighths times a number is greater
than or equal to 12. Find the number.
Ex. 9 Two and one half times a number is less than one
and one fifth. Find the number.
Solve. Then graph the solution.
3
Ex. 7  2 y  22
Ex. 6  11.78  1.9f
5
 11.78  1.9f
5
  5  13

 22   

  y/
1.9
1.9
 13  5 <
 13 
 6.2  f
110
y
f  6.2
13
O
–6.2

•
110
13
Write an inequality and then solve.
Ex. 8 Negative three eighths times a number is greater
>
than or equal to 12.
3
 n  12
8
4
 8
  8   3 n  12




/
3

 3 8 < 
n  32
Ex. 9 Two and one half times a number is less than one and
<
one fifth.
1
1
2 n 1
2
5
 2 5 n  6  2 
 
 
55
 5 2
12
n
25
If you multiply or divide each side
of an inequality by a positive
number, the direction of the
inequality symbol is unchanged.
If you multiply or divide each side
of an inequality by a negative
number, the direction of the
inequality symbol is reversed.