Download 6-6 Solving Inequalities Involving Absolute Value

6-6 Solving Inequalities Involving
Absolute Value
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
GO LA
An “AND” type of compound inequality will graph as ....
An “OR” type of compound inequality will graph as ....
Which type of compound inequality, “AND” or “OR”,
has a greater number of solutions? Why?
“Or” because it graphs as opposite rays that continue
to infinity.
An absolute-value inequality is an inequality that has one
of these forms:
ax  b  c
ax  b  c

When the absolute value is
on the left, the Less than
symbol represents the
“AND” type of inequality.
It graphs as a line segment
and has less – fewer
solutions.
I should copy
the above
notes in my
notebook!
ax  b  c
ax  b  c

When the absolute value is
on the left, the Greater
than symbol represents
the “OR” type of
inequality. It graphs as
two opposite rays and has
a greater number of
solutions.
Identify the inequality as an “AND” type or “OR” type.
Then identify the graph as a line segment or opposite rays.
GO
Inequality
Type
x 2  9
OR
opposite rays
Graph
LA
x 4
AND
line segment
LA
3x  6  8
AND
line segment
GO
3x  4  5
OR
opposite rays
BEFORE you identify the type of compound inequality you
must isolate the absolute value.
An absolute-value inequality is an inequality that
has one of these forms:
ax  b  c
ax  b  c

“AND” type
ax  b  c
ax  b  c

“OR” type
To solve an absolute-value inequality, write the two
related inequalities – a positive inequality and a negative
inequality.
When you write the related inequality for the negative
value, reverse the inequality symbol.
Solve. Then graph the solution.
 4x  6 /
 4
Write the inequality.
<4

4
Isolate the absolute value on one
x6 1
side of the inequality sign.
x  6  1 and x  6  1
Write the positive related
6 6
66
inequality.
x  7
x  5 and
Dorelated
not identify
Write the negative
the type of
inequality;
 7  x  5
inequality until
Solve each inequality.
the absolute
O
O
is isolated!
Write as a single value
inequality.
–7
–5
Graph.
LA
Solve. Then graph the solution.
 4x  6 /
 4
Write the inequality.
<4

4
Isolate the absolute value on one
x6 1
side of the inequality sign.
1 x6 1
Write the positive related
6
6 6
inequality.
Dorelated
not identify
Write the negative
 7  x  5
the type of
inequality;
O
O
Solve.
Graph.
inequality until
the absolute
value is isolated!
I should copy
the above
notes in my
notebook!
–7
–5
LA
Solve. Then graph the solution.
 4x  6 /
 4
Write the inequality.
>4

4
Isolate the absolute value on one
x6 1
side of the inequality sign.
or x  6  1
x  6  1 or
Write the positive related
6 6
66
inequality.
x  7
x  5
Write the negative related
inequality;
GO
Solve each inequality.
Reposition.
Graph.
O
–7
O
–5
Solve. Then graph the solution.
Example 1
x 4
Example 2
x 8
Solve. Then graph the solution.
Example 1
LA
x 4
x  4 and x   4
4x 4
•
–4
•
4
Example 1
LA x  4
4
x4
•
•
–4
4
Note: Less than symbol represents the “AND”
type of inequality and will graph as a line segment.
Solve. Then graph the solution.
Example 2
GO
x 8
xx  88 or
or xx 8
•
–8
•
8
Note: Greater than symbol represents the “OR”
type of inequality and will graph as opposite rays.
Solve. Then graph the solution.
x  9  1

Absolute value cannot be less than zero (cannot
be negative)! Thus there are no values that will
be less than negative one.
Solve. Then graph the solution.
x  9  1
all real numbers
•
0
Absolute value will be zero or greater (it cannot be
negative). Thus x can be any real number and the
absolute value will be greater than negative one.
Solve. Then graph the solution.
Example 3
x9 1
Example 4
x  3  5  1
Example 5
x  12  5
Example 6
 2 3x  9  4  8
Example 7
3a  4
5
2
Example 8
x  6  7
Example 9
3  x  1  8
Example 3 Solve. Then graph the solution.
GO
x9 1
x  9  1 or
9 9
x  10 or
•
8
x  9  1
x  9  1
9 9
x8
•
10
Note: Greater than symbol represents the “OR”
type of inequality and will graph as opposite rays.
Example 4 Solve. Then graph the solution.
x  3  5  1
5 5
GO
x 3  4
x  3  4 or x  3   4
3 3
3 3
x  7 or
x  1
Isolate the absolute
value on one side of
the inequality sign.
O
–1
O
7
Example 5 Solve. Then graph the solution.
x  12  5

Absolute value cannot be less than zero (cannot
be negative)! Thus there are no values that will
be less than negative one.
Example 6 Solve. Then graph the solution.
 2 3x  9  4  8
4 4
 2 3x  9 /
  12
< 2
2
3x  9  6
 6 3x  9  6
9
9 9
 15  3x   3
3
3
3
 5  x  1
•
–5
•
–1
LA
Example 7 Solve. Then graph the solution.
3a  4
 5 LA
2
3a  4
5
5
2
 3a  4  2  5 2

5

2


 2 
 10  3a  4  10
4 4
4
 14  3a  6
3
3 3
14

 a2
3
•
14

3
•
2
Example 8 Solve. Then graph the solution.
x  6  7
all real numbers
•
0
Absolute value will be zero or greater (it cannot be
negative). Thus x can be any real number and the
absolute value will be greater than negative one.
Example 9 Solve and then graph.
GO
3  x  1  8
3 x 1  8
x4 8
 x  4  8 or  x  4  8
4 4
4 4
x  4
 x  12
1 1
1 1
or
x  4
x  12
•
4
•
12
When solving an absolute-value inequality:
The less than symbol represents the “AND” type of
inequality. It graphs as a line segment and has less
(fewer) solutions.
The greater than symbol represents the “OR” type
of inequality. It graphs as two opposite rays and
has a greater number of solutions.
GO LA
6-A12 Pages 332-333 # 8–16,23–26,46-51.