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6-6 Solving Inequalities Involving Absolute Value
Algebra 1 Glencoe McGraw-Hill Linda Stamper
GO LA
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 “AND” type of compound inequality will graph as ....An “OR” type of compound inequality will graph as ....
An absolute-value inequality is an inequality that has one of these forms:
cbax cbax cbax cbax
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.
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.I should
copy the above notes
in my notebook!
Identify the inequality as an “AND” type or “OR” type. Then identify the graph as a line segment or opposite rays.
4x
Inequality
Type Graph
AND line segment
92x OR opposite rays
54x3 OR opposite rays
86x3 AND line segment
GOLA
LA
GO
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:
cbax cbax cbax cbax
“AND” type
“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.
Do not identify the
type of inequality until the absolute value is isolated!
Solve. Then graph the solution.
16x
Write the positive related inequality.
16x
Write the inequality.
Write the negative related inequality;
and 16x
LA46x4
Isolate the absolute value on one side of the inequality sign.
4 4
–7 –5
Solve each inequality.O O
Write as a single inequality.
5x 6 66 6
5x7
Graph.
7x and
</
Do not identify the
type of inequality until the absolute value is isolated!
Solve. Then graph the solution.
16x
Write the positive related inequality.
16x
Write the inequality.
Write the negative related inequality;
LA46x4
Isolate the absolute value on one side of the inequality sign.
4 4
–7 –5Solve.O O
6 66
5x7
Graph.
I should copy the
above notes in my
notebook!
</
1
Solve. Then graph the solution.
16x
Write the positive related inequality.
16x
Write the inequality.
Write the negative related inequality;
ro 16x
GO
46x4
Isolate the absolute value on one side of the inequality sign.
4 4
–7 –5
Solve each inequality.O O
Reposition.
5x 6 66 6
Graph.
7x
or
>/
5x 7x
Solve. Then graph the solution.
4x
Example 1 8x
Example 2
Solve. Then graph the solution.
4x
4x 4x4
–4 4
4x and
• •
Note: Less than symbol represents the “AND” type of inequality and will graph as a line segment.
LA
Example 1 4x
4x
–4 4
• •
LA
Example 1
4
Solve. Then graph the solution.
8x
8x
–8 8
8x ro
• •
GO
or 8x
Example 2
8x
Note: Greater than symbol represents the “OR” type of inequality and will graph as opposite rays.
Solve. Then graph the solution.
19x
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.
19x
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.
all real numbers
0•
Solve. Then graph the solution.Example 3
Example 4
Example 6
19x
153x
849x32
Example 7 52
4a3
Example 8 76x
Example 5 512x
Example 9 81x3
Example 3 Solve. Then graph the solution.
19x
19x
8 10
19x or
• •
Note: Greater than symbol represents the “OR” type of inequality and will graph as opposite rays.
or8x
10x 8x
10x or
9 99 9
GO
19x
Example 4 Solve. Then graph the solution.
153x
43x
7x
–1 7
43x or
O O
1x
Isolate the absolute value on one side of the inequality sign. 43x
or 1x 7x or3 3 3 3
5 5 GO
Example 5 Solve. Then graph the solution.
512x
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. 849x32
12 9x32 2 2
/
69x3 69x3
3x315
1x5
–5 –1• •
4 4
99
3 3
LA
96
3
Example 7 Solve. Then graph the solution.
52
4a3
52
4a3
104a310
2a314
2• •
2 2
3 3
LA
522
4a35
446a314
5
314
4
3
Example 8 Solve. Then graph the solution.
76x
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.
all real numbers
0•
Example 9 Solve and then graph.
1
GO
44
81x3 81x3
84x
84xor84x
4x 14x
4412x
1 112x or
12• •4
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.
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