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Solving Compound and
Absolute Value Inequalities
Solving Compound and Absolute Value Inequalities
 Solve compound inequalities.
 Solve absolute value inequalities.
1) compound inequality
2) intersection
3) union
Solving Compound and Absolute Value Inequalities
A compound inequality consists of two inequalities joined by the word and
or the word or.
Solving Compound and Absolute Value Inequalities
A compound inequality consists of two inequalities joined by the word and
or the word or.
To solve a compound inequality, you must solve each part of the inequality.
Solving Compound and Absolute Value Inequalities
A compound inequality consists of two inequalities joined by the word and
or the word or.
To solve a compound inequality, you must solve each part of the inequality.
The graph of a compound inequality containing the word “and” is the
intersection of the solution set of the two inequalities.
Solving Compound and Absolute Value Inequalities
A compound inequality divides the number line into three separate regions.
Solving Compound and Absolute Value Inequalities
A compound inequality divides the number line into three separate regions.
x
y
z
Solving Compound and Absolute Value Inequalities
A compound inequality divides the number line into three separate regions.
x
y
z
The solution set will be found:
in the blue (middle) region
Solving Compound and Absolute Value Inequalities
A compound inequality divides the number line into three separate regions.
x
y
z
The solution set will be found:
in the blue (middle) region
or
in the red (outer) regions.
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word and is true if and only if (iff), both
inequalities are true.
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word and is true if and only if (iff), both
inequalities are true.
Example:
x  1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word and is true if and only if (iff), both
inequalities are true.
Example:
x  1
x
-5
x2
-4
-3
-2
-1
0
1
2
3
4
5
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word and is true if and only if (iff), both
inequalities are true.
Example:
x  1
x
-5
x2
x  1
and
x2
-4
-3
-2
-1
0
1
2
3
4
5
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word and is true if and only if (iff), both
inequalities are true.
Example:
x  1
and
x2
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true if one or more, of the
inequalities is true.
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true if one or more, of the
inequalities is true.
Example:
x 1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true if one or more, of the
inequalities is true.
Example:
x 1
x
-5
x3
-4
-3
-2
-1
0
1
2
3
4
5
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true if one or more, of the
inequalities is true.
Example:
x 1
x
-5
x3
x 1
or
x3
-4
-3
-2
-1
0
1
2
3
4
5
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true if one or more, of the
inequalities is true.
Example:
x 1
or
x3
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
A compound inequality divides the number line into three separate regions.
x
y
z
The solution set will be found:
in the blue (middle) region
or
in the red (outer) regions.
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |a| < 4 to mean that the distance between a and 0 on a number
line is less than 4 units.
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |a| < 4 to mean that the distance between a and 0 on a number
line is less than 4 units.
To make |a| < 4 true, you must substitute numbers for a that are fewer than 4 units
from 0.
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |a| < 4 to mean that the distance between a and 0 on a number
line is less than 4 units.
To make |a| < 4 true, you must substitute numbers for a that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |a| < 4 to mean that the distance between a and 0 on a number
line is less than 4 units.
To make |a| < 4 true, you must substitute numbers for a that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |a| < 4 to mean that the distance between a and 0 on a number
line is less than 4 units.
To make |a| < 4 true, you must substitute numbers for a that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
Notice that the graph of |a| < 4 is the same
as the graph a > -4 and a < 4.
4
5
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |a| < 4 to mean that the distance between a and 0 on a number
line is less than 4 units.
To make |a| < 4 true, you must substitute numbers for a that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
Notice that the graph of |a| < 4 is the same
as the graph a > -4 and a < 4.
All of the numbers between -4 and 4 are less than 4 units from 0.
The solution set is
{ a | -4 < a < 4 }
5
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |a| < 4 to mean that the distance between a and 0 on a number
line is less than 4 units.
To make |a| < 4 true, you must substitute numbers for a that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Notice that the graph of |a| < 4 is the same
as the graph a > -4 and a < 4.
All of the numbers between -4 and 4 are less than 4 units from 0.
The solution set is
{ a | -4 < a < 4 }
For all real numbers a and b, b > 0, the following statement is true:
If |a| < b then, -b < a < b
Solving Compound and Absolute Value Inequalities
A compound inequality divides the number line into three separate regions.
x
y
z
The solution set will be found:
in the blue (middle) region
or
in the red (outer) regions.
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
You can interpret |a| > 2 to mean that the distance between a and 0 on a number
line is greater than 2 units.
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
You can interpret |a| > 2 to mean that the distance between a and 0 on a number
line is greater than 2 units.
To make |a| > 2 true, you must substitute numbers for a that are more than 2 units
from 0.
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
You can interpret |a| > 2 to mean that the distance between a and 0 on a number
line is greater than 2 units.
To make |a| > 2 true, you must substitute numbers for a that are more than 2 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
You can interpret |a| > 2 to mean that the distance between a and 0 on a number
line is greater than 2 units.
To make |a| > 2 true, you must substitute numbers for a that are more than 2 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
You can interpret |a| > 2 to mean that the distance between a and 0 on a number
line is greater than 2 units.
To make |a| > 2 true, you must substitute numbers for a that are more than 2 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
Notice that the graph of |a| > 2 is the same
as the graph a < -2 or a > 2.
4
5
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
You can interpret |a| > 2 to mean that the distance between a and 0 on a number
line is greater than 2 units.
To make |a| > 2 true, you must substitute numbers for a that are more than 2 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Notice that the graph of |a| > 2 is the same
as the graph a < -2 or a > 2.
All of the numbers not between -2 and 2 are greater than 2 units from 0.
The solution set is
{a|a>2
or a < -2 }
Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)
You can interpret |a| > 2 to mean that the distance between a and 0 on a number
line is greater than 2 units.
To make |a| > 2 true, you must substitute numbers for a that are more than 2 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Notice that the graph of |a| > 2 is the same
as the graph a < -2 or a > 2.
All of the numbers not between -2 and 2 are greater than 2 units from 0.
The solution set is
{a|a>2
or a < -2 }
For all real numbers a and b, b > 0, the following statement is true:
If |a| > b then, a < -b
or
a>b
Solving Compound and Absolute Value Inequalities
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Using Glencoe’s Algebra 2 text,
© 2005