1. Solving Compound and
Absolute Value Inequalities

2. Warm Up
1)Three –fourths of the difference between a number
and six is no more than the quotient of that number
and four
2.
4x + 1 > x - 2
2
3
4. Glynn has to drive 450 miles. His car has an 18
gallon gas tank and he would like to make the trip on
one tank of gas. What is the minimum miles per
gallon his car would have to get to make the trip on
one tank? Write an inequality to show .

3. Warm Up
5. Solve for x if ax + by = c
6. -3(2x - 3) + 7x = -4(x - 5) + 6x – 2
7. Simplify: -6 - (-3) + (-2) * 4 =
8.
|
|
10. -2 x - 7 - 4 = -22

4. Solving Compound and Absolute Value Inequalities
A compound inequality consists of two inequalities joined
by the word and or the word or.
A. Conjunctions: Two inequalities joined by the word ‘and’.
For example: -1 < x and x < 4; This can also be written -1 < x < 4
To solve a compound inequality, you must solve each part
of the inequality separately. Conjunctions are solved when
both parts of the inequality are true -1 < x < 4; x > -1 < 4
x
-1
4
The graph of a compound inequality containing the word ‘and’
is the intersection of the solution set of the two inequalities. The
Intersection is the solution for the compound inequality.

5. Solving Compound and Absolute Value Inequalities
All compound inequalities divide the number line into
three separate regions.
x
y
z
A compound inequality containing the word and is true
if and only if (iff), both inequalities are true.

6. 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

7. 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
x
1
2
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
-5
-4
-3
-2
-1
0
1
2
3
4
5

8. 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
x
1
2
x
-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
1
and
x
x
-5
2
x

9. 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

10. Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true
one or more, of the inequalities is true.
Example:
x 1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5

11. Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true
one or more, of the inequalities is true.
Example:
x 1
x 3
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
-5
-4
-3
-2
-1
0
1
2
3
4
5

12. Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true
one or more, of the inequalities is true.
Example:
x 1
x 3
x
-5
-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 1
or
x 3
x

13. Solving Compound and Absolute Value Inequalities
A compound inequality containing the word or is true
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

14. 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.

15. 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.

16. 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.

17. 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

18. Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |x| < 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 x that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5

19. Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |x| < 4 to mean that the distance between
ax and 0 on a number line is less than 4 units.
To make |x| < 4 true, you must substitute numbers for x that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Notice that the graph of |x| < 4 is the same
as the graph x > -4 and x < 4.

20. Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (<)
You can interpret |x| < 4 to mean that the distance between xa and 0 on a number
line is less than 4 units.
To make |x| < 4 true, you must substitute numbers for x that are fewer than 4 units
from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Notice that the graph of |x| < 4 is the same
as the graph x > -4 and x < 4.
All of the numbers between -4 and 4 are less than 4 units from 0.
The solution set is { x | -4 < x < 4 }

21. 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

22. 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.

23. Solving Compound and Absolute Value Inequalities
Solve an Absolute Value Inequality (>)

24. 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.

25. 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.

26. 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

27. 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

28. 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

29. 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 }

30. 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

31. Solving Compound and Absolute Value Inequalities

32. Credits
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Using Glencoe’s Algebra 2 text,
© 2005