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```x3  9
Solve Absolute
Value Inequalities
2x4 8
© 2011 The Enlightened Elephant
Quick Intro Video
https://www.youtube.com/watch?v=XCeH
QgXRzFU
First, let’s think about
what absolute value
inequalities are really
asking us to find!
What does
x 4
x 4
All the numbers whose distance
from zero is greater than 4.
4
-4
x  4
really mean?
or
x4
***Notice that you need to have two inequalities
to represent the distances that are greater than
4 from zero.
What does
x 4
x 4
really mean?
All the numbers whose distance
from zero is less than 4.
x  4
-4
and
4
x4
***Notice that you need to have two inequalities
to represent the distances that are less than 4
from zero.
However, since these inequalities must happen at
the same time, it should be written as
4 x  4
OK, so how do we solve
more difficult problems?
Solve and graph 2 x  4  8
Step 1: Isolate the absolute value.
x4  4
Step 2: Set up two inequalities.
x4  4
x  4  4
Step 3: Solve the inequalities.
x 8
x0
Step 4: Graph the solutions.
0
8
Let’s practice!
You Try!
Step 1: Solve and graph. 4 x  2
Step 2:
Step 3:
Step 4:
 1  11
x2 3
x23
x  2  3
x 1
x  5
Step 5:
-5
Final answer: -5<x<1
1
Set up two
inequalities!
You Try!
x3  9
x  3  9
x 3  9
Solve and graph.
x6
x  12
-12
Final answer:
6
x6
or
x  12
Isolate the absolute
value first!
You Try!
Solve and graph.
3 2 x  1  21
2x 1  7
2 x 1  7
x4
2x 1  7
x  3
-3
Final answer:
4
x4
or
x  3
You Try!
Solve and graph.
Isolate the absolute
value first!
3x  6  3  15
3x  6  12
3x  6  12 3x  6  12
x6
-2
Final answer:
x  2
6
2 x  6
Isolate the absolute
value first!
You Try!
Solve and graph.
3 2 x  1  8  23
2x 1  5
2x  1  5
x2
-3
Final answer:
2x 1  5
x  3
2
3  x  2
You Try!
Solve and graph.
Isolate the absolute
value first!
2 x  8  12
Final answer: NO SOLUTION!
Absolute values are distances, which
cannot be negative.
You can choose any value for x and the left
side of the inequality will always be
positive.
However, positive numbers are NEVER
less than negative numbers.
You Try!
Solve and graph.
Isolate the absolute
value first!
6x 12  10
Final answer: ALL REAL NUMBERS!
Absolute values are distances, which
cannot be negative.
You can choose any value for x and the left
side of the inequality will always be
positive.
Positive numbers are ALWAYS greater
than negative numbers.
Success!
Nice job!
```
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