Download Absolute Value

ABSOLUTE VALUE
INEQUALITIES
A FISH STORY
Definition of Absolute Value
Remember the definition of absolute value – the distance a
point is away from zero on the number line.
If it involves a greater than sign, you are looking for all
the points more than a given distance from zero.
Definition of Absolute Value
x 5
x  5 | x  5
)
-5
(
+5
The numbers less than – 5 are more than 5 units away from zero in the negative direction.
The numbers greater than 5 are more than 5 units away from zero in the positive direction.
Definition of Absolute Value
Notice the pattern
x 5
x  5 | x  5
Kill the bars, Kill the bars.
flip it
and neg it
Absolute Value >
When you see an absolute value inequality, picture someone
telling a story about fishing.
If it involves a greater than sign, picture a story about a
really BIG fish.
Absolute Value >
Now picture the number line looking like this guy!
I call this “shooting arrows”
Absolute Value >
Whenever you see an “absolute value greater than”
problem, the graph of the answer will be “shooting arrows”
Let’s look at an example
Find x: x  2  5
x  2  5 | x  2  5
2  2 |  2  2
x  3 |
)
-3
x7
(
7
Definition of Absolute Value
Remember the definition of absolute value – the distance a
point is away from zero on the number line.
If it involves a less than sign, you are looking for all the
points less than a given distance from zero.
Definition of Absolute Value
x 5
x  5 | x  5
(
-5
)
+5
The negative numbers greater than – 5 are less than 5 units away from zero.
The positive numbers less than 5 are less than 5 units away from zero.
Definition of Absolute Value
Notice the pattern:
x 5
x  5 | x  5
Kill the bars, Kill the bars.
flip it
and neg it
Absolute Value <
Find x:
x2 5
x  2  5 | x  2  5
2  2 |  2  2
x  3 |
(
-3
x7
)
7
Absolute Value <
When you see an absolute value inequality, picture someone
telling a story about fishing.
If it involves a less than sign, picture a story about a really
tiny fish.
Absolute Value >
Now picture the number line looking like this guy!
(
)
I call this the “barbell”
Absolute Value <
Whenever you see an “absolute value less than” problem,
the graph of the answer will be a “barbell”
(
)
Trick Questions
By Definition, absolute values are always positive.
If you have a question like
3x  4  0 or 5x  2  3
The answer is always No Solution or

Trick Questions
By Definition, absolute values are always positive.
If you have a question like
3x  4  0 or 5x  2  3
The answer is always All Real Numbers or
(, )
Things to
remember
Remember!
The absolute value sign is not parentheses.
In complicated problems, do not distribute.
Isolate the absolute value
3 5 x  4  6  12
3 5 x  4  12  6
3 5x  4  6
and then solve.
5x  4  2
Remember!
Expect 2 answers
5x  4  2
5 x  4  2 | 5 x  4  2
5x  2 |
5x  6
2
6
x |
x
5
5
Remember!
By definition, absolute values are always positive.
If you have an absolute > the graph is shooting arrows.
If its > 0 or > - it’s really shooting arrows - the whole
line.
(, )
If you have an absolute < the graph is a barbell.
If its < 0 or < - the barbell’s so tiny it disappears!
