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```Algebra – 3.3
Distance and Absolute Value
DISTANCE
• What’s the distance between these
points?
•
•
•
•
(-1, -3) and (4, -3)
(-1, -3) and (-1, 9)
(2, 1) and (-3, 1)
(2, 1) and (2, -16)
DISTANCE
• Can distance be negative?
• How do these compare:
– Distance from 1 to 4
– Distance from 4 to 1
• The easiest way to calculate distance on a
number line is to do subtraction
• But which number do you subtract?
– “What is the distance between a and b?”
– If a > b, then a – b. If b > a, then b – a.
• I don’t like that! It’s wordy!
• There’s a better way.
Absolute Value
DEFINITION:
If x and y are numbers,
then “the absolute value of (x-y)”,
written |x – y|
is the distance between x and y.
Example: |8 – 3| = 5 and |3 – 8| = 5
since 8 and 3 are 5 units apart
• Ex:
|1 – (-5)| =
• Ex: |-6 – 0| =
• Ex: |(-2) – 6| =
• Ex: |4 – 10| =
• Ex: |(-3) – (-5)| =
Special case
• What does THIS mean?
|8 + 10|
|8|
Theorem
The absolute value or a number x (written: |x|)
is its distance from 0 on the number line.
PROOF:
|x – 0| is the distance between x and 0.
x–0=x
By substitution, |x – 0| = |x|
Therefore, |x| is the distance between x and 0.
Algebraic representation
|x| =
x, if x≥0
-x, if x<0
```