The absolute value of a number is the distance between the number and zero on the real number line. Distances are measured as positive units (or zero units). Consequently, absolute value is never negative. Absolute value answers the question "How far from zero?", but not the question "In which direction from zero?". The notation used for absolute value is two vertical bars.
Absolute value should be treated as a grouping symbol in the order of operations. Be sure to evaluate, or simplify, any values within the absolute value before evaluating the absolute value.
For example:  4  6  =  2  = 2
Examples: 
1.  − 3  = 3 
2. −  − 7  = − 7


3.  0  = 0 
4.  − 5 ^{2 } = 25


5. −  (− 8)^{2 } = − 64 
6. −  − 3 ^{2 }= − 9 (see note below)


7.  3  +  − 7  −  − 2  = 8 
8.  − 8  + ( − 9) +  − 14  + ( − 2) = 11 

9. 
10.  − 3 + 6 ^{2 }+  − 3  = 12 


In question #6, you can hink of it as
− (  − 3  •  − 3  ) = −1 ( 3 • 3 ) = − 9
Think of the negative sign out front as multiplying by −1.
By order of operations, exponents are done before multiplication. 
Things to Consider:
Measuring Distance:
 8 − 3  = 5 and  3 − 8  = 5
The expression  a − b  represents the distance from a to b on the number line. This distance is the same when measured forward from 3 to 8, or backward from 8 to 3.

Properties:
 3 x 7  =  3  x  7 
1.  a • b  =  a  •  b  (multiply)
2.  a / b  =  a  /  b  (divide)
3.  a + b  ≠  a  +  b  (add)
4.  a  b  ≠  a    b  (subtract)
5.  a^{n}  =  a ^{n} (power)
6.
(called the Triangle Inequality)

Square Root Definition:

Graph:

Since absolute value always yields a positive result, or zero, the graph of absolute value plots only yvalues that are positive, or zero. The graph resides above the xaxis (plus the origin), in quadrants I and II.
y =  x  

Negating absolute value creates yvalues that
are negative or zero. The graph resides in
quadrants III and IV (plus the origin). 
Subtracting 4 from x moves the absolute value
graph horizontally 4 units to the right. The
yvalues remain positive or zero. 

For help with absolute value
on your calculator,
click here. 


