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.

absgraphMBRB

 

statement
Algebraically speaking,


Abs Rule

While it is understood that absolute value yields a positive result (or zero), you must be a bit more careful when stating this concept using a variable, such as x. One never knows the value which may replace the variable (such as x) as it could be positive, or it could be negative.
ruleabsa


bullet Examples:
1.  | − 3 | = 3
2.   − | − 7 | = − 7
 
3.   | 0 | = 0
4.   | − 5 |2 = 25
 
5.   − | (− 8)2 | = − 64
6.   − | − 3 |2 = − 9
 
7.   | 3 | + | − 7 | − | − 2 | = 8
8.   | − 8 | + ( − 9) + | − 14 | + ( − 2) = 11
 
9.  absex12
10.  | − 3 + 6 |2 + | − 3 | = 12


bullet
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. | an | = | a |n (power)
6. abstrineq(called the Triangle Inequality)
Square Root Definition:
abssqrtdef
squareroot2a


bullet Graph:
 
Since absolute value always yields a positive result, or zero, the graph of absolute value plots only y-values that are positive, or zero. The graph resides above the x-axis (plus the origin), in quadrants I and II.

y = | x |    arrow

absgraph1
absgraph2
Negating absolute value creates y-values that
are negative or zero. The graph resides in
quadrants III and IV (plus the origin).
graphover
Subtracting 4 from x moves the absolute value
graph horizontally 4 units to the right. The
y-values remain positive or zero.

ti84c
For help with absolute value
on your calculator,
click here.

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