Absolute Value
Absolute Value
-7
-10
-5
+7
0
5
10
 A number line has many functions. Previously, we learned
that numbers to the right of zero are positive and numbers
to the left of zero are negative. By putting points on the
number line, we can graph values.
 If one were to start at zero and move seven places to the
right, this would represent a value of positive seven.
 If one were to start at zero and move seven places to the
left, this would represent a value of negative seven.
Absolute Value
-7
-10
-5
+7
0
5
10
 Both of these numbers, positive seven (+7) and negative
seven (-7), represent a point that is seven units away from
the origin.
 The absolute value of a number is the distance between
that number and zero on a number line. Absolute value is
shown by placing two vertical bars around the number as
follows:
| 5 | The absolute value of five is five.
| -3 | The absolute value of negative three is three.
ABSOLUTE VALUE
• A numbers distance from zero on the
number line.
-5 -4 -3 -2 -1
0
1 2
3
4
5
3
3
12 
12
0
5 
5
7 
-7
5.4 
5.4
 16 
-16
10  10
 23 
-23
0
Opposites vs Absolute Value
Given Number
Opposite
Absolute Value
8
-8
8 8
-24
+24
24  24
-3.5
+3.5
35
.  35
.
1
4
2
1
1
4 4
2
2
1
4
2
Solve each equation below.
1)
x  10
x = 10 or -10
2) x  4
x = 4 or - 4
3) x  0
x=0
4) x  5
“no solution”
5)  x  14
x = 14 or - 14
3
6) t 
4
3
3
t = 4 or - 4
Solve each equation below.
1) 9  2 
4)
-3
x=7
2)  8  3 
x = 11
3)   17 
x = -17
3
5)
 3 2 
x = -5
6) 10  1  2 
t = 11
Determine whether each statement is true always,
sometimes, or never for all real numbers.
1) x  x
sometimes
2)  x  x
always
3) x   x
sometimes
4) x  0
never
5)  x   x
sometimes
6) x   x
always
Velocity vs. Speed
Velocity - Indicates speed and direction.
Speed - The absolute value of velocity.
Example:
A helicopter descends at 50 feet/second.
A) What is its velocity?
B) What is its speed?
-50 ft./sec.
+50 ft./sec.
Counterexamples
To prove a statement true, it must be proven true for all
examples - difficult!
Counterexample:
An example that proves
a statement false.
Statement: All pets are furry.
Counterexample: Goldfish.
Statement:
Counterexample:
x x
x0
Determine whether each statement is true or false
for all real numbers. If it is false, find a counterexample that proves it is false.
1) x  0
False, x = 0
2) x  x
False, x = -7
3)  x   x
False, x = -5
4) x   x
True
5) x   x
True
6) x  x
True