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Absolute Value
The absolute value of a number is its distance from zero.
The symbol for absolute value is
6 = 6
-5 = 5
Absolute Value is always positive or zero.
Graphing Absolute Value
X = 6 Where are the numbers that are 6 units from zero?
20 31 54 76-1-4 -2-3-5-6-7
20 31 54 76-1-4 -2-3-5-6-7
Graph:
OR
{ 6, -6}
Graphing Absolute Value
X 3 Where are the numbers that are more than 3 units
from zero?
20 31 54 76-1-4 -2-3-5-6-7
Graph:
To the right of 3To the left of -3
Do you want the 3? NODo you want the -3? NO
OR
X < -3 OR X > 3
Graphing Absolute Value
X ≤ 3 Where are the numbers that are less than 3 units
from zero?
20 31 54 76-1-4 -2-3-5-6-7
Graph:
To the left of 3To the right of -3
Do you want the 3? YesDo you want the -3? Yes
20 31 54 76-1-4 -2-3-5-6-7
AND at the same time
-3 X 3
or
= positive #
If > 0
Could represent an expression that has a positive value or a negative value.
= positive #
If < 0
-( ) = positive #
X = 3 If X > 0
X = 3 or
If X< 0
-(X)= 3A B
An expression that represents any real number except 0
-1(X)= 3
-1 -1
X = -30-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
B
AB
Put together
A
X = 3 If X > 0
X = 3 or
If X< 0
-(X)= 3A B
{ 3, -3}
or
> positive #
If > 0
Could represent an expression that has a positive value or a negative value.
> positive #
If < 0
An expression that represents any real number except 0
X 4 If X > 0
X 4 or
If X< 0
-(X) 4A B
-( ) > positive #
-1(X) 4
-1 -1
X -40-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
B
AB
Put together
A
X 4
X 4 or -(X) 4A B
IF YOU MULTIPLY OR DIVIDE BOTH SIDES OF AN INEQUALITY BY A NEGATIVE TURN THE INEQUALITY SIGN AROUND
X < -4 OR X > 4
or
< positive #
If > 0
Could represent an expression that has a positive value or a negative value.
< positive #
If < 0
X 2 If X > 0
X 2 and
If X< 0
-(X) 2A B
-( ) < positive #
An expression that represents any real number except 0
-1(X) 2
-1 -1
X -20-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
B
ABWhat’s the same?
A
X 2 If X > 0
X 2 and
If X< 0
-(X) 2A B
IF YOU MULTIPLY OR DIVIDE BOTH SIDES OF AN INEQUALITY BY A NEGATIVE TURN THE INEQUALITY SIGN AROUND
-2 X 2
OR
= Positive #
= Positive #
- ( ) = Positive #
OR
> Positive #
> Positive #
- ( ) > Positive #
AND
< Positive #
< Positive #
- ( ) < Positive #
2X -3 > 1
2X -3 > 1 or -(2X -3) > 1-2X +3 > 1
-2 -2X < 1
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
B
ABPut together
A
A B
+3 +32X 42 2X > 2
-3 -3-2X > -2
IF YOU MULTIPLY OR DIVIDE BOTH SIDES OF AN INEQUALITY BY A NEGATIVE TURN THE INEQUALITY SIGN AROUND
>
X < 1 OR X > 2
5X -3 = 7
5X -3 = 7 or -(5X -3) = 7-5X +3 = 7
-5 -5X = -.8
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
B
A BPut together?
A
A B
+3 +35X 105 5 X= 2
-3 -3-5X = 4=
{ 2, -.8}
3X +6 9
3X +6 9 and -(3X +6) 9-3X - 6 9
-3 -3X -5
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
0-5 -4 -3 -2 -1 1 2 3 4 5
B
ABWhat’s the same?
A
A B
-6 -63X 33 3X 1
+6 +6-3X 15
-5 X 1
Graphing Absolute Value
2X - 3 = - 6Absolute value is always a positive number or zero.
20 31 54 76-1-4 -2-3-5-6-7
Graph:
Absolute value will never equal a negative number.
NO SOLUTION
Graphing Absolute Value
X - 5 < -2Absolute value is always a positive number or zero.
20 31 54 76-1-4 -2-3-5-6-7
Graph:
Since absolute value is always positive or zero, it cannot be less
than a negative number.
NO SOLUTION
Graphing Absolute Value
3X+2 ≥ -2Absolute value is always a positive number or zero.
20 31 54 76-1-4 -2-3-5-6-7
Graph:
Since absolute value is always positive or zero, it will ALWAYS be greater than ANY negative number.
ALL Real Numbers will have an Absolute Value ≥ -2
R
ABSOLUTE VALUE
0-5 -4 -3 -2 -1 1 2 3 4 5
SYMBOL:
Distance from zero
OPPOSITES or ADDITIVE INVERSES
Have the same absolute value
-2 = 2 2 = 2
-1.5 1.5
