1. 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.

2. Where are the numbers that are 6 units from zero?
Graphing Absolute Value
Graph:
Where are the numbers that are 6 units from zero? X = 6 OR 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7
{ 6, -6}

3. Where are the numbers that are more than 3 units from zero?
Graphing Absolute Value
Graph: X  3
Where are the numbers that are more than 3 units from zero? OR
To the left of -3
To the right of 3 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7
Do you want the -3? NO
Do you want the 3? NO
X < -3 OR X > 3

4. Where are the numbers that are less than 3 units from zero?
Graphing Absolute Value
Graph: X ≤ 3
Where are the numbers that are less than 3 units from zero?
To the right of -3
AND at the same time
To the left of 3 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7
Do you want the 3?
Yes
Do you want the -3?
Yes 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7
-3  X  3

5. An expression that represents any real number except 0
 = positive #
Could represent an expression that has a positive value or a negative value.
If > 0
If < 0 or
= positive #
-( ) = positive #
X = 3
If X > 0
If X< 0 A B or
X = 3
-(X)= 3

6. or X = 3 X = 3 -(X)= 3 A B -1(X)= 3 -1 -1 X = -3 A B AB If X > 0-5 -4 -3 -2 -1 1 2 3 4 5 B -5 -4 -3 -2 -1 1 2 3 4 5
AB -5 -4 -3 -2 -1 1 2 3 4 5
{ 3, -3}
Put together

7. An expression that represents any real number except 0
  > positive #
Could represent an expression that has a positive value or a negative value.
If > 0
If < 0 or
> positive #
-( ) > positive #
X   4
If X > 0
If X< 0 A B or
X  4
-(X)  4

8. or X   4 X  4 -(X)  4 A B -1(X)  4 -1 -1 X  -4 A B AB
IF YOU MULTIPLY OR DIVIDE BOTH SIDES OF AN INEQUALITY BY A NEGATIVE TURN THE INEQUALITY SIGN AROUND
-1(X)  4 -1 -1
X  -4 A -5 -4 -3 -2 -1 1 2 3 4 5 B -5 -4 -3 -2 -1 1 2 3 4 5
AB -5 -4 -3 -2 -1 1 2 3 4 5
X < -4 OR X > 4
Put together

9. An expression that represents any real number except 0
  < positive #
Could represent an expression that has a positive value or a negative value.
If > 0
If < 0 or
< positive #
-( ) < positive #
X   2
If X > 0
If X< 0 A B
and
X  2
-(X)  2

10. and X   2 X  2 -(X)  2 A B -1(X)  2 -1 -1 X  -2 A B AB
If X > 0
X  2
and
If X< 0
-(X)  2 A B
IF YOU MULTIPLY OR DIVIDE BOTH SIDES OF AN INEQUALITY BY A NEGATIVE TURN THE INEQUALITY SIGN AROUND
-1(X)  2 -1 -1
X  -2 A -5 -4 -3 -2 -1 1 2 3 4 5 B -5 -4 -3 -2 -1 1 2 3 4 5
AB -5 -4 -3 -2 -1 1 2 3 4 5
-2  X  2
What’s the same?

11.  = Positive #   > Positive #   < Positive #OR
- ( ) = Positive #
  > Positive #
> Positive # OR
- ( ) > Positive #
  < Positive #
< Positive #
AND
- ( ) < Positive #

12. or 2X -3  > 1 A B 2X -3 > 1 -(2X -3) > 1 -2X +3 > 1
IF YOU MULTIPLY OR DIVIDE BOTH SIDES OF AN INEQUALITY BY A NEGATIVE TURN THE INEQUALITY SIGN AROUND 2X
> 4
-2X > -2 2 2 -2 -2 X
> 2
X < 1 A -5 -4 -3 -2 -1 1 2 3 4 5 B -5 -4 -3 -2 -1 1 2 3 4 5
AB -5 -4 -3 -2 -1 1 2 3 4 5
X < 1 OR X > 2
Put together

13. or 5X -3  = 7 A B 5X -3 = 7 -(5X -3) = 7 -5X +3 = 7 +3 +3 -3 -3 5X = 5X = 10
-5X = 4 5 5 -5 -5 = X 2
X =
-.8 A -5 -4 -3 -2 -1 1 2 3 4 5 B -5 -4 -3 -2 -1 1 2 3 4 5
A  B -5 -4 -3 -2 -1 1 2 3 4 5
{ 2, -.8}
Put together?

14. and 3X +6   9 A B 3X +6  9 -(3X +6)  9 -3X - 6  9 -6 -6 +6 +6 3X
+6 +6 3X  3
-3X  15 3 3 -3 -3  X 1
X  -5 A -5 -4 -3 -2 -1 1 2 3 4 5 B -5 -4 -3 -2 -1 1 2 3 4 5
AB -5 -4 -3 -2 -1 1 2 3 4 5
-5  X  1
What’s the same?

15. Graphing Absolute Value
Absolute value is always a positive number or zero.
Graph:
2X - 3 =
- 6
Absolute value will never equal a negative number.
NO SOLUTION 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7

16. Absolute value is always a positive number or zero.
Graphing Absolute Value
Absolute value is always a positive number or zero.
Graph:
X - 5 -2
<
Since absolute value is always positive or zero, it cannot be less than a negative number.
NO SOLUTION 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7

17. Absolute value is always a positive number or zero.
Graphing Absolute Value
Absolute value is always a positive number or zero.
Graph:
3X+2 -2 ≥
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 2 3 1 5 4 7 6 -1 -4 -2 -3 -5 -6 -7 R

18.   ABSOLUTE VALUE SYMBOL: Distance from zero
 
SYMBOL:
Distance from zero -5 -4 -3 -2 -1 1 2 3 4 5
-1.5
1.5
OPPOSITES or ADDITIVE INVERSES
Have the same absolute value
-2 = 2
2 = 2