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.

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}

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

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

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

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

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

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

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

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?

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

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 +3 +3 -3 -3 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

or 5X -3  = 7 A B 5X -3 = 7 -(5X -3) = 7 -5X +3 = 7 +3 +3 -3 -3 5X = +3 +3 -3 -3 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?

and 3X +6   9 A B 3X +6  9 -(3X +6)  9 -3X - 6  9 -6 -6 +6 +6 3X -6 -6 +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?

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

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

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

  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