2.5 Absolute Value Equations

Recall: 

        The notation for the absolute value of 7 is 7 .

Definitions:

        Absolute Value:   The absolute value of a number is its distance from 0 on a number line.  The absolute value of a number is always positive. 

        If  x = k  then x = k or x = -k.   There are two solutions to an absolute value equation.

How To Solve Absolute Value Equations

1.      Remember you are looking for two solutions.  You must get rid of the absolute value symbol before you can find the solutions.  Each solution comes from a different equation.  We will refer to these equations as case 1 and case 2.

                                                example:           3x + 2 = 4

CASE 1

the positive case

What is inside the absolute value symbols is equal to a positive 4.

3x + 2 = 4

3x = 2

x = 2/3

 

CASE 2

the negative case

     What is inside the absolute value symbols is equal to a negative 4.

3x + 2 = - 4

3x = - 6

x  = - 2

 

  2.   Some absolute value equations have two absolute value symbols.  To solve these you still have two cases.    

 example:    x - 3= 2x + 1

CASE 1

the equal case

What is inside the 2 absolute value symbols are equal to each other.

x - 3 = 2x + 1

-4 = x  

 

CASE 2

the opposite case

What is inside the 2 absolute value symbols

are the opposite of each other.

-(x - 3) = 2x + 1

-x +3 = 2x + 1

2 = 3x

2/3 = x

OR

x - 3 = -(2x + 1)

x - 3 = -2x - 1

3x = 2/3

 x=2/3

 

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This page was created by Susan Peterson and updated on Wednesday August 13, 2003.