To solve an absolute value equation, isolate the
absolute value on one side of the equal sign, and establish two
cases:
Case 1:
Set the
expression inside the
absolute value symbol equal to
the
other given expression. 
Case 2:
Set the
expression inside the
absolute value symbol equal to
the
negation of the other given
expression 
. . .
and always CHECK your answers.
The two cases create "derived"
equations. These derived equations may not always be
true equivalents to the original equation.
Consequently, the roots of the derived equations MUST BE
CHECKED in the original equation so that you do not list
extraneous roots as answers.

Your graphing calculator can be used to solve
absolute value equations and/or double check your answers.


How to use
your
TI83+/84+ graphing calculator with absolute
value equations.
Click calculator. 


Example 1:
(Two cases with two solutions)
Case 1:

Case 2:

Answer:
x = 16, x = 4 
Check:

Check:

The solutions are 16 or 4.
On a number line, these value are each 6 units away from 10. 
Example 2:
(No
solution)
As soon as you isolate the absolute value expression,
you observe:
There is no need to work out the two cases in
this problem. Absolute value is NEVER equal to a negative value. This equation is never true. The
answer is the
empty set .
Example 3:
(Two
cases with one solution)
Case 1:

Case 2:

Answer:
x = 1 
Check:
Not an answer! 
Check:

The check shows that x =
3 is NOT a solution to this absolute value equation. There
is only ONE answer, x = 1.
Always check!!!! 
Example 4:
A machine fills Quaker Oatmeal containers with 32
ounces of oatmeal. After the containers are filled, another
machine weighs them. If the container's weight differs from
the desired 32 ounce weight by more than 0.5 ounces, the container
is rejected. Write an equation that can be used to find the
heaviest and lightest acceptable weights for the Quaker Oatmeal
container. Solve the equation.
Solution: Let x =
the weight of the container.
Case 1:

Case 2:

Answer:
x = 31.5 ounces (lightest)
x = 32.5 ounces (heaviest) 
When setting up a word problem
involving absolute value, remember that absolute value can
represent "distance" from a given point.
The
difference between the answer (x) and the desired
point (32) is placed under the absolute value symbol.
This absolute value is then set equal to the desired
"distance" (0.5). 
