Solving an absolute value inequality
problem is similar to solving an absolute value equation.
Start by isolating the absolute value
on one side of the inequality symbol, then follow the rules below:
If the symbol is > (or >)
:
(or)
If a > 0, then the solutions to
are
x > a
or
x <  a. 

If a < 0,
all real numbers
will satisfy
. 
Think about it:
absolute value is always positive (or zero), so, of course, it is greater
than any
negative number. 

If the symbol is < (or
<)
:
(and)
If a > 0, then the
solutions to
are x < a
and x
>  a.
Also written:
 a < x <
a. 

If a < 0, there is
no solution to
. 
Think
about it: absolute
value is always positive (or zero),
so, of course, it cannot be less than
a negative
number. 

R E M E M B E R:
When working with any absolute value
inequality,
you must create two cases.
If <, the connecting word is "and".
If >, the connecting word is "or".
To set up the two cases:
x < a
Case 1: Write the problem
without the absolute value sign, and solve the inequality.


x > a
Case 2: Write the problem
without the absolute value sign,
reverse the inequality, negate the value NOT under
the absolute value, and solve the inequality. 


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


How to use
your
TI83+ graphing calculator with absolute
value inequalities.
Click calculator. 


Example 1:
(solving with "greater than")
Solve:
Case 1:


or 
Case 2:




Note that there are two parts
to the solution and that the connecting word is "or".


Example 2:
(solving with "less than or equal to")
Solve:
Case 1:


and 
Case 2:



Note that there are two parts
to the solution and that the connecting word is "and". 

also written as:


Example 3:
(isolating the absolute value first)
Solve:
Case 1:


and 
Case 2:



Note that the absolute value
is isolated before the solution begins. 

also written as:



Example 4:
(compound inequalities) 
Separate a compound
inequality into two separate problems. 

Solve:


Case 1:


or

Case 2:


Case 1:


and

Case 2:


x > 4 or x
< 6 
8 < x < 6 
Now, where do the solutions
overlap???

8 < x < 6 as well as
4 < x < 6 

Example 5:
(all values work)
Solve:
Case 1:


or

Case 2:




You already know the answer! Absolute
value is ALWAYS positive (or zero), so it is always > 3.
All values work! 


x
> 7 or x < 1
Answer:


Example 6:
(no values work)
Solve:
Case 1:


and

Case 2:




You already know the answer!
Absolute value is ALWAYS positive (or zero). It is NEVER < 6.
No values work! 


x
< 7 and x > 5 ???
Answer:
(the empty set) 

Example 7:
(word problem)
At the Brooks Graphic Company, the
average starting salary for a new graphic designer is $37,600, but
the actual salary could differ from the average by as much $2590.

The absolute value
represents the set of all points x that are less
than b units away from a. 

b.) 
Case 1:


and
Case 2:




a.) Write an absolute value inequality to describe this
situation.
b.) Solve the inequality to find the range of the starting
salaries. 
Solution:
a.)
the difference between the average and the salary < $2590 
