Absolute Value Inequalities
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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
TI-83+ graphing calculator  with absolute value inequalities.
Click calculator.


Example 1:
(solving with "greater than")

Solve: 

Case 1:

or

Case 2:

 

x < 15   or   x > 25

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:


$35,010 <  x  < $40,190

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