Rules for removing absolute value bars:

1. For cases of  where  is positive:

Rewrite as

 

2. For cases of 

Rewrite as

 

3. For cases of  where  is positive:

Rewrite as

 OR 

4. For cases of 

Rewrite as

 OR 




This is case 4, so to remove the absolute value
bars:



Rewrte that as

 OR 

Solve each

 OR 

 OR 

 OR 

Mark the points  and  on the number line.
Notice that to plot  you mark a point 
of the way between  and  



We mark them with solid circles because the problem has 
and/or .  If it had been just  and/or 
you would have left the circles open


 ------------@-----------------------@----------
 -4   -3   -2   -1    0    1    2    3    4    5   
     
For the  part we shade left of 


<============@-----------------------@----------
 -4   -3   -2   -1    0    1    2    3    4    5

And since the solution is that OR  we shade
right of :


<============@-----------------------@===========>
 -4   -3   -2   -1    0    1    2    3    4    5

Now for the interval notation, the left shaded part is

(,-]

and we union that with the right shaded part [3,oo)

and the final answer in interval notation is

        (,] U [3,)

Edwin