To remove the absolute value bars of

      |Ax + B| < C  , where C is a positive number, write this:

    
        -C < Ax + B < C

Then solve that for x in the middle.  The solution set will
usually be of the form:

            (D, E) 

Your example 4:

          |2x + 1| < 5

      -5 < 2x + 1 < 5

Add -1 to all three sides:

        -6 < 2x < 4

Divide through by 2 (That will NOT reverse the < signs)

         -3 < x < 2

Graph:

---------o===================o--------
-5  -4  -3  -2  -1   0   1   2   3   4

Interval notation:

         (-3, 2)


 [x] +6 < 10

Your example 6:

          |x| + 6 < 10

Isolate the |x| by subtracting 6 from both sides:

             |x| < 4


           -4 < x < 4

Graph:

-----o===============================o--------
-5  -4  -3  -2  -1   0   1   2   3   4   5   6

Interval notation:

         (-4, 4)


---------------------------------------

To remove the absolute value bars of

      |Ax + B| > C  , where C is a positive number, write this:

    
   Ax + B < -C  OR  Ax + B > C

Then solve that for x in each part, leaving the word "OR" between them

The solution usually will be of this form:

      (, D) U (E, )

Your example 2:

              |x - 5| > 2 

       x - 5 < -2 OR x - 5 > 2

Add 5 to borth sides of both parts:

            x < 3 OR x > 7

Graph:

<====================o---------------o===========>
-2  -1   0   1   2   3   4   5   6   7   8   9  10

Solution set:

     (, 3) U (7,)

Edwin