SOLUTION: Solve absolute value inequality and graph solution set. 1 > 1/2|6 - x| - 3/4

Solve absolute value inequality and graph solution set. 
1>1/2|6-x|-3/4

     1            3 
1 > ---|6 - x| - ---
     2            4

Write the 1 as 1/1 so every term will have a fraction:

 1     1            3 
--- > ---|6 - x| - ---
 1     2            4

The LCD = 4 so multiply every term by 4 written as 4/1.
That is put

 4 
---·
 1

In front of every term:

 4   1     4   1            4   3 
---·--- > ---·---|6 - x| - ---·---
 1   1     1   2            1   4

Now cancel the 2 into the 4 and the 4 into the 4

           2                1
 4   1     4   1            4   3 
---·--- > ---·---|6 - x| - ---·---
 1   1     1   2            1   4
               1                1    

And all you have left is

     4 > 2|6 - x| - 3

Isolate the term with the absolute value:

Add 3 to both sides:

    7 > 2|6 - x|

Solve for the absolute value

    7
   --- > |6 - x|
    2

It's easier to see if the absolute value is on
the left.  So we can write the above as

              7   
   |6 - x| < ---
              2

The absolute value of 6 - x will be less than 7/2
if and only if 6 - x is between -7/2 and +7/2.
That is:

   7             7   
- --- < 6 - x < ---
   2             2       

Clear of fractions by multiplying all three sides
by 2

-7 < 12 - 2x < 7

Add -12 to all three sides

 -19 < -2x < -5

Divide all three sides by -2, remembering to reverse
the inequalities when dividing by a negative number.

 19/2 > x > 5/2

It is easier to see this is we write this inequality
smallest to largest:

 5/2 < x < 19/2

  2.5 < x < 9.5

The graph:

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

In interval notation the solution is (5/2, 19/2)

Edwin