Solve the inequality |1/2 x-3| lt= 4

To solve this inequality, find where the expression under absolute value sign is non-negative and where it is negative.
1/2 x - 3 gt= 0  for  x gt= 6  and  1/2 x - 3 lt 0  for  x lt 6.
 
Therefore for  x gt= 6  we obtain  |1/2 x - 3| = 1/2 x - 3 lt= 4,  i.e.  1/2 x lt= 7,  x lt= 14.  Thus  x in [6, 14].
 
For  x lt 6  we obtain  |1/2 x - 3| = -(1/2 x - 3) lt= 4,  i.e.  1/2 x - 3 gt= -4,  1/2 x gt= -1,  x gt= -2. Thus  x in [-2, 6).
 
Combining the results for x lt 6 and x gt= 6 we obtain that  x in [-2, 14]. This is the answer.

Solve |1/2 x -3| lt=4
First, the absolute value means
-4lt= 1/2 x -3 lt=4
Now continue to solve the inequality by adding 3 to all sides.
-1lt= 1/2 x lt=7
Multiply by 2 .
-2lt= x lt=14
Therefore x is on the closed interval [-2,14]