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]`