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

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Borys Shumyatskiy eNotes educator| Certified Educator

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.

Charlie Hooper eNotes educator| Certified Educator

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]`

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