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

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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.

Approved by eNotes Editorial Team
An illustration of the letter 'A' in a speech bubbles

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]`

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial