Question

How do you solve the inequality `2<=abs(4-x)<7` and write your answer in interval notation?

Answer 1

`-3 < x <= 2` OR `6<=x<11`

Explanation:

First, recall the rules of absolute values in inequality expressions.

Absolute value is less than something rule
If `abs(A) < B`, that's the same as saying `-B < A < B`

Absolute value is greater than something rule
If `B<=abs(A)`, that's the same as saying either `B<=A` or `A<=-B`

The inequality we are given can be broken into two parts.

First part: Less than rule
`abs(4-x)<7`

Applying the "less than rule" gives
`-7<4-x<7`

Subtract `4` from all three sides
`-11 < -x < 3`

Divide by `-1` and flip the inequality signs
`-3 < x < 11`

Second part: Greater than rule
`2<=abs(4-x)`

Applying the "greater than rule" gives
Either `2<=4-x` or `4-x<=-2`

Solving both for `x` gives

Either `x<=2` or `x>=6`

Finally, combine all solutions into one

The solution: `-3 < x < 11` gives the most-negative and most-positive outside boundaries.

The solution: Either `x<=2` or `x>=6` gives the inner boundaries.

ANSWER: `-3 < x <= 2` OR `6<=x<11`

Here is what the graph would look like: