Question

How do you solve `abs(8-3n)<=18`?

Answer 1

Every absolute value expression has two domains, one that is positive and one that is negative. Write both inequalities and find the respective ranges.

Explanation:

We write definition of the given absolute value expression as two equations with two different domain restrictions:

`|8 - 3n| = 8 - 3n; 8 - 3n >= 0`
and
`|8 - 3n| = 3n- 8; 8 - 3n < 0`

Simplify the domain restrictions:

`|8 - 3n| = 8 - 3n; n < 8/3`
and
`|8 - 3n| = 3n- 8; n >=8/3`

Use the above to create two inequalities with the two domain restrictions:

`8 - 3n <= 18; n < 8/3`
and
`3n - 8 <= 18; n >=8/3`

Solving the ranges:

`n > 10/3; n < 8/3` [1]
and
`n <= 10/3; n >=8/3` [2]

The two inequalities in [1] contradict so it must be discarded but [2] does not have a contradiction; we can write it as:

`8/3 <= n <= 10/3`