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

Sep 25, 2016

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:

$\frac{8}{3} \le n \le \frac{10}{3}$