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

1 Answer
Sep 25, 2016

Answer:

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#