How many solutions does the equation have? –|3n + 6| − 5 = 3

Question

How many solutions does the equation have?

^–|3n + 6| − 5 = 3

Answer 1

The idea is to isolate the absolute value expression first.

-|3n + 6| - 5 = 3

-|3n + 6| = 8 (add 5 to both sides)

|3n + 6| = -8 (multiply (or divide) both sides by -1)

At this point is becomes clear that there are no solutions because the absolute value of any number is nonnegative. No matter what number you plug in for n, the absolute value of 3n + 6 must be ≥ 0, so can never give a result of -8.

Suppose the original problem was:

-|3n + 6| + 5 = 3

-|3n + 6| = -2

|3n + 6| = 2

At this point the technique is to realize that the expression inside the absolute value symbol must have a value of either 2 or -2, so:

3n + 6 = 2 OR 3n + 6 = -2

3n = -4 OR 3n = -8

n = -4/3 OR n = -8/3

So in this latter case there would be two solutions.