How do you know if an absolute value inequality is "No Solution" or not?

Answer 1

You have to try to solve the inequality and then conclude at some point that there are no solutions. Sometimes you can use some inequalities that always hold, like:

|a| >= 0

|a+b| <= |a| + |b|

etc.

If you can rearrange an inequality so that it contradicts these standard inequalities, it cannot have a solution.