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Let assume you are given this problem:
prove that equation | x | = -5 has no solution.
It is very simple to prove.
Since the absolute value | x | is always NON-NEGATIVE, | x | can not be negative.
Hence, the given equation with the negative right side HAS NO SOLUTIONS.
PROVED.
In the same way, you can prove that these equations have no solutions :
| x | + 5 = 3;
| x + 1 | = -1;
| 2x + 1 | + 2 = 1,
and many other similar equations.