**Quote:**

In the question, it is not mentioned that "*a*" or "*b*" is a constant …

*a* and

*b* are variables.

**Quote:**

… " |*a*| + |*b*| = 0 " does not guarantee that *a* and *b* are "0" …

Any absolute value is non-negative. If

*a* and

*b* were not zeroes, then |

*a*| would be positive and |

*b*| would be positive.

positive + positive > 0

That would contradict the equation. Even if only one variable was not zero, the left side would still be positive. So the both variables can possess only one value, 0.

**Quote:**

… It could mean *a* = -*b*.

If to plug in

*a* = -

*b* we will get |-

*b*| + |

*b*| as the left side of the equation. It is not necessarily 0. Plug any non-zero value for

*b*, for example

*b* = 2 yields |-2| + |2| = 2 + 2 = 4. So the conclusion "It could mean

*a* = -

*b*" is not correct.

If to solve the equation |-

*b*| + |

*b*| = 0:

The absolute values |-

*b*| and |

*b*| are the same (equal), so the equation transforms into

2|

*b*| = 0

*b* = 0