Absolute value doubts

Nevermind, totally misread OP

there is nothing wrong. we can derive |a| = |b| from a = |b| i.e.
a = |b| => |a| = |b| (but not the other way around).

you just need to consider the range of numbers that the first equation implies for 'a'. in the first equation (a = |b|) the right side ( |b| ) is 0 or greater than 0, therefore the left side (a) is also 0 or greater, so "a" can't be a negative number
(a >= 0).


let me explain in a different way:

1) original equation:
a = |b| implies that:
I. 'b' can be anything
II. a is 0 or greater than 0 i.e. 'a' cant be negative (a>=0).
for example:
b= -0.6 and a is 0.6
b = 0 and a is 0
b = 10 and a is 10

2) the new equation:
|a| = |b| says that absolute value of 'a' is equal to absolute value of 'b'.
why is this ok? because from the first equation we already know that 'a' is not negative, and since absolute value of non-negative numbers are the same as the number, the new equation true also.

remember always pay attention to range of values a variable can take. for example in X = 1/Y you should beware that y can never be 0.

Thanks amostofi1999. i get it now. After reading your explanation, i also figured out why the other way round (that is a=|b| derived from |a|=|b|) is not correct.

if you have difficulty understanding, i have to recommendations for you regarding absolute value function. 1- try to really understand and learn the definitions deeply; dont just memorize a bunch of formulas 2- at first, learn the concept in your language. best source is probably official high school math or other books that are teaching the subject to first time learners.