|a|=|b|, which of the following must be true :

Responding to a pm:

Neither method needs to be used here. Just think of the definition of mod we use to remove the mod sign.

|x| = x if x >= 0 and |x| = -x if x < 0

We don't know whether a and b are positive or negative. |a|=|b| when absolute values of both a and b are the same. The signs can be different or same. There are 4 cases: a and b are positive, a is positive b is negative, a is negative b is positive, a and b are negative.
For a must be true question, the relation should hold in every case.

1. a=b
Doesn't hold when a and b have opposite signs. e.g. a = 5, b= -5

2.|a|=-b
Doesn't hold when b is positive because -b will become negative while left hand side is always non negative. e.g. a = 5, b = 5
\(|5| \neq -5\)

3.-a=-b
Doesn't hold when a and b have opposite signs. e.g. a = 5, b = -5
\(-5 \neq 5\)

Answer (E)