Question

|a+b|=|a|+|b| if and only if ab>=0

Answer 1

i presume, u r asking to prove it, if not plz do comment.
First take the equation ab>=0.....(i)
and i am gonna take 4 cases to prove our main equation which is |a+b|= |a| + |b|.....(ii)
CASE 1. LET a=b=0
now put in (i), 0.0>=0
which is true
now put these value in (ii)
|0+0|= |0| + |0| => 0=0
it satisfies.

CASE 2. LET a=-2, b =-2
now put in (i), -2*-2=4>=0
which is true
now put these values in (ii)
|-2+(-2)| = |-2| + |-2|
=> |4|= 2+ 2
=> 4=4
it satisfies.

CASE 3. LET a =2, b=4
now put it in (i), 2*4=8>=0
which is true
now put these values in (ii)
|2+4|=|2|+|4|
=> |6|= 2+ 4
=> 6=6
it satisfies.

CASE 4. LET a= -2, b =4
now put it in (i),-2*4>=-8
which is not true
now put these values in (iil
|-2+4|=|-2| + |4|
=>2=6
it does not satify

HENCE, with the help of above 4 cases we can say that |a+b|=|a|+|b| will only true if ab>=0.

Answer 2

to prove   | a + b |  = | a | + | b |   iff   a b >= 0

1) Let  ab = 0  . So  a= 0 or  b = 0  or both.
    LHS = | a + 0| = | a| + 0|  = RHS     TRUE
   OR   = | 0 + b | = | 0 | + | b | = RHS    TRUE

2)  Let  ab > 0
     So  a & b are both positive or  both negative.

     (i)  a >0,   b>0.      LHS = a + b  = a + b        TRUE
     (ii)  a <0, b<0.     |b| = -b,  |a| = -a
                  LHS = -(a+b)  = RHS = -a - b    TRUE

3)  Let  a b <0
     So  a & b have opposite signs.

     (i)  a > 0,  b<0,    b = - |b|
         LHS= | a - |b|  |       RHS = a + |b|
               LHS= RHS only if   b =0.

     (ii)  a <0,  b>0.    a = - |a|
         LHS = |  - |a|  + b |       RHS = |a| + b
             LHS = RHS only if  a = 0.

Hence, LHS = RHS  only if   ab >= 0.  Proved.