|z+8| = |z-3|

To remove the absolute value bars | |, we break it into two
equations, 

(1) where the expressions inside that absolute values have THE
SAME sign,

and

(2) where the expressions inside that absolute values have OPPOSITE
signs.

We examine case (1), which is

   z + 8 = z - 3
      
Subtract z from both sides and we have
      
       8 = -3

Which is false and shows that case (1) has no solution. 

So we examine case (2), which is found by multiplying one
side by (-1), so they will have opposite signs:

   z + 8 = -1(z - 3)
   z + 8 = -z + 3
      
Add z to both sides and we have
      
  2z + 8 = 3
      
Subtracting 8 from both sides:

      2z = -5

Dividing both sides by 2:

       z = 

So this absolute value equation has only one solution, since 
case (1) turned out to have no solution.

Edwin