a|x-1| + b|x-2| + c|x-3| +d|x-4| =5

Find condition on a,b,c,d such that the equation has infinitely many solutions.

For x
a(1-x) + b(2-x) + c(3-x) +d(4-x) =5,
(a+2b+3c+4d) - (a+b+c+d)*x =5,
if a+2b+3c+4d=5 and a+b+c+d=0, then it will have infinite solutions as any value of x less than 1 will satisfy the equation.
Similarly you can solve for other regions, you will get
a+2b+3c+4d=5 and a+b+c+d=0
-a+2b+3c+4d=5 and -a+b+c+d=0
-a-2b+3c+4d=5 and -a-b+c+d=0
-a-2b-3c+4d=5 and -a-b-c+d=0
-a-2b-3c-4d=5 and -a-b-c-d=0