We can solve the equation by graphing y1=|8-x| and y2=|5x+8| and finding the intersections:

Click here to see the graph.

The solutions are 0 and -4.

Indeed, for x = 0 we have:

|8-0|=|5*0+8|

8 = 8

And for x = -4

|8-(-4)|=|5*(-4)+8|

|12| = |-12|

12 = 12