Can someone explain this solution to a basic circuit analysis question?
The solution starts off with a mistake. i_x is not 0.5 ma.: it can't possibly be
so. The problem was probably meant to have the arrow pointing to the left or
have i_x be -0.5 ma. Knowing that, the problem is a simple Ohm's law problem. 
The voltage across R is 9 volts, so the voltage across the right 10k resistor is
6 volts, and the current through it is 0.6 ma. going right to left. That current
splits with 0.5 ma going to the left 10k resistor and 0.1 ma going down through
R. R is 90k.