Resistances calculations
1. Your calculation for R2 contains a sign error
Double-check your equation
$$
-v_{gs} + i_dR_2 - 6 = 0
$$
There is a sign error in there. It looks like you're using KVL; double-check and be very detail-oriented in how you apply KVL. Correct the sign error and you will get the correct answer.
2. Your calculation for R1 is correct -- actually more correct than than the &quot;solution&quot;!
By your calculations you end up with a value of 4.8kΩ. In the answer key they just round this to 5kΩ:
$$
R_1=\frac{6V-1.2V}{1mA}=4.8k\Omega \approx 5k\Omega
$$