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Let x be the "feel good" hours

Then 8- x  must be the "tired" hours

And   

rate in  feel good hrs * feel good hrs  + rate in tired hrs * tired hrs  = total distance

20 * x  +    12 * (8 - x)  = 122     simplify

20x  + 96  - 12x  =  122

8x + 96  = 122     subtract 96 from both sides

8x  =  26       divide both sides by 8

x  = 26/8  =  13 / 4  hrs   of  "feel good"....LOL!!!