His Error: He set the expressions 10x-1 and 59 equal to each other. Those two angles are not congruent.
He should have added the two expressions to get 180
(10x-1) + (59) = 180
10x+(-1+59) = 180
10x+58 = 180
10x+58-58 = 180-58
10x = 122
10x/10 = 122/10
x = 12.2