There are really two equations in one: when 10-3z is greater than or equal to zero, 10-3z=9, so 3z=1 and z=1/3. When 10-3z is less than zero, then 3z-10=9, so 3z=19 and z=19/3 (6.33). Now we look at the preconditions: 10-3z=>0 implies 10=>3z or z<=10/3 (3.33). This precondition is consistent with the result z=1/3 (0.33) because 0.33<3.33. The other precondition is 10-3z<0, so 10<3z and z>3.33. This precondition is consistent with the result because 6.33>3.33. The two answers are z=0.33 or 6.33. Both answers fit the original absolute equation.
Another way to solve this is to square both sides: (10-3z)^2=81. When you take the square root of both sides: 10-3z=+9, so 10-3z=-9 (z=6.33) or 10-3z=9 (z=0.33).