What's the [solution for] 9|x-2| - 10 >= -64 ?

I thought it was all real numbers, but my teacher said it was no solution since absolute value can't equal a negative.

I agree with your solution; your teacher might have been thinking about a different situation.

The inequality symbol ≥ means greater than OR equal.

As long as the left-hand side evaluates to a number that is greater than -64, the inequality is true. This happens for any value of x.

If you solved the inequality by hand, you ought to have reached this:

|x - 2| ≥ -6

This shows that x can be any Real number because |x-2| is non-negative, and any non-negative number is greater than -6.