For a triangle exists, the side lengths of the triangle must be such that the sum of the two shorter sides must be greater than the third side.
This also is equivalent to any two sides must have a sum greater than the third side.
So
7+10 > x^2, => x^2 < 17 => x < sqrt(17) (maximum)
7+x^2 > 10, => x^2 >3 => x > sqrt(3)
Therefore
sqrt(3) < x < sqrt(17)
If x must be an integer,
2< x < 4, or x = {2,3,4}