smc

We first notice that the sides $10$ and $24$, can be part of $2$ different right triangles, one with sides $10,24,26$, and the other with a leg somewhere between $21$ and $22$. We now notice that if $x$ is less than or equal to $21$, one of the angles is obtuse, and that the same is the same for any value of $x$ above $26$. Thus the only integer values of $x$ that fit the conditions, are $x=22,23,24,\text{and }25.$ So, the answer is $\boxed{4}$