Obtuse-Angled Triangles Count

By triangle inequality, $|15 - 8| < x < 15 + 8 \implies 7 < x < 23$.

For the triangle to be obtuse, the square of the longest side must be strictly greater than the sum of squares of the other two sides.

Case 1: $15$ is the longest side ($x \le 15$): $15^2 > 8^2 + x^2 \implies 225 > 64 + x^2 \implies x^2 < 161$ Since $x > 7$, integer values of $x$ can be $8, 9, 10, 11, 12$. Total = 5 values.

Case 2: $x$ is the longest side ($x > 15$): $x^2 > 8^2 + 15^2 \implies x^2 > 64 + 225 = 289 \implies x > 17$ Since $x < 23$, integer values of $x$ can be $18, 19, 20, 21, 22$. Total = 5 values.

Total possible integer values for $x = 5 + 5 = 10$.