“Is it possible to construct a triangle with the side lengths of 7, 8, and 15?”

In the usual sense of things (Euclidean Geometry), no.

You could have a ‘degenerate triangle‘, but that’s not exactly a triangle in the way we usually think about a triangle.

8 + 7 = 15.

So basically you want the sum of the two smaller sides to be larger (even slightly larger) than the largest side to get a triangle that isn’t degenerate, a triangle in the way we usually think about triangles.