Is it possible to build a gapless rectangle with non-integer side lengths using rectangles each with two integer side length and two non-integer side length? The rectangles are not required to be the same size.
No. What you are asking is not possible. Because if an integer and a non-integer pair up on two opposite sides of the new rectangle, then two of the sides will have non-integer values, but the other two will be forced to have integer values.
Given: Rectangle with area of 5N (n is a non-integer number) & Rectangle with area of 4X (x is a non-integer number)
Proven impossible by the following screenshot: