Here's how I would do it. Since x< y, y- x> 0. Since q< 1, subtracting q from each part, 0< 1-q. Since 0< q, -q< 0 and 1- q< 1. Setting p= 1- q, we also have 0< p< 1. multiplying each part by the positive number y- x, 0(y-x)< p(y-x)< 1(y-x) or 0< py- px< y- x. Adding x to each part, x< py- px+ x< y. -px+ x= (1- p)x= qx and py= (1-q)y so that is x< (1- q)y+ qx< y.
(I first did that using "q" rather than "p" and arrived at x< qy+ (1-q)x< y which is true but not what is wanted. That when I switched to p= 1- q and did the same proof.)