Let $h(4x-1) = 2x + 7$.  For what value of $x$ is $h(x) = x$?
First, we find an expression for $h(x)$.  From our definition of $h$, we have $h(4y-1) = 2y+7$.  So, if we let $x=4y-1$, so that $y = (x+1)/4$, we have  \[h(x) = 2\cdot\frac{x+1}{4} + 7 = \frac{x+1}{2} + 7.\] Setting this equal to $x$ gives \[x =\frac{x+1}{2} + 7.\] Multiplying both sides by 2 gives $2x = x+1 + 14$, so $x = \boxed{15}$.