How work out the length of this side? 
This is probably so basic, but I just cannot see it.  If you do not know that the left side is $5x$ and are only given $3x$ and $4x$, how do you deduce $5x$?

Are you sure we are not given that the dotted side if 5x?  What actually is the exercise we are supposed to solve?
We can conclude that if the triangle is right then the side is 5x.  And if we are given that the side if 5x we can conclude the triangle is right.
But if we are given neither we can't conclude either.
If we knew the length of the bold arrow we could use trigonometry to figure out the angle and third side.  And given that the triangle is right or the third side is 5x, we could calculate the length of the bold arrow.
But given know of those we can't conclude anything.
Looking at the image, it'd be my interpretation that we are given the side is 5x and we are asked to determine the length of the bold arrow.
... which I'd do by noting:  $4x^2 + 3x^2 = 5x^2$ so triangle is right.
So area of triangle is $1/2*4x*3x = 6x^2$.  But area of triangle is $1/2*bold*5x = 6x^2$ so $bold = 12x/5$
