The area of the rectangle is $6\times 8=48$ square units, so the unshaded area is $\frac{1}{4}\times 48=12$ square units. Therefore $(\frac{1}{2}\times x\times 2)+(\frac{1}{2}\times (6-x)\times 8)=12$, that is, $x+24-4x=12$, so $x=4$.