#### Answer

$x = 10.4~in$

#### Work Step by Step

Let the length of side $a = 12.0~in$ and let angle $A = 70.4^{\circ}$
Let angle $B = 54.8^{\circ}$. We can find the length of the side opposite angle $B$, which is the length of $x$:
$\frac{x}{sin~B} = \frac{a}{sin~A}$
$x = \frac{a~sin~B}{sin~A}$
$x = \frac{(12.0~in)~sin~(54.8^{\circ})}{sin~(70.4^{\circ})}$
$x = 10.4~in$