Answer
$v = 5.48~m/s$
The particle is moving in the +x direction.
Work Step by Step
In part (b), we found that from $x=0$ to $x=7.0~m$, the work done on the particle by the force is $~~30.0~J$
Since the particle started at rest, the kinetic energy of the particle will be equal to the work done on the particle by the force.
We can find the speed of the particle at $x = 7.0~m$:
$K = W$
$\frac{1}{2}mv^2 = W$
$v^2 = \frac{2W}{m}$
$v = \sqrt{\frac{2W}{m}}$
$v = \sqrt{\frac{(2)(30.0~J)}{2.00~kg}}$
$v = 5.48~m/s$
Since the work done on the particle is positive, the particle is moving in the +x direction.
