What keeps me from solving this question is that the values that a can take are not given, hence we don't know where in the distribution of X it'll be located. It if were, let's say on the left side of µ, then I'd find P(X<a) and times that by two but since I don't know where it's located it might as well be on the right side and then P(X<a) would not give half of alpha...
The answer to this question is 2phi(-(15-a)/1.5)
$\alpha$ is not needed. They ask you to find the p-value, that is the area of the two tails of the gaussian
I suppose you know that $a\sim N(\mu;\frac{\sigma^2}{n})$ and thus your p-value is
$$\mathbb{P}[Z >\frac{|a-15|}{\sqrt{\frac{4.5^2}{9}}}]=\mathbb{P}\Bigg[Z >\frac{|a-15|}{1.5}\Bigg]$$
..and this is exaclty what you solution states (it is just a matter of writing this probability in terms of $\Phi$ but I think you can do it by yourself)
After calculating it, if you p-value is lower than the fixed $\alpha$ you will reject your null hypothesis, if it will be higher you will not reject it.