[UG]

x is ethanoic acid

[Borek]

(3.88 x 10-4 - 3.88 x 10-3 x) / x = 1.75 x 10^-5 x
Calculating gives x = 0.0999549 M which would give [H⁺] = 4.51 x 10^-5 M and pH = 4.3
Does this mean the approximation is not valid?

I am afraid so. Unfortunately, that also means you should solve full 3^rd degree equation. Correct result doesn't have to be far from what you have calculated, but it is hard to predict how far it is. From my calculations pH is around 3.3.

[UG]

But the answer says that the solubility is 0.130 mol L^-1 and pH is equal to 4.23 

[Borek]

Unfortunatelty that most likely means they used wrong approximation and have not checked its validity later.

Try to use equations you already have and the answer given as real to calculate concentrations of all ions and check if all mass balances and reaction quotients have correct values. My bet is that you will find they are inconsistent with question data.

[UG]

Hmm... you're right, the answers come out all funny. So the way to solve this is to just try and find a way to express an equation using only one of the 4 variables?

[Borek]

Yes. Unfortunately that leads to a 3rd degree polynomial.