Analyze the logical form of the following statement. (How To Prove It, Velleman) Part d of question 3 in section 2.1 asks us to analyze the logical form and state the free variables of

If there is a number $x$ such that $x^2+5x=w$ and there is a number $y$ such that $4-y^2=w$, then $w$ is between -10 and 10.

My original answer was 
$\forall w[\exists x(x^2+5x=w) \land \exists y(4-y^2=w) \implies (w>-10) \land (w<10)]$ with no free variables.
However, the answer here doesn't have the $\forall w$ and thus states that $w$ is a free variable.
Which one of these answers is correct and why? If the correct answer is not the one I stated, when do you know to have a free variable over using a $\forall$?
Thank you in advance for your help.
 A: The core problem here is not one of symbolic logic, but one of understanding ordinary mathematical prose. Doing that is more complex than most people (even mathematicians) realize, because it is very context dependent.
Usually, when we write something of the form

If there is a number $x$ such that $x^2+5x=w$ and there is a number $y$ such that $4-y^2=w$, then $w$ is between -10 and 10.

we do indeed intend to assert that this claim is true for every value of $w$, and then it is appropriate to represent the claim symbolically with a $\forall w$. This is especially the case if what is written is something like

Theorem 1234. If there is a number $x$ such that $x^2+5x=w$ and there is a number $y$ such that $4-y^2=w$, then $w$ is between -10 and 10.

However, we can also write something like the above in the middle of an argument where we're already speaking about some particular $w$. And in that case the argument that $-10<w<10$ may depend on something else we have already concluded about this $w$, and is not stated as an explicit assuption.,
In that context it is important to formalize the claim without quantifying $w$.
The decision really can't be made without knowing the context, and exercises that ask you to do such translation without context are misleading at best. Teaching students to be pedantic about stating quantifications explicitly is all well and fine -- especially because one needs to be aware of whether the $w$ is intended to be arbitrary or not -- but it should be accompanied by an acknowledgement that actual published mathematical prose is rarely that pedantic.
