I'm solving exercise 3 in page 63 of the book "How to prove it" by Daniel J. Velleman. I don't know if my solutions are correct so I post those solutions here and want you to tell me if I'm correct, and a form of reasoning that allows me to solve problems like that.
Analyze the logical forms of the following statements. The universe of discourse is $\mathbb{R}$. What are the free variables in each statement?
a) "Every number that is larger than x is larger than y"
Solution: $\forall z [(z>x) \implies (z > y)]$. The free variables are x and y. z is a bound variable.
b) For every number a, the equation $ax^2+4x-2=0$ has at least one solution iff a $\gt$ -2.
Solution: $\forall a$ (the equation $ax^2+4x-2=0$ has at least one solution $\iff a \ge -2$. a is a bound variable and x is not a variable at all.
c) All solutions of the inequality $x^3 - 3x \lt 3$ are smaller than 10.
Solution: let S(x) stand for "x is a solution of the inequality $x^3 - 3x \lt 3$". $\forall x (S(x) \implies x \lt 10$. x is a bound variable and there are no free variables at all.
d) 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.
Solution: $[\exists x (x^2+5=w) \land \exists y (4-y^2=w)] \implies -10 \lt w \lt 10$. w is a free variable. x and y are bound variables.
Everything looks good except (b), which should be $$\forall a \, (\,(\exists x \,\, (ax^2+4x-2=0)) \iff a \ge -2) $$ so $a$ and $x$ are both bound variables.