I'm not sure on the following parts, I insert what I'm not sure on in () brackets.

(a) Let $\displaystyle a$ and $\displaystyle b$ be ratonal numbers satisfying $\displaystyle a < b$. To which of the following three sets does the number $\displaystyle \frac{2a +3b}{5}$ belong:

$\displaystyle A = \{x \in \mathbb{Q} : x<a\},

B = \{x \in \mathbb{Q} : a<x<b\},

C = \{x \in \mathbb{Q} : b<x\}$

(It seems fairly obvious that it's in set B but I'm not sure exactly why...)

(b) Let $\displaystyle X$ be a set and let $\displaystyle \leq$ be a total order on $\displaystyle X$. We say that $\displaystyle X$ is dense if for any two $\displaystyle x,y \in X$ with $\displaystyle x < y$ there exists $\displaystyle z \in X$ such that $\displaystyle x<z<y$

Consider the set

$\displaystyle A = \{\frac{a}{5^n} : a,n \in \mathbb{Z}, n \geq 0\}$

Is A dense?

(it looks as though it is dense because for every two elements you can incerease n so that you find an element between them but I wasn't 100% sure on this.)

(c) Prove that $\displaystyle r=1$ is the only positive rational number such that $\displaystyle r + \frac{1}{r}$ is an integer.

(I'm not sure what to do here, I tried to suppose that $\displaystyle r + \frac{1}{r}$ is an integer and that $\displaystyle r = \frac{a}{b}$ and then $\displaystyle \frac{a^2 +b^2}{ab}$ is an integer but that this only happens when $\displaystyle a=b=1$, is any of that right?)

(d) How many positive real numbers x are there such that $\displaystyle x + \frac{1}{x}$ is an integer? Answer should be one of: 0,1,2,3,..., countable infinite, uncountably infinite.

(x=1 is the only rational from the previous part so its certainly not 0. But are there irrationals x such that $\displaystyle x + \frac{1}{x}$ is an integer? I don't know...)

Thanks for any help!