I just started reading some book where it explicitly says that  

$\left\{2x : x \in\mathbb{Z}, |x| < 4\right\} =$ $\left\{−6,−4,−2,0,2,4,6\right\}$

But that's how I tried to solve it and I had different answer:

$2x: x \in\mathbb{Z}$ which means, *the set of all things in the form $2x$ such that $x$ is an element of $\mathbb{Z}$*  

$\left\{..., -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, ...\right\}$

Then we have $|x| < 4$, which scales down the previous set to a range of only $[-2, 2]$, so if the set contains $\left\{-6, -4, 4, 6\right\}$ they're all proved to be wrong because their absolutes are all $≥ 4$.

So the correct set should actually only be $\left\{-2, 0, 2\right\}$ according to my understanding.
[![Book Of Proof by Richard Hammack: Page 16](https://i.stack.imgur.com/tI8ZA.png)](https://i.stack.imgur.com/tI8ZA.png)

But Page 16 of Book Of Proof by Richard Hammack has a different solution with no explanation

I'm quite new to these set expressions so please tell me if I interpreted anything wrong.