Compound absolute value with different left and right values

Question

I'm not sure how to find the solution set for a compound absolute value with different left and right values. Here is an example:

A = { 2 ≤ |x| < 4 : x ε integers }

My thinking is to create two inequality statements. Such as:

|x| ≥ 2 and |x| < 4

Solving for each:

-2 ≥ x ≥ 2 and -4 < x < 4

Since |x| cannot be a negative number, would my solution set be:

{ 0, 1, 2, 3 } ?

Thanks in advance

Answer 1

We want $2\le |x|<4$. That means $|x|=2$ or $|x|=3$, which means $x=-2,2,3,-3$.

Answer 2

Note that $|x|\ge 2$ is not the same as $-2\ge x\ge 2$ (there is no such $x$) and that $$|x|\ge 2\iff x\le -2\ \ \text{or}\ \ x\ge 2.$$