How to show $x\in \{y\in\mathbb R: 2^{k-2}\leq |y|\leq 2^{k+1}\}\Leftrightarrow k=j, j+1, j+2$?

Question

I need help to show the following:

Let $x\in \mathbb R$ such that $2^j\leq |x|\leq 2^{j+1}$ for some integer $j\geq 1$. Then $$x\in \{y\in\mathbb R: 2^{k-2}\leq |y|\leq 2^{k+1}\}\Leftrightarrow k=j, j+1, j+2?$$

Any help will be valuable. Thanks.

Answer 1

Hint: Note that $\{y\in\mathbb{R}\mid 2^{k-2}\leq |y|\leq 2^{k+1}\}$ is equal to $$\{y\in\mathbb{R}\mid 2^{k-2}\leq |y|\leq 2^{k-1}\}\cup\{y\in\mathbb{R}\mid 2^{k-1}\leq |y|\leq 2^k\}\cup\{y\in\mathbb{R}\mid 2^k\leq |y|\leq 2^{k+1}\}.$$