The problem asks to solve the inequality $| |x| - 2 | \le 1$.

1. Problem Description

The problem asks to solve the inequality $| |x| - 2 | \le 1$.

2. Solution Steps

We have the inequality $| |x| - 2 | \le 1$. This inequality is equivalent to the following compound inequality:

$-1 \le |x| - 2 \le 1$.

Adding $2$ to all parts of the inequality gives:

$-1 + 2 \le |x| - 2 + 2 \le 1 + 2$, which simplifies to

$1 \le |x| \le 3$.

This inequality is equivalent to two separate inequalities:

$|x| \ge 1$ and $|x| \le 3$.

The inequality $|x| \ge 1$ means $x \ge 1$ or $x \le -1$.

The inequality $|x| \le 3$ means $-3 \le x \le 3$.

Combining these inequalities, we have:

$x \ge 1$ and $-3 \le x \le 3$, which implies $1 \le x \le 3$.

Also, $x \le -1$ and $-3 \le x \le 3$, which implies $-3 \le x \le -1$.

Therefore, the solution is $-3 \le x \le -1$ or $1 \le x \le 3$. In interval notation, the solution is $[-3, -1] \cup [1, 3]$.

3. Final Answer

The solution is $x \in [-3, -1] \cup [1, 3]$.