1. The problem is to understand and simplify expressions involving double absolute values, such as $| |x| |$. 2. The absolute value function $|x|$ returns the non-negative value of $x$, meaning $|x| \geq 0$ for any real number $x$. 3. Applying the absolute value twice, $| |x| |$, means taking the absolute value of a non-negative number $|x|$. Since $|x|$ is already non-negative, the outer absolute value does not change the value. 4. Therefore, $| |x| | = |x|$. 5. In simple terms, the double absolute value is equivalent to a single absolute value because the absolute value function is idempotent: applying it multiple times does not change the result beyond the first application.