To solve the equation [tex]\(\left|\left| |x|-1 \right|-2\right| = 3\)[/tex], we need to carefully handle the absolute value functions step-by-step.
### Step-by-Step Solution:
1. Innermost Absolute Value:
[tex]\[
y = |x| - 1
\][/tex]
2. Next Absolute Value:
[tex]\[
z = |y| - 2 = ||x|-1| - 2
\][/tex]
3. Setting up the full equation using the outer absolute value:
[tex]\[
\left|\left| |x|-1 \right|-2\right| = 3
\][/tex]
Breaking it down, we have:
[tex]\[
|z|=3 \implies z = 3 \text{ or } z = -3
\][/tex]
4. Solution for [tex]\(z = 3\)[/tex]:
[tex]\[
||x|-1|-2 = 3
\][/tex]
This can be further broken down into two cases:
[tex]\[
||x|-1| - 2 = 3 \text{ or } ||x|-1| - 2 = -3
\][/tex]
Case 1:
[tex]\[
||x|-1| - 2 = 3 \implies ||x|-1| = 5
\][/tex]
This further splits into two sub-cases:
[tex]\[
|x| - 1 = 5 \implies |x| = 6 \implies x = 6 \text{ or } x = -6
\][/tex]
and
[tex]\[
|x| - 1 = -5 \text{ which has no solution since } |x|-1 \geq 0.
\][/tex]
Case 2:
[tex]\[
||x|-1| - 2 = -3 \implies ||x|-1| = -1
\][/tex]
This also has no solution since [tex]\(||x|-1 \geq 0\)[/tex].
Therefore, from solving [tex]\(||x|-1|-2 = 3\)[/tex], we have:
[tex]\[
x = 6 \text{ or } x = -6
\][/tex]
5. Solution for [tex]\(z = -3\)[/tex]:
[tex]\[
||x|-1|-2 = -3
\][/tex]
This case also has no solution since [tex]\(|\cdot| - 2 \geq -2\)[/tex].
### Conclusions
By examining the cases, we find that the only valid solutions are:
- [tex]\( x = 6 \)[/tex]
- [tex]\( x = -6 \)[/tex]
Hence, the number of solutions for the equation [tex]\(\left|\left| |x|-1 \right|-2\right| = 3\)[/tex] is [tex]\(2\)[/tex].
### Answer:
[tex]\[
\boxed{2}
\][/tex]
