The solution set of the inequation `|[|x|-7]|-5< 0,` is ... ([*] denotes the greatest integer function )

Options

A. `[3, 12)`
B. `(-12, -3]`
C. `(-12, 12)`
D. `(-12, -3] cup [3, 12)`

Explanation

To solve the inequation \( |||x| - 7| - 5 < 0 \), we will break it down step by step.

### Step 1: Simplify the Inequation
We start with the given inequation:
\[ |||x| - 7| - 5 < 0 \]
This can be rewritten as:
\[ |||x| - 7| < 5 \]

### Step 2: Remove the Absolute Value
The expression \( |||x| - 7| < 5 \) implies:
\[ -5 < |x| - 7 < 5 \]
This gives us two inequalities to solve:
1. \( |x| - 7 < 5 \)
2. \( |x| - 7 > -5 \)

### Step 3: Solve the First Inequality
From the first inequality \( |x| - 7 < 5 \):
\[ |x| < 12 \]
This means:
\[ -12 < x < 12 \]

### Step 4: Solve the Second Inequality
From the second inequality \( |x| - 7 > -5 \):
\[ |x| > 2 \]
This means:
\[ x < -2 \quad \text{or} \quad x > 2 \]

### Step 5: Combine the Results
Now we have two sets of inequalities:
1. From \( |x| < 12 \): \( -12 < x < 12 \)
2. From \( |x| > 2 \): \( x < -2 \) or \( x > 2 \)

We need to find the intersection of these two sets.

### Step 6: Find the Intersection
The intervals from the inequalities are:
- From \( -12 < x < 12 \): This is the interval \( (-12, 12) \).
- From \( x < -2 \) or \( x > 2 \): This gives us two intervals \( (-\infty, -2) \) and \( (2, \infty) \).

Now, we find the intersection:
1. For \( x < -2 \): The intersection with \( (-12, 12) \) gives \( (-12, -2) \).
2. For \( x > 2 \): The intersection with \( (-12, 12) \) gives \( (2, 12) \).

### Final Solution
Combining both intervals, we have:
\[ x \in (-12, -2) \cup (2, 12) \]

### Conclusion
The solution set of the inequation \( |||x| - 7| - 5 < 0 \) is:
\[ (-12, -2) \cup (2, 12) \]