Solve `||x-2|-3|lt 5 `

Explanation

To solve the inequality \( ||x - 2| - 3| < 5 \), we will break it down step by step.

### Step 1: Remove the outer absolute value
We start with the inequality:
\[ ||x - 2| - 3| < 5 \]
This can be rewritten as:
\[ -5 < |x - 2| - 3 < 5 \]
Since the left side of the inequality \(-5 < |x - 2| - 3\) is always true (absolute values are non-negative), we only need to focus on the right side:
\[ |x - 2| - 3 < 5 \]

### Step 2: Solve the right side of the inequality
Now, we add 3 to both sides:
\[ |x - 2| < 8 \]

### Step 3: Remove the absolute value
The inequality \( |x - 2| < 8 \) implies:
\[ -8 < x - 2 < 8 \]

### Step 4: Solve the compound inequality
Now, we can solve the compound inequality by adding 2 to all parts:
\[ -8 + 2 < x < 8 + 2 \]
This simplifies to:
\[ -6 < x < 10 \]

### Step 5: Final solution
Thus, the solution to the inequality \( ||x - 2| - 3| < 5 \) is:
\[ x \in (-6, 10) \]