solve `||x-2|-1|lt 2`
`"(a) (2,5) (b)(-1, 5) (c) (-2,-1) (d) (0,5)"`

Options

A. `(2,5)`
B. `(-1,5)`
C. `(-2,-1)`
D. `(0,5)`

Answer

To solve the inequality \( ||x-2|-1| < 2 \), we will break it down step by step.

### Step 1: Remove the outer absolute value
The expression \( ||x-2|-1| < 2 \) implies that:
\[ -2 < |x-2|-1 < 2 \]
This can be split into two separate inequalities:
1. \( |x-2|-1 < 2 \)
2. \( |x-2|-1 > -2 \)

### Step 2: Solve the first inequality
Starting with the first inequality:
\[ |x-2|-1 < 2 \]
Adding 1 to both sides gives:
\[ |x-2| < 3 \]
This means:
\[ -3 < x-2 < 3 \]
Adding 2 to all parts of the inequality results in:
\[ -1 < x < 5 \]

### Step 3: Solve the second inequality
Now, we solve the second inequality:
\[ |x-2|-1 > -2 \]
Adding 1 to both sides gives:
\[ |x-2| > -1 \]
Since the absolute value is always non-negative, this inequality is always true. Therefore, it does not impose any additional restrictions on \( x \).

### Step 4: Combine the results
From the first inequality, we found:
\[ -1 < x < 5 \]
The second inequality does not restrict \( x \) further. Thus, the solution set is:
\[ (-1, 5) \]

### Step 5: Determine the intersection with the condition from the absolute value
However, we must consider the nature of the absolute value \( |x-2| \). The expression \( |x-2| \) is defined for all \( x \), but we also need to ensure that \( |x-2| \) is greater than 0, which leads us to:
\[ x \neq 2 \]
Thus, we need to exclude \( x = 2 \) from our solution set.

### Final Solution
The solution set is:
\[ (-1, 2) \cup (2, 5) \]
This can be expressed in interval notation as:
\[ (-1, 5) \text{ excluding } 2 \]

### Answer
The closest option to our solution is:
- (b) (-1, 5)