`||x+3|-5|=2`

Answer

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

### Step 1: Remove the outer absolute value
The equation \( ||x+3|-5|=2 \) implies two cases:
1. \( |x+3|-5 = 2 \)
2. \( |x+3|-5 = -2 \)

### Step 2: Solve the first case
For the first case \( |x+3|-5 = 2 \):
\[ |x+3| = 2 + 5 = 7 \]
This gives us two sub-cases:
1. \( x+3 = 7 \)
2. \( x+3 = -7 \)

**Sub-case 1:**
\[ x + 3 = 7 \implies x = 7 - 3 = 4 \]

**Sub-case 2:**
\[ x + 3 = -7 \implies x = -7 - 3 = -10 \]

### Step 3: Solve the second case
For the second case \( |x+3|-5 = -2 \):
Since the absolute value cannot be negative, this case does not yield any solutions.

### Step 4: Solve the sub-cases from the first case
Now we have the solutions from the first case:
1. \( x = 4 \)
2. \( x = -10 \)

### Step 5: Check the second part of the first case
Now we need to check the second part of the first case:
From \( |x+3| = 7 \), we already solved this and found:
- \( x = 4 \)
- \( x = -10 \)

### Step 6: Solve the second part of the first case
Now we will solve for \( |x+3| = 5 \) from the second case:
This gives us two sub-cases:
1. \( x+3 = 5 \)
2. \( x+3 = -5 \)

**Sub-case 1:**
\[ x + 3 = 5 \implies x = 5 - 3 = 2 \]

**Sub-case 2:**
\[ x + 3 = -5 \implies x = -5 - 3 = -8 \]

### Final Solutions
Combining all the solutions, we have:
1. \( x = 4 \)
2. \( x = -10 \)
3. \( x = 2 \)
4. \( x = -8 \)

Thus, the final solutions are \( x = 4, -10, 2, -8 \).

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