`||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
We start by recognizing that 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 \( |x+3|-5 = 2 \)
Adding 5 to both sides gives us:

\[ |x+3| = 7 \]

Now, we can remove the absolute value, which leads to two sub-cases:

1. \( x + 3 = 7 \)
2. \( x + 3 = -7 \)

#### Sub-case 1: \( x + 3 = 7 \)
Subtracting 3 from both sides:

\[ x = 4 \]

#### Sub-case 2: \( x + 3 = -7 \)
Subtracting 3 from both sides:

\[ x = -10 \]

### Step 3: Solve the second case \( |x+3|-5 = -2 \)
Adding 5 to both sides gives us:

\[ |x+3| = 3 \]

Again, we can remove the absolute value, leading to two more sub-cases:

1. \( x + 3 = 3 \)
2. \( x + 3 = -3 \)

#### Sub-case 1: \( x + 3 = 3 \)
Subtracting 3 from both sides:

\[ x = 0 \]

#### Sub-case 2: \( x + 3 = -3 \)
Subtracting 3 from both sides:

\[ x = -6 \]

### Step 4: Compile all solutions
From the cases and sub-cases, we have the following solutions:

\[ x = 4, -10, 0, -6 \]

### Final Answer
The solutions to the equation \( ||x+3|-5|=2 \) are:

\[ x = -10, -6, 0, 4 \]