Solve |4-|x-1||=3

Explanation

To solve the equation \( |4 - |x - 1|| = 3 \), we will break it down step by step.

### Step 1: Set up the equation
We start with the equation:
\[ |4 - |x - 1|| = 3 \]

### Step 2: Break down the absolute value
The absolute value equation can be split into two cases:
1. \( 4 - |x - 1| = 3 \)
2. \( 4 - |x - 1| = -3 \)

### Step 3: Solve the first case
For the first case:
\[ 4 - |x - 1| = 3 \]
Subtract 4 from both sides:
\[ -|x - 1| = 3 - 4 \]
\[ -|x - 1| = -1 \]
Multiplying both sides by -1 gives:
\[ |x - 1| = 1 \]

### Step 4: Break down the absolute value again
For \( |x - 1| = 1 \), we have two sub-cases:
1. \( x - 1 = 1 \)
2. \( x - 1 = -1 \)

#### Sub-case 1:
\[ x - 1 = 1 \implies x = 2 \]

#### Sub-case 2:
\[ x - 1 = -1 \implies x = 0 \]

### Step 5: Solve the second case
Now, we solve the second case:
\[ 4 - |x - 1| = -3 \]
Subtract 4 from both sides:
\[ -|x - 1| = -3 - 4 \]
\[ -|x - 1| = -7 \]
Multiplying both sides by -1 gives:
\[ |x - 1| = 7 \]

### Step 6: Break down the absolute value again
For \( |x - 1| = 7 \), we have two sub-cases:
1. \( x - 1 = 7 \)
2. \( x - 1 = -7 \)

#### Sub-case 1:
\[ x - 1 = 7 \implies x = 8 \]

#### Sub-case 2:
\[ x - 1 = -7 \implies x = -6 \]

### Step 7: Compile all solutions
From both cases, we have the solutions:
- From the first case: \( x = 2, 0 \)
- From the second case: \( x = 8, -6 \)

Thus, the final solutions are:
\[ x = -6, 0, 2, 8 \]

### Final Answer:
The solutions to the equation \( |4 - |x - 1|| = 3 \) are:
\[ x = -6, 0, 2, 8 \]