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

Explanation

To solve the equation |4 - |x - 1|| = 3, we can break it down into two cases based on the properties of absolute values. Here’s the step-by-step solution:

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

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

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

Now, we can break this absolute value into two equations:
1. \( x - 1 = 1 \)
2. \( x - 1 = -1 \)

Solving these:
1. \( x - 1 = 1 \) gives \( x = 2 \)
2. \( x - 1 = -1 \) gives \( x = 0 \)

### Step 4: Solve Case 2
For the second case:
\[ 4 - |x - 1| = -3 \]
Subtract 4 from both sides:
\[ -|x - 1| = -7 \]
Multiply both sides by -1:
\[ |x - 1| = 7 \]

Now, we can break this absolute value into two equations:
1. \( x - 1 = 7 \)
2. \( x - 1 = -7 \)

Solving these:
1. \( x - 1 = 7 \) gives \( x = 8 \)
2. \( x - 1 = -7 \) gives \( x = -6 \)

### Step 5: Combine Solutions
Now, we combine all the solutions from both cases:
- From Case 1: \( x = 2 \) and \( x = 0 \)
- From Case 2: \( x = 8 \) and \( x = -6 \)

Thus, the complete solution set is:
\[ x = -6, 0, 2, 8 \]

### Final Answer
The values of \( x \) are:
\[ -6, 0, 2, 8 \]

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