The number of real roots of the equation `|2-|1-|x|||=1` is

Options

A. `1`
B. `3`
C. `5`
D. `6`

Answer

To find the number of real roots of the equation \( |2 - |1 - |x|| | = 1 \), we will solve it step by step.

### Step 1: Analyze the outer absolute value
We start with the equation:
\[ |2 - |1 - |x|| | = 1 \]
This implies two cases:
1. \( 2 - |1 - |x|| = 1 \)
2. \( 2 - |1 - |x|| = -1 \)

### Step 2: Solve the first case
For the first case:
\[ 2 - |1 - |x|| = 1 \]
Subtracting 2 from both sides gives:
\[ -|1 - |x|| = -1 \]
This simplifies to:
\[ |1 - |x|| = 1 \]
This again gives us two sub-cases:
1. \( 1 - |x| = 1 \)
2. \( 1 - |x| = -1 \)

#### Sub-case 1.1: \( 1 - |x| = 1 \)
\[ -|x| = 0 \implies |x| = 0 \implies x = 0 \]

#### Sub-case 1.2: \( 1 - |x| = -1 \)
\[ -|x| = -2 \implies |x| = 2 \]
This gives:
\[ x = 2 \quad \text{or} \quad x = -2 \]

### Step 3: Collect solutions from the first case
From the first case, we have found the solutions:
\[ x = 0, \quad x = 2, \quad x = -2 \]

### Step 4: Solve the second case
Now we consider the second case:
\[ 2 - |1 - |x|| = -1 \]
Subtracting 2 from both sides gives:
\[ -|1 - |x|| = -3 \]
This simplifies to:
\[ |1 - |x|| = 3 \]
This again gives us two sub-cases:
1. \( 1 - |x| = 3 \)
2. \( 1 - |x| = -3 \)

#### Sub-case 2.1: \( 1 - |x| = 3 \)
\[ -|x| = 2 \implies |x| = -2 \quad \text{(not possible)} \]

#### Sub-case 2.2: \( 1 - |x| = -3 \)
\[ -|x| = -4 \implies |x| = 4 \]
This gives:
\[ x = 4 \quad \text{or} \quad x = -4 \]

### Step 5: Collect solutions from the second case
From the second case, we have found the solutions:
\[ x = 4, \quad x = -4 \]

### Step 6: Combine all solutions
Combining all the solutions from both cases, we have:
\[ x = 0, \quad x = 2, \quad x = -2, \quad x = 4, \quad x = -4 \]

### Conclusion
The total number of real roots of the equation \( |2 - |1 - |x|| | = 1 \) is **5**.

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