The value of `x` satisfying `|2-|2x-4|| = 1` are distinct and can ve ordered s `a lt b lt c lt d` which of the followin inequalities holds good?

Options

A. `0 lt a lt b lt c lt d`
B. `a lt 0 lt b lt c lt d`
C. `a lt b lt 0 lt c lt d`
D. `a lt b lt c lt 0 lt d`

Explanation

To solve the equation \( |2 - |2x - 4|| = 1 \), we will break it down step by step.

### Step 1: Remove the outer absolute value
The equation \( |2 - |2x - 4|| = 1 \) can be interpreted in two cases based on the definition of absolute value.

**Case 1:**
\[ 2 - |2x - 4| = 1 \]

**Case 2:**
\[ 2 - |2x - 4| = -1 \]

### Step 2: Solve Case 1
From Case 1, we have:
\[ 2 - |2x - 4| = 1 \]
Subtract 2 from both sides:
\[ -|2x - 4| = -1 \]
Multiply both sides by -1:
\[ |2x - 4| = 1 \]

### Step 3: Remove the inner absolute value for Case 1
Now we will break this down into two more cases.

**Subcase 1.1:**
\[ 2x - 4 = 1 \]
Add 4 to both sides:
\[ 2x = 5 \]
Divide by 2:
\[ x = \frac{5}{2} \]

**Subcase 1.2:**
\[ 2x - 4 = -1 \]
Add 4 to both sides:
\[ 2x = 3 \]
Divide by 2:
\[ x = \frac{3}{2} \]

### Step 4: Solve Case 2
From Case 2, we have:
\[ 2 - |2x - 4| = -1 \]
Subtract 2 from both sides:
\[ -|2x - 4| = -3 \]
Multiply both sides by -1:
\[ |2x - 4| = 3 \]

### Step 5: Remove the inner absolute value for Case 2
Again, we will break this down into two more cases.

**Subcase 2.1:**
\[ 2x - 4 = 3 \]
Add 4 to both sides:
\[ 2x = 7 \]
Divide by 2:
\[ x = \frac{7}{2} \]

**Subcase 2.2:**
\[ 2x - 4 = -3 \]
Add 4 to both sides:
\[ 2x = 1 \]
Divide by 2:
\[ x = \frac{1}{2} \]

### Step 6: Collect all solutions
From both cases, we have the following solutions:
1. \( x = \frac{5}{2} \)
2. \( x = \frac{3}{2} \)
3. \( x = \frac{7}{2} \)
4. \( x = \frac{1}{2} \)

### Step 7: Order the solutions
Now we will order these solutions:
- \( \frac{1}{2} \)
- \( \frac{3}{2} \)
- \( \frac{5}{2} \)
- \( \frac{7}{2} \)

Thus, we can denote them as \( a = \frac{1}{2}, b = \frac{3}{2}, c = \frac{5}{2}, d = \frac{7}{2} \).

### Step 8: Determine the inequality
The ordered values give us the inequality:
\[ a < b < c < d \]
Which translates to:
\[ \frac{1}{2} < \frac{3}{2} < \frac{5}{2} < \frac{7}{2} \]