The number of integral values of `x` satisfying the equation `|x-|x-4||=4` is

Options

A. `5`
B. `7`
C. `9`
D. infinite

Explanation

To solve the equation \( |x - |x - 4|| = 4 \), we will break it down into cases based on the properties of absolute values.

### Step 1: Analyze the inner absolute value
The expression \( |x - 4| \) can be split into two cases:
1. Case 1: \( x - 4 \geq 0 \) (i.e., \( x \geq 4 \))
2. Case 2: \( x - 4 < 0 \) (i.e., \( x < 4 \))

### Step 2: Solve Case 1: \( x \geq 4 \)
In this case, \( |x - 4| = x - 4 \). Thus, the equation becomes:
\[ |x - (x - 4)| = 4 \]
This simplifies to:
\[ |4| = 4 \]
This is always true, meaning any \( x \geq 4 \) is a solution.

### Step 3: Solve Case 2: \( x < 4 \)
In this case, \( |x - 4| = 4 - x \). Thus, the equation becomes:
\[ |x - (4 - x)| = 4 \]
This simplifies to:
\[ |2x - 4| = 4 \]
Now, we need to solve the absolute value equation \( |2x - 4| = 4 \) which gives us two subcases:
1. Subcase 1: \( 2x - 4 = 4 \)
- Solving this gives:
\[ 2x = 8 \implies x = 4 \]
However, this solution does not satisfy \( x < 4 \).

2. Subcase 2: \( 2x - 4 = -4 \)
- Solving this gives:
\[ 2x = 0 \implies x = 0 \]
This solution satisfies \( x < 4 \).

### Step 4: Combine the solutions
From Case 1, we have all integral values \( x \geq 4 \) (which includes \( 4, 5, 6, \ldots \) and so on). From Case 2, we found \( x = 0 \).

### Step 5: Count the integral solutions
The integral solutions satisfying the equation are:
- From Case 1: All integers \( x \geq 4 \) (infinitely many)
- From Case 2: The integer \( x = 0 \)

Thus, the total number of integral values of \( x \) satisfying the equation is infinite.

### Final Answer
The number of integral values of \( x \) satisfying the equation \( |x - |x - 4|| = 4 \) is infinite.

---