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 analyze the expression by considering two cases based on the value of \( x \).

### Step 1: Case 1 - \( x < 4 \)

In this case, \( |x - 4| = 4 - x \) because \( x - 4 \) is negative. Therefore, we can rewrite the equation:

\[ |x - (4 - x)| = 4 \]

This simplifies to:

\[ |x + x - 4| = 4 \quad \Rightarrow \quad |2x - 4| = 4 \]

Now, we can break this absolute value into two equations:

1. \( 2x - 4 = 4 \)
2. \( 2x - 4 = -4 \)

#### Solving the first equation:

\[ 2x - 4 = 4 \quad \Rightarrow \quad 2x = 8 \quad \Rightarrow \quad x = 4 \]

Since \( x < 4 \) in this case, \( x = 4 \) is not a valid solution.

#### Solving the second equation:

\[ 2x - 4 = -4 \quad \Rightarrow \quad 2x = 0 \quad \Rightarrow \quad x = 0 \]

Since \( 0 < 4 \), this is a valid solution.

### Step 2: Case 2 - \( x \geq 4 \)

In this case, \( |x - 4| = x - 4 \) because \( x - 4 \) is non-negative. Therefore, we can rewrite the equation:

\[ |x - (x - 4)| = 4 \]

This simplifies to:

\[ |4| = 4 \]

This statement is always true, meaning any \( x \geq 4 \) is a solution.

### Step 3: Summary of Solutions

From Case 1, we found one solution: \( x = 0 \).

From Case 2, we found that all \( x \geq 4 \) are solutions.

### Step 4: Integral Values

The integral solutions are:
- From Case 1: \( x = 0 \)
- From Case 2: All integers \( x \) such that \( x \geq 4 \) (i.e., \( 4, 5, 6, \ldots \))

### Conclusion

The integral values of \( x \) satisfying the equation are \( 0 \) and all integers starting from \( 4 \) upwards. Therefore, the total number of integral values of \( x \) is infinite.

### Final Answer

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

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