Number of solutions of the equation `|2-|x||=x +4` is a.0 b.1 c.2 d.Infinite

Options

A. 0
B. 1
C. 2
D. Infinite

Explanation

To find the number of solutions for the equation \( |2 - |x|| = x + 4 \), we will analyze it step by step.

### Step 1: Understand the equation
We start with the equation:
\[ |2 - |x|| = x + 4 \]
This equation involves absolute values, which means we need to consider different cases based on the value of \( x \).

### Step 2: Break down the absolute value
We can break down the absolute value \( |2 - |x|| \) into two cases based on the value of \( |x| \).

**Case 1:** \( |x| \leq 2 \)

In this case, \( 2 - |x| \geq 0 \), so:
\[ |2 - |x|| = 2 - |x| \]
Thus, the equation becomes:
\[ 2 - |x| = x + 4 \]
Rearranging gives:
\[ - |x| - x = 2 \]
This can be rewritten as:
\[ -|x| - x = 2 \]
If \( x \geq 0 \), then \( |x| = x \):
\[ -2x = 2 \implies x = -1 \]
If \( x < 0 \), then \( |x| = -x \):
\[ -(-x) - x = 2 \implies 0 = 2 \text{ (no solution)} \]

**Case 2:** \( |x| > 2 \)

In this case, \( 2 - |x| < 0 \), so:
\[ |2 - |x|| = |x| - 2 \]
Thus, the equation becomes:
\[ |x| - 2 = x + 4 \]
Rearranging gives:
\[ |x| - x = 6 \]
If \( x \geq 0 \), then \( |x| = x \):
\[ 0 = 6 \text{ (no solution)} \]
If \( x < 0 \), then \( |x| = -x \):
\[ -x - x = 6 \implies -2x = 6 \implies x = -3 \]

### Step 3: Summary of solutions
From the analysis:
- From Case 1, we found one solution: \( x = -1 \).
- From Case 2, we found another solution: \( x = -3 \).

### Conclusion
The total number of solutions to the equation \( |2 - |x|| = x + 4 \) is **2**.

Thus, the answer is:
**c. 2**

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