If the equation `|2 - x| - |x + 1|= k` has exactly one solution, then number of integral values of `k` is (A) 7 (B) 5 (C) 4 (D) 3

Options

A. 7
B. 5
C. 4
D. 3

Explanation

To solve the equation \( |2 - x| - |x + 1| = k \) and find the number of integral values of \( k \) for which this equation has exactly one solution, we can follow these steps:

### Step 1: Define the function
Let \( f(x) = |2 - x| - |x + 1| \). We need to analyze this function in different intervals based on the critical points where the expressions inside the absolute values change.

### Step 2: Identify critical points
The critical points occur when the expressions inside the absolute values are zero:
- \( 2 - x = 0 \) gives \( x = 2 \)
- \( x + 1 = 0 \) gives \( x = -1 \)

Thus, we will consider three intervals:
1. \( x < -1 \)
2. \( -1 \leq x < 2 \)
3. \( x \geq 2 \)

### Step 3: Evaluate \( f(x) \) in each interval

1. **For \( x < -1 \)**:
\[ f(x) = (2 - x) - (-(x + 1)) = 2 - x + x + 1 = 3 \]

2. **For \( -1 \leq x < 2 \)**:
\[ f(x) = (2 - x) - (x + 1) = 2 - x - x - 1 = 1 - 2x \]

3. **For \( x \geq 2 \)**:
\[ f(x) = (-(2 - x)) - (x + 1) = -2 + x - x - 1 = -3 \]

### Step 4: Summary of the function
Now we have:
- \( f(x) = 3 \) for \( x < -1 \)
- \( f(x) = 1 - 2x \) for \( -1 \leq x < 2 \)
- \( f(x) = -3 \) for \( x \geq 2 \)

### Step 5: Analyze the function for intersections with \( k \)
To have exactly one solution for \( f(x) = k \), the horizontal line \( y = k \) must intersect the graph of \( f(x) \) at exactly one point.

- For \( k = 3 \): intersects at \( x < -1 \) (one solution).
- For \( k = 1 \): intersects at \( x = 0 \) (one solution).
- For \( k = 0 \): intersects at \( x = 0.5 \) (one solution).
- For \( k = -1 \): intersects at \( x = 1 \) (one solution).
- For \( k = -2 \): intersects at \( x = 1.5 \) (one solution).
- For \( k = -3 \): intersects at \( x \geq 2 \) (one solution).

### Step 6: Identify integral values of \( k \)
The integral values of \( k \) for which there is exactly one solution are:
- \( k = 3 \)
- \( k = 1 \)
- \( k = 0 \)
- \( k = -1 \)
- \( k = -2 \)

Thus, the integral values of \( k \) are \( 3, 1, 0, -1, -2 \), which gives us a total of **5 integral values**.

### Final Answer
The number of integral values of \( k \) is **5**.

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