If `-4lex lt 2 ` then ||x+2|-3 lies in the inerval

Options

A. (1,3]
B. [1,3]
C. [0,3]
D. `[0,oo)`

Explanation

To solve the problem, we need to find the range of the expression \( ||x + 2| - 3| \) given the inequality \( -4 \leq x < 2 \).

### Step-by-Step Solution:

1. **Understanding the given range for x**:
We have the inequality:
\[ -4 \leq x < 2 \]
This means \( x \) can take values from \(-4\) to \(2\) (not including \(2\)).

2. **Transforming the inequality**:
We want to analyze the expression \( |x + 2| \). First, we can find the minimum and maximum values of \( x + 2 \):
- When \( x = -4 \):
\[ x + 2 = -4 + 2 = -2 \]
- When \( x \) approaches \( 2 \) (but does not include it):
\[ x + 2 \to 2 + 2 = 4 \]
Therefore, the range of \( x + 2 \) is:
\[ -2 \leq x + 2 < 4 \]

3. **Finding the range of \( |x + 2| \)**:
The absolute value function \( |x + 2| \) will affect the range:
- The minimum value of \( |x + 2| \) occurs at \( x = -4 \):
\[ |x + 2| = |-2| = 2 \]
- The maximum value occurs as \( x + 2 \) approaches \( 4 \):
\[ |x + 2| \to |4| = 4 \]
Thus, the range of \( |x + 2| \) is:
\[ 2 \leq |x + 2| < 4 \]

4. **Subtracting 3 from \( |x + 2| \)**:
Now we need to find the range of \( ||x + 2| - 3| \):
- The minimum value of \( |x + 2| - 3 \) occurs when \( |x + 2| = 2 \):
\[ |x + 2| - 3 = 2 - 3 = -1 \]
- The maximum value occurs when \( |x + 2| \) approaches \( 4 \):
\[ |x + 2| - 3 \to 4 - 3 = 1 \]
Therefore, the range of \( |x + 2| - 3 \) is:
\[ -1 < |x + 2| - 3 < 1 \]

5. **Finding the range of \( ||x + 2| - 3| \)**:
Now we apply the absolute value:
- The minimum value of \( ||x + 2| - 3| \) occurs at \( |x + 2| - 3 = -1 \):
\[ ||x + 2| - 3| = |-1| = 1 \]
- The maximum value occurs at \( |x + 2| - 3 = 1 \):
\[ ||x + 2| - 3| = |1| = 1 \]
Hence, the range of \( ||x + 2| - 3| \) is:
\[ 0 \leq ||x + 2| - 3| < 3 \]

6. **Final range**:
Therefore, we conclude that:
\[ ||x + 2| - 3| \text{ lies in the interval } [0, 3) \]

### Conclusion:
The expression \( ||x + 2| - 3| \) lies in the interval \([0, 3)\).