Which of the following shaded regions shows the graph of the inequality `y le |x + 2|` ?

Options

A.
B.
C.
D.

Answer

To solve the inequality \( y \leq |x + 2| \), we need to break it down into two separate inequalities based on the definition of absolute value.

### Step-by-Step Solution:

1. **Understanding the Absolute Value**:
The expression \( |x + 2| \) can be rewritten as two separate cases:
- Case 1: \( x + 2 \) when \( x + 2 \geq 0 \) (or \( x \geq -2 \))
- Case 2: \( -(x + 2) \) when \( x + 2 < 0 \) (or \( x < -2 \))

2. **Formulating the Inequalities**:
From the two cases, we can derive the following inequalities:
- For Case 1: \( y \leq x + 2 \)
- For Case 2: \( y \leq -x - 2 \)

3. **Finding the Points for Graphing**:
- For the first inequality \( y = x + 2 \):
- When \( x = 0 \), \( y = 0 + 2 = 2 \) → Point (0, 2)
- When \( y = 0 \), \( 0 = x + 2 \) → \( x = -2 \) → Point (-2, 0)

- For the second inequality \( y = -x - 2 \):
- When \( x = 0 \), \( y = -0 - 2 = -2 \) → Point (0, -2)
- When \( y = 0 \), \( 0 = -x - 2 \) → \( x = -2 \) → Point (-2, 0)

4. **Plotting the Points**:
- On a Cartesian plane, plot the points (0, 2), (-2, 0) for the line \( y = x + 2 \) and (0, -2), (-2, 0) for the line \( y = -x - 2 \).

5. **Drawing the Lines**:
- Draw the line for \( y = x + 2 \) and shade below it since the inequality is \( y \leq x + 2 \).
- Draw the line for \( y = -x - 2 \) and shade below it since the inequality is \( y \leq -x - 2 \).

6. **Identifying the Shaded Region**:
- The solution to the inequality \( y \leq |x + 2| \) will be the region that is below both lines.

7. **Choosing the Correct Option**:
- Compare the shaded regions in the provided options (A, B, C, D) and identify which one matches the region below both lines.

### Conclusion:
After analyzing the options, we find that option D correctly represents the shaded region that satisfies the inequality \( y \leq |x + 2| \).

---