Given the system `x^2 = 4 and y^3 = -8`, which one of the following is NOT necessarily true?

Options

A. `y lt 0`
B. `x lt 5`
C. y is an integer
D. `x gt y`

Answer

To solve the problem, we need to analyze the equations given and the options provided.

1. **Given Equations**:
- \( x^2 = 4 \)
- \( y^3 = -8 \)

2. **Finding Values of \( x \)**:
- From \( x^2 = 4 \), we can take the square root of both sides:
\[ x = \pm 2 \]
- Therefore, the possible values for \( x \) are \( 2 \) and \( -2 \).

3. **Finding Values of \( y \)**:
- From \( y^3 = -8 \), we can take the cube root of both sides:
\[ y = -2 \]
- Thus, the only value for \( y \) is \( -2 \).

4. **Evaluating the Options**:
- **Option A**: \( y < 0 \)
- Since \( y = -2 \), which is indeed less than \( 0 \), this statement is **true**.

- **Option B**: \( x < 5 \)
- The possible values for \( x \) are \( 2 \) and \( -2 \). Both values are less than \( 5 \), so this statement is **true**.

- **Option C**: \( y \) is an integer
- Since \( y = -2 \), which is an integer, this statement is **true**.

- **Option D**: \( x > y \)
- When \( x = 2 \), \( 2 > -2 \) is true.
- When \( x = -2 \), \( -2 \) is equal to \( -2 \), which means \( x \) is not greater than \( y \) in this case.
- Therefore, this statement is **not necessarily true** since it can be equal.

5. **Conclusion**:
- The option that is NOT necessarily true is **Option D**: \( x > y \).

### Final Answer:
The option that is NOT necessarily true is **D: \( x > y \)**.