Question

16
paul says that \\( \frac{x}{y} \\) can result in either a positive or negative quotient, where \\( x \\) and \\( y \\) are integers and \\( y \
eq 0 \\).
is paul correct? explain why or why not and use examples to support your answer.
respond in the space provided.
paul is

Explanation:

Step1: Recall Sign Rules for Division

The sign of a quotient $\frac{x}{y}$ (where $y
eq0$) depends on the signs of $x$ and $y$. The rule is: if $x$ and $y$ have the same sign (both positive or both negative), the quotient is positive; if $x$ and $y$ have different signs (one positive, one negative), the quotient is negative.

Step2: Positive Quotient Example

Let $x = 4$ and $y = 2$ (both positive). Then $\frac{x}{y}=\frac{4}{2}=2$, which is positive. Another example: $x=-4$ and $y = -2$ (both negative). Then $\frac{x}{y}=\frac{-4}{-2}=2$, also positive.

Step3: Negative Quotient Example

Let $x = 4$ and $y=-2$ (different signs). Then $\frac{x}{y}=\frac{4}{-2}=-2$, which is negative. Another example: $x=-4$ and $y = 2$ (different signs). Then $\frac{x}{y}=\frac{-4}{2}=-2$, also negative.

Step4: Conclusion

Since we can find examples where $\frac{x}{y}$ is positive (same - sign $x$ and $y$) and negative (different - sign $x$ and $y$), Paul's statement is correct.

Answer:

Paul is correct. The sign of $\frac{x}{y}$ (with $y
eq0$) depends on the signs of $x$ and $y$. If $x$ and $y$ have the same sign (both positive or both negative), $\frac{x}{y}$ is positive (e.g., $\frac{4}{2}=2$, $\frac{-4}{-2}=2$). If $x$ and $y$ have different signs (one positive, one negative), $\frac{x}{y}$ is negative (e.g., $\frac{4}{-2}=-2$, $\frac{-4}{2}=-2$). Thus, $\frac{x}{y}$ can be positive or negative.