If the values of x and y are negative, which of the following values must be positive?

Rationale

x/y is a positive value when both x and y are negative.

When both x and y are negative numbers, the division of x by y results in a positive value since the negative signs cancel each other out. Thus, x/y must be positive.

A (x²-y²) YOUR ANSWER

The expression x² - y² can be rewritten as (x - y)(x + y). Since both x and y are negative, x - y is positive (as the larger negative number subtracted from a smaller negative number yields a positive result), but x + y is negative (as adding two negative numbers results in a negative). Hence, x² - y²'s sign is indeterminate and cannot be guaranteed to be positive.

C (x+y) YOUR ANSWER

Adding two negative numbers, x and y, results in a value that is also negative. Therefore, x + y cannot be positive, as the sum of two negative values will always yield a negative result.

D (x-y) YOUR ANSWER

The expression x - y also results in a positive value because subtracting a larger negative number (y) from a smaller negative number (x) produces a positive outcome. However, it is not universally guaranteed to be positive depending on the specific values of x and y, making it an unreliable choice.

Conclusion

In scenarios where both x and y are negative, the only expression that must consistently yield a positive value is x/y. The other options, while they may sometimes be positive under specific conditions, do not universally satisfy the requirement of being positive for all negative values of x and y. Thus, x/y stands out as the only guaranteed positive result.