Question 7

The value of x/y must be positive if both x and y are negative.

When both x and y are negative, their quotient x/y results in a positive value since dividing two negative numbers yields a positive number.

A) x²-y²

The expression x²-y² can take various values depending on the magnitudes of x and y. Since both x and y are negative, x² and y² are positive, but x²-y² could be positive, negative, or zero, depending on the specific values of x and y. Therefore, x²-y² is not guaranteed to be positive.

B) x/y

This option is correct because dividing two negative numbers (x and y) always results in a positive number. Thus, x/y must be positive when both x and y are negative.

C) x+y

The sum x+y will also be negative, as adding two negative values results in a negative value. Therefore, this option cannot be positive if both x and y are negative.

D) x-y

The expression x-y will yield a negative result because subtracting a negative number (y) from another negative number (x) effectively increases the negativity. Hence, x-y cannot be positive.

Conclusion

The only option that must be positive when both x and y are negative is x/y, as it is the only expression guaranteed to yield a positive value based on the mathematical rule that the quotient of two negative numbers is positive. All other options either do not consistently yield a positive result or are definitively negative under the given conditions.