Question 14

|y - x| is the greatest number when y < x < 0

In the context of the inequality y < x < 0, the expression |y - x| will yield the largest value compared to the other options. This is due to the fact that both y and x are negative, and the absolute value of the difference between them will be greater than the absolute values of either variable alone.

A) x + y

The expression x + y is not the greatest because both x and y are negative, which means their sum will also be negative. Since we are looking for the greatest number, this option cannot be valid as it will always yield a value less than zero.

B) |x|

The absolute value |x| represents the non-negative value of x. While |x| is positive, it is less than |y - x| because |y - x| accounts for the distance between two negative values, which can yield a larger result than the absolute value of x alone.

C) |y|

Similar to |x|, the absolute value |y| is also positive but represents the magnitude of y. Since y is less than x and both are negative, |y| will not be the greatest number compared to |y - x|, which encompasses the difference between y and x.

D) |y - x|

This option is correct as |y - x| captures the absolute difference between the two negative values y and x. Given the relationship y < x < 0, this expression results in a positive value larger than the absolute values of x or y alone, making it the greatest number among the options.

E) [Blank]

Conclusion

In conclusion, |y - x| is the greatest number because it reflects the largest distance between the two negative numbers y and x. The other options either result in negative numbers or are smaller positive values, failing to surpass the absolute difference represented by |y - x|. Thus, D is definitively the correct choice.