If x < y < 0, which of the following must be true?

x and y are both negative numbers. Imagine the number line: y is to the left of 0 and x is to the left of y.
e.g. x = -2 and y = -1
or
x = -1/2 and y = -1/3
and so on.
The absolute value of x will be greater than absolute value of y to make x "more negative".

Since x and y are both negative, x/y is positive. Since absolute value of x is greater than absolute value of y, x/y > 1

So both I and II will hold.

III may not hold if y is even. e.g. x = -3, y = -2 so x^y = (-3)^(-2) = 1/9 (not negative)

Answer (D)


You can prove I and II algebraically too.

x and y are negative.

x < y
Multiply both sides by -1 to get -x > -y
Since x and y are negative, |x| = -x and |y| = -y
So |x| > |y|

x < y
Divide both sides by y (which is negative) to get
x/y > 1 (the inequality sign flips)