If x and y are integers such that x<0<y, and z is non

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

Given: \(x<0<y\) and \({0}\leq{z}\).

Evaluate each option:

A. \(x^2<y^2\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

B. \(x+y=0\) --> not necessarily true, for example: \(x=-2\) and \(y=1\);

C. \(xz<yz\) --> not necessarily true, if \(z=0\) then \(xz=yz=0\);

D. \(xz=yz\) --> not necessarily true, it's true only for \(z=0\);

E. \(\frac{x}{y}<z\) --> as \(x<0<y\) then \(\frac{x}{y}=\frac{negative}{positive}=negative<0\) and as \({0}\leq{z}\) then \(\frac{x}{y}<0\leq{z}\) --> always true.

Answer: E.