inequality symbols question (beginning algebra) Please help me with this problem:
"In each of the following exercises $x$ and $y$ represent any two whole numbers. As you know, for these numbers exactly one of the statements $x < y, x = y$, or $x > y$ is true.
Which of these is the true statement for each of the following exercises?"

My reasoning is:
e. Since $y$ is not greater than $x$ and $y$ is not equal to $x$ then from this follows that $у$ is less than $x$, which is equivalent to the statement $x > y$ (from the above). So my answer: $x > y$ is the only true statement.
f. Since $y$ is not less than $x$ and $y$ is not equal to $x$ then from this follows that $y$ is greater than $x$, which is equivalent to the statement $x < y$ (from the above).
So my answer: $x < y$ is the only true statement.
Am I right?
(on the image in red color are the answers suggested by the book (i have instructor's copy for self-study) which contradicts with my reasoning...)
P.S. sorry for my English

Your reasoning is correct, but your reasoning actually does not contradict the answers suggested in the book. 
As voldemort pointed out in a comment, $x>y$ is logically equivalent to $y<x$ [e.g., $3>2$ and $2<3$]. This is due to the symmetric nature of the relations $<$ and $>$. That is, $x>y$ and $y<x$ are logically the same; also, $x<y$ is logically the same as $y>x$. In case you are interested, note that even though the relations $>$ and $<$ have the property of symmetry, they do not possess the property of reflexivity; that is, $x<x$ is not a valid statement. 
