(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|
The explanation given for 2 is
(2) SUFFICIENT: If -y does not equal |y|, then y must be positive (and -y must be negative). Since -x < -y, we know that -x is also negative, so x is also positive. The point (x, y) is therefore in the first quadrant.
I am unable to understand the following:
If -y does not equal |y|, then y must be positive (and -y must be negative).
let x=5
|-x|=x
=>
|-5|=5
=> x is positive.
let x=-5
|-x|=x
=>
|-(-5)|=5
-x=-(-5)=5
=> -x = |-x|
here x is negative..
I am getting really confused with this stuff. Please help me understand the above in quotes.
I however solved it the following way:
-x < -y < |y|
=>
x>y>-|y|
|y|>=0
so
-|y|<=0
=>
x>y>=0
hence x and y are positive and are in 1st quadrant