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

Statement 1: (x+2)y>(y+2)x
xy+2y>xy+2x
2y>2x
y>x

We know that 0<x<y

True.


2) (x-y)/x<0
x-y<0
x<y

We know that 0<x<y

True


3) 2x<x+y
x<y

True.

All the three statements are true.

Let's examine each statement individually:

A) (x + 2)/(y + 2) > x/y
Since y is POSITIVE, we can safely take the given inequality and multiply both sides by y to get: (y)(x+2)/(y+2) > x
Also, if y is POSITIVE, then (y+2) is POSITIVE, which means we can safely multiply both sides by (y+2) to get: (y)(x+2) > x(y+2)
Expand: xy + 2y > xy + 2x
Subtract xy from both sides: 2y > 2x
Divide both sides by 2 to get: y > 2
Perfect! This checks out with the given information that says 0 < x < y
So, statement A is TRUE


B) (x - y)/x < 0
Let's use number sense here.
If x < y, then x - y must be NEGATIVE
We also know that x is POSITIVE
So, (x - y)/x = NEGATIVE/POSITIVE = NEGATIVE
In other words, it's TRUE that (x - y)/x < 0
Statement B is TRUE


C) 2x/(x + y) < 1
More number sense...
If x is positive, then 2x is POSITIVE
If x and y are positive, then x + y is POSITIVE
If x < y, then we know that x + x < x + y
In other words, we know that 2x < x + y
If 2x < x + y, then the FRACTION 2x/(x + y) must be less than 1
Statement C is TRUE


Answer: E

Cheers,
Brent

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