Question

For all values of $x$ and $y$ if $x<y<(x+y)$, which of the following must be negative? Indicate all possible options.

• A. $-2x$
• B. $-y$
• C. $x-y$
• D. $2x-y$
• E. $3x$

Hint:

When solving inequality-based problems, first deduce which variable must always be positive or negative. Then test each option systematically under those conditions.

Answer 1

The Correct Option is B

Step 1: Analyze the inequality.
We are given: \[ x<y<x+y \] From $y<x+y$, we conclude that $y>0$.
Thus, $y$ is always positive, and $x$ may be either negative or positive, but smaller than $y$.
Step 2: Check each option.
- (A) $-2x$: If $x<0$, then $-2x>0$ (positive). If $x>0$, then $-2x<0$ (negative). Therefore, this is not always negative.
- (B) $-y$: Since $y>0$, then $-y<0$. This is always negative.
- (C) $x-y$: Since $x<y$, subtracting gives $x-y<0$. This is always negative.
- (D) $2x-y$: Consider $x=5$, $y=6$. Then $2x-y = 10-6=4>0$, not always negative.
- (E) $3x$: If $x<0$, then $3x<0$. If $x>0$, then $3x>0$. So not always negative.

Step 3: Conclude the must-be-negative expressions.
The only expressions guaranteed to be negative under the given condition are: \[ -y \quad \text{and} \quad x-y \]
Final Answer: \[ \boxed{-y \text{ and } x-y} \]