Question

If \(y - x > x + y\), where "x" and "y" are integers, which of the following must be true?
I. (OCR error, likely x < 0)
II. xy < 0
III. y < 0

• A. I only
• B. II only
• C. I and II only
• D. I and III only
• E. II and III only.

Hint:

When testing "must be true" statements, always try to find a counterexample. If you can find even one case where the statement is false while the original condition is true, then the statement is not a "must be true" consequence.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are given an inequality and need to determine which of the three statements must be true as a consequence. The first step is to simplify the given inequality. 
Step 2: Detailed Explanation:  
Simplify the inequality \(y - x>x + y\): 
Subtract \(y\) from both sides: 
\[ -x>x \] 
Add \(x\) to both sides: 
\[ 0>2x \] 
Divide by 2: 
\[ 0>x \quad \text{or} \quad x<0 \] 
This simplification shows that \(x\) must be a negative integer. Let's evaluate the three statements based on this result. 
Statement I: The OCR for this statement is corrupted , but given the result \(x<0\), it is highly probable that the intended statement was simply x<0. Based on our simplification, this statement must be true. 
Statement II: "xy<0" 
This statement means that \(x\) and \(y\) must have opposite signs. We know \(x\) is negative. For this statement to be true, \(y\) must be positive. However, the original inequality places no restrictions on \(y\). For example, if \(x = -1\) and \(y = -5\), the original inequality holds: \(-5 - (-1)>-1 + (-5) \rightarrow -4>-6\), which is true. In this case, \(xy = (-1)(-5) = 5\), which is not less than 0. Therefore, statement II does not have to be true. 
Statement III: "y<0" 
As shown in the counterexample for Statement II, \(y\) can be negative (\(y=-5\)). But can \(y\) be positive? Let \(x = -2\) and \(y = 3\). The inequality holds: \(3 - (-2)>-2 + 3 \rightarrow 5>1\), which is true. In this case, \(y\) is positive. Since \(y\) can be either positive or negative, this statement does not have to be true. 
Step 3: Final Answer: 
Only statement I (\(x<0\)) must be true. Therefore, the correct option is (A).