Question

the sign of ( x ) is negative and the sign of ( y ) is positive. if the mean of ( x ) and ( y ) is a negative number, which of the following statements is true?
a ( y>x )
b ( x + y = 0 )
c ( |x| = |y| )
d ( |y|>|x| )
e none of the above

Explanation:

Step1: Recall the formula for the mean

The mean of \(x\) and \(y\) is \(\frac{x + y}{2}\). Given that \(\frac{x + y}{2}<0\), then \(x + y<0\).

Step2: Analyze the relationship between \(x\) and \(y\) based on their signs

Since \(x<0\) (let \(x=-a,a > 0\)) and \(y>0\) (let \(y = b,b>0\)), the inequality \(x + y<0\) becomes \(-a + b<0\), or \(b < a\). In terms of absolute values, \(|x|=a\) and \(|y|=b\). So \(|y|<|x|\).

• Option A: \(y>x\) is always true because \(y>0\) and \(x < 0\), but this does not address the condition of the mean being negative.
• Option B: If \(x + y=0\), then \(\frac{x + y}{2}=0\), which contradicts \(\frac{x + y}{2}<0\).
• Option C: If \(|x|=|y|\), then \(x=-y\) and \(x + y = 0\), \(\frac{x + y}{2}=0\), which contradicts \(\frac{x + y}{2}<0\).
• Option D: If \(|y|>|x|\), then \(y+x>0\) (since \(y>0,x<0\)), and \(\frac{x + y}{2}>0\), which contradicts \(\frac{x + y}{2}<0\).

Answer:

E. None of the above