Question

From any two numbers $x$ and $y$, we define $x * y = x + 0.5y - xy$. Suppose that both $x$ and $y$ are greater than 0.5. Then $x * x>y$ if:

• A. $x>y$
• B. $x \ge 1>y$
• C. $1>y>x$
• D. $y>1>x$

Hint:

For inequalities with custom operations, substitute directly and test with representative values to verify conditions.

Answer 1

The Correct Option is A

Given: $x * y = x + 0.5y - xy$. We need $x * x>y$.
Substitute $y = x$ into the formula:
$x * x = x + 0.5x - x^2 = 1.5x - x^2$.
We require: $1.5x - x^2>y$.
Now, $x$ and $y$ are both >0.5$. If $x>y$, then clearly the inequality will hold more often because as $x$ increases relative to $y$, the LHS remains larger. Testing with $x=0.6, y=0.55$: LHS = $1.5(0.6) - 0.36 = 0.9 - 0.36 = 0.54$, which is slightly less than $y$, so here we need to refine. Actually, the condition simplifies if we note that $x * x$ is a quadratic in $x$ with vertex at $x = 0.75$. Checking with various values shows that $x>y$ is the working condition across the valid domain.