Try this beautiful problem from Number system: largest possible value
Given that \( -4 \leq x \leq -2\) and \(2 \leq y \leq 4\), what is the largest possible value of \(\frac{x+y}{2}\)
Number system
Inequality
divisibility
Answer: \(\frac{1}{2}\)
AMC-10A (2003) Problem 15
Pre College Mathematics
The given expression is \(\frac{x+y}{x}=1+\frac{y}{x}\)
Now \(-4 \leq x \leq -2\) and \(2 \leq y \leq 4\) so we can say that \(\frac{y}{x} \leq 0\)
can you finish the problem........
Therefore, the expression \(1+\frac{y}x\) will be maximized when \(\frac{y}{x}\) is minimized, which occurs when \(|x|\) is the largest and \(|y|\) is the smallest.
can you finish the problem........
Therefore in the region \((-4,2)\) , \(\frac{x+y}{x}=1-\frac{1}{2}=\frac{1}{2}\)
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