Given that -4 ≤ x ≤ -2 and 2 ≤ y ≤ 4, what is the largest possible value of (x + y)/(x)?

2004 AMC 10A Problem 15 — Solution

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15.

Given that $-4 \le x \le -2$ and $2 \le y \le 4,$ what is the largest possible value of $\dfrac{x + y}{x}?$

$-1$

$-\dfrac{1}{2}$

$0$

$\dfrac{1}{2}$

$1$

Write $\dfrac{x + y}{x} = 1 + \dfrac{y}{x}.$ Here $\dfrac{y}{x} \lt 0,$ so the expression is largest when $\left|\dfrac{y}{x}\right|$ is smallest.

That happens with $y = 2$ and $x = -4,$ giving $1 + \dfrac{2}{-4} = 1 - \dfrac{1}{2} = \dfrac{1}{2}.$

Thus, the correct answer is D.