2004 AMC 10A Problem 15

Problem 15 of 25IntermediateAlgebra

Given that

-4 \le x \le -2

and

2 \le y \le 4,

what is the largest possible value of

\dfrac{x + y}{x}?

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Solution

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.

Concepts: inequality · optimization · algebraic manipulation

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