If x and y are integers such that -5 \le x \le -1 and 2 \le y \le 6, what is the minimum p… — GMAT

If $x$ and $y$ are integers such that $-5 \le x \le -1$ and $2 \le y \le 6$ , what is the minimum possible value of $\frac{x - y}{x + y}$ ?

Answer: -11

Answer

The minimum possible value of the expression is -11.

The correct answer is -11 because the numerator

x - y

is negative for all allowed values of

x

and

y

. To minimize a negative fraction, the denominator

x + y

must be positive and minimized (equal to 1), while the numerator must be as negative as possible. Choosing

x = -5

and

y = 6

gives a numerator of

-11

and a denominator of

1

, producing the minimum value of -11.

Step-by-Step Solution

Evaluate the sign of the numerator

Since

x

is negative (

x \le -1

) and

y

is positive (

y \ge 2

x - y

is always negative (

x - y \le -3

Subtracting a positive integer from a negative integer yields a negative result.

Determine the conditions for minimizing a negative fraction

To obtain the minimum (most negative) value, the denominator

x + y

must be a positive integer and as small as possible.

A negative numerator divided by a positive denominator yields a negative quotient. Dividing by a smaller positive number yields a quotient with larger absolute value, making it smaller (more negative).

Identify the smallest positive denominator and solve for variables

The smallest positive integer for

x + y

1

. Setting

y = 1 - x

gives

x - y = 2x - 1

Since

x

and

y

are integers,

x + y

must be an integer.

Maximize the magnitude of the negative numerator

Using the lowest boundary

x = -5

gives

y = 6

. The value of the expression is

\frac{-5 - 6}{-5 + 6} = -11

Evaluating at

x = -5

and

y = 6

gives numerator

-11

and denominator

1

, resulting in

-11

Key Concept

Minimizing algebraic fractions involving signed numbers

Question

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