Let x, y, and z be integers such that -5 \le x \le 5, -5 \le y \le 5, and -5 \le z \le 5. … — GMAT

Let $x$ , $y$ , and $z$ be integers such that $-5 \le x \le 5$ , $-5 \le y \le 5$ , and $-5 \le z \le 5$ . If these integers satisfy all of the following conditions:
1. $x^3 y z^2 < 0$
2. $\frac{x}{z} < -1$
3. $x + y \le 1$

What is the maximum possible value of the expression $x + 2y - z$ ?

Answer: 5

Answer

The maximum possible value of the expression

x + 2y - z

is 5.

By analyzing the given inequalities,

x

and

y

must have opposite signs, and

x

and

z

must have opposite signs, which means

y

and

z

must have the same sign. To maximize

x + 2y - z

, we examine the scenario where

x

is negative, while

y

and

z

are positive. Taking

z = 1

(the smallest positive integer),

y = 5

(the largest positive integer within the given range), and

x = -4

satisfies

x + y \le 1

\frac{x}{z} = -4 < -1

, and

x^3 y z^2 = -320 < 0

. This yields

x + 2y - z = -4 + 10 - 1 = 5

Step-by-Step Solution

Analyze product and quotient inequalities to deduce the relative signs of

x

y

, and

z

x

and

y

have opposite signs,

x

and

z

have opposite signs, and

|x| > |z| \ge 1

. Consequently,

y

and

z

share the same sign.

Because

z^2 > 0

for non-zero

z

x^3 y z^2 < 0

requires

x^3 y < 0

. Also

\frac{x}{z} < -1

implies

\frac{x}{z}

is negative and has magnitude greater than 1.

Evaluate the sign cases to determine which case yields a larger value for

x + 2y - z

Case A (

x < 0, y > 0, z > 0

) allows positive contributions from

2y

and

-z

, whereas Case B (

x > 0, y < 0, z < 0

) bounds the expression below 2.

In Case B,

y

and

z

are negative, so

2y \le -2

suppresses the sum.

Apply integer domain bounds and inequality constraints to maximize

x + 2y - z

in Case A.

The maximum value is 5, achieved when

x = -4

y = 5

, and

z = 1

Setting

z = 1

(smallest positive integer) and

y = 5

(largest positive integer) with

x = -4

satisfies

x + y \le 1

and all problem conditions.

Key Concept

Deduction of variable signs from inequality products and quotients, combined with integer range optimization.

Estimated Time:2m 0s

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