If x and y are real numbers that satisfy the absolute value inequalities |2x - 6| \le 4 an… — GRE General Test

If $x$ and $y$ are real numbers that satisfy the absolute value inequalities $|2x - 6| \le 4$ and $|y + 3| \le 5$ , which of the following values could be equal to the product $xy$ ? Indicate all such values.

A
$-45$
$-35$ Answer
$0$ Answer
$8$ Answer
E
$15$

Answer

The possible values for the product

xy

are

-35

0

, and

8

Solving

|2x - 6| \le 4

yields

1 \le x \le 5

, and solving

|y + 3| \le 5

yields

-8 \le y \le 2

. The product

xy

attains its minimum at

5 \times (-8) = -40

and its maximum at

5 \times 2 = 10

. Since

xy

can take any value in the continuous interval

[-40, 10]

, the values

-35

0

, and

8

are all valid choices.

Step-by-Step Solution

Solve the inequality

|2x - 6| \le 4

for

x

-4 \le 2x - 6 \le 4 \implies 2 \le 2x \le 10 \implies 1 \le x \le 5

Unfold the absolute value into a compound inequality to determine the valid range for

x

Solve the inequality

|y + 3| \le 5

for

y

-5 \le y + 3 \le 5 \implies -8 \le y \le 2

Unfold the absolute value into a compound inequality to determine the valid range for

y

Determine the minimum and maximum possible values of the product

xy

Evaluating the extreme product combinations of endpoints:

1 \times (-8) = -8

1 \times 2 = 2

5 \times (-8) = -40

, and

5 \times 2 = 10

. Thus,

-40 \le xy \le 10

The continuous product of two real intervals

[a, b]

and

[c, d]

spans from the minimum endpoint product to the maximum endpoint product.

Select all options that fall within the interval

[-40, 10]

The values

-35

0

, and

8

lie within

[-40, 10]

, while

-45

and

15

fall outside.

Any real number within the closed interval

[-40, 10]

can be formed by valid choices of

x

and

y

Key Concept

Determining the range of a product from two independent absolute value inequalities.

Question

Answer

Step-by-Step Solution

Key Concept

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