If x and y are real numbers such that |x - 3| \le 2 and |y + 1| \le 4, which of the follow… — GRE General Test

If $x$ and $y$ are real numbers such that $|x - 3| \le 2$ and $|y + 1| \le 4$ , which of the following could be the value of $|x - y|$ ? Select all such values.

A
$-2$
$0$ Answer
$5$ Answer
$10$ Answer
E
$12$

Answer

The possible values of

|x - y|

are 0, 5, and 10.

Solving the inequalities yields

1 \le x \le 5

and

-5 \le y \le 3

. The expression

|x - y|

represents the distance between

x

and

y

on the number line. Because the intervals overlap between 1 and 3,

x

and

y

can be equal, making the minimum distance 0. The maximum distance occurs at the extreme points

x = 5

and

y = -5

, giving a distance of

|5 - (-5)| = 10

. Therefore, any value from 0 to 10 inclusive is possible, making 0, 5, and 10 valid values.

Step-by-Step Solution

Solve the absolute value inequality for

x

-2 \le x - 3 \le 2 \implies 1 \le x \le 5

Unwrapping

|x - 3| \le 2

gives the bounds for

x

on the real number line.

Solve the absolute value inequality for

y

-4 \le y + 1 \le 4 \implies -5 \le y \le 3

Unwrapping

|y + 1| \le 4

gives the bounds for

y

on the real number line.

Determine the minimum and maximum possible values for

|x - y|

Minimum value is 0 (since the intervals

[1, 5]

and

[-5, 3]

overlap at

[1, 3]

). Maximum value is

|5 - (-5)| = 10

. Thus,

0 \le |x - y| \le 10

The absolute value

|x - y|

represents the distance between

x

and

y

on the number line, which can take any real value from 0 to 10.

Evaluate the given choices against the range

[0, 10]

The values 0, 5, and 10 fall within

[0, 10]

, whereas

-2

is impossible for absolute values and

12

exceeds the maximum bound.

Determines which specific options are valid outcomes for

|x - y|

Key Concept

Absolute value as distance on the real number line and range of differences between bounded real variables

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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