If x and y are real numbers such that |x - 4| \le 3 and |y + 2| \le 5, what is the maximum… — GRE General Test

If $x$ and $y$ are real numbers such that $|x - 4| \le 3$ and $|y + 2| \le 5$ , what is the maximum possible value of $|x - y|$ ?

Answer: 14

The maximum possible value of

|x - y|

is 14.

Solving

|x - 4| \le 3

gives the closed interval

[1, 7]

for

x

. Solving

|y + 2| \le 5

gives the closed interval

[-7, 3]

for

y

. The maximum possible value of

|x - y|

is the maximum distance between a point in

[1, 7]

and a point in

[-7, 3]

, which is

7 - (-7) = 14

Determine the range of possible values for

x

1 \le x \le 7

The inequality

|x - 4| \le 3

represents all numbers within distance 3 of 4 on the number line.

Determine the range of possible values for

y

-7 \le y \le 3

The inequality

|y + 2| \le 5

represents all numbers within distance 5 of -2 on the number line.

Find the maximum distance between any point

x

[1, 7]

and any point

y

[-7, 3]

The maximum absolute difference

|x - y|

occurs between the upper endpoint of the

x

-interval (

7

) and the lower endpoint of the

y

-interval (

-7

Absolute value inequalities as distance intervals on the real number line and maximizing differences between bounded variables

Question