If |x - 3| ≤ 5; |y + 5| ≤ 9; and |2z - 6| ≤ 14, then find the difference between the maximum and minimum possible values of (x + y + z)^2.

To find the difference between the maximum and minimum possible values of ( x + y + z ) 2 , we need to begin by examining each absolute value inequality separately:

Solve ∣ x − 3∣ ≤ 5 :
This inequality implies:
− 5 ≤ x − 3 ≤ 5
Adding 3 to all sides, we get:
− 2 ≤ x ≤ 8

Solve ∣ y + 5∣ ≤ 9 :
This inequality implies:
− 9 ≤ y + 5 ≤ 9
Subtracting 5 from all sides, we get:
− 14 ≤ y ≤ 4

Solve ∣2 z − 6∣ ≤ 14 :
This inequality implies:
− 14 ≤ 2 z − 6 ≤ 14
Adding 6 to all sides:
− 8 ≤ 2 z ≤ 20
Dividing the whole inequality by 2:
− 4 ≤ z ≤ 10


Now we will determine the range of x + y + z . The goal is to find both the maximum and minimum values of which will allow us to then compute ( x + y + z ) 2 .

Maximum x + y + z :

To maximize, use the maximum value for each variable:

x = 8 , y = 4 , z = 10
Therefore, the maximum sum is:


x + y + z = 8 + 4 + 10 = 22

Minimum x + y + z :

To minimize, use the minimum value for each variable:

x = − 2 , y = − 14 , z = − 4
Therefore, the minimum sum is:


x + y + z = − 2 − 14 − 4 = − 20

Calculate ( x + y + z ) 2 :

Maximum squared value: ( 22 ) 2 = 484

Minimum squared value: ( − 20 ) 2 = 400


Thus, the difference between the maximum and minimum possible values of ( x + y + z ) 2 is:
484 − 400 = 84
Therefore, the answer is 84. This is the difference between the maximum and minimum possible values of ( x + y + z ) 2 .

Answered by danjohnbrain | 2025-07-22