Official ExplanationCorrect Answer: DFirst representing the points x and y on the number line: x < -10 and y > 6.

Then, since the expressions talk about |x+4|, |x-4|, |y+4| and |y-4|, representing these distances on the number line as well.

|x+4| represents the distance of x from -4 on the number line

|x-4| represents the distance of x from 4

|y+4| represents the distance of y from -4

|y-4| represents the distance of y from 4

I. |x+4| + | y + 4| + | y-4| + |x – 4| > 32From the diagram it’s clear that:

|x+4| > 6

|y + 4| >10

|y – 4| > 2

|x – 4| 14

Adding all these we get Expression 1. Therefore, is true always.

II. |x – 4| + |y + 4| - | x + 4| - | y – 4| < 0From the diagram, we can see that |x – 4| - |x + 4| = 8

And, |y + 4| - | y – 4| = 8

Adding the 2 equations, we get:

|x – 4| + |y + 4| - | x + 4| - | y – 4| = 16

So, expression 2 is NOT TRUE.

III. |x + 4| + |y + 4| - |x – 4| - |y – 4| = 0From the diagram, we can see that |x + 4| - |x – 4| = -8

And, |y + 4| - | y – 4| = 8

Adding the 2 equations, we get:

|x + 4| + |y + 4| - |x – 4| - |y – 4| = 0

So, expression 3 is TRUE.

What if non integers are considered? Will you solution work for that as well?