Re: Inequality and absolute value questions from my collection

7. |x+2|=|y+2| what is the value of x+y?

I. (x+2) = (y+2) ===> x-y = 0 (x = y)
OR
II. (x+2) = -y-2 ===> x + y = -4

(1) xy<0
if xy < 0 then x,y ≠ 0 and either x OR y is negative.

This is a VERY tricky problem!

To start, let's look at just |x+2|=|y+2|. #1) tells us that either x OR y is negative and that neither = 0. Of course, |x+2| must = |y+2| so we can try a few different values for x and y.

|x+2|=|y+2|
x= -8, y= 4
|-8+2| = |4+2|
|-6|=|6|
6=6 Valid

x=2, y=-6
|2+2| = |-6+2|
|4| = |-4|
4=4 Valid

If we were told that xy>0, we wouldn't be able to solve. Now, back to the stem:

(x+2) = (y+2) ===> x-y = 0 (x = y)
OR
(x+2) = -y-2 ===> x + y = -4

x and y don't = 0. In scenario 1 we have x-y=0 or (x=y) #1) says that xy<0 meaning either x or y is negative and therefore x ≠ y.

(2) x>2 y<2

|x+2|=|y+2| tells us that x+2 must equal y+2. #2.) rules out the possibility of scenario #1) which means the only other possibility (|x+2|=|y+2| is stated as fact so one of the scenarios must hold true) is that x+y=-4. This can be confirmed by plugging in x values greater than 2 and y values less than -2 to get 0x+y=-4.
SUFFICIENT

(D)