Questions involving pure number properties and absolute value are really tough.

Start always from what you do know and the stem says to you.

Now, \(y > 0\) and \(xz > 0\), which means that y is positive and x and y can be both positive or negative.

At the same time\(y > x\) or\(y < x\). At this point, pick numbers that satisfy all the conditions you consider.

A) \(x=1,y=5,z=9\)

B)\(y=1,x=2,z=5\)

testing them in all answer choices and you can quickly see the only C is always false.

Hope this helps

How y > 0 and xy > 0, means that both could be positive or negative? For y>0, it means that x must be positive in order for xy to be greater that 0?

Sorry I made a mistake. \(xz > 0\) as the stems says.

Regards

What is the significance of this - |x| < |y + 2| < |z| -in the question ..?

well need to increase speed on this!
if I take y =2 ,x = 1, z = 3 --> the condition is not even upheld
This means y has to be less than 1 but positive
so y =1, x = 2, z = 3 --> so D holds true and so does E. A and B are possible but not a 'must'. Hence a,b,d,e.
Kudos if you like this explanation

Assume y=10.
We also know x and z are either both negative or both positive, since xz>0.
And then we have the big inequality given by the problem.

a. y=10, x=11, z=13 --> yes, possible
b. y=10, x=5, z=13 --> yes, possible
c. If x and z are both negative, the large inequality in the problems tells us that that z needs to be further from zero than x, in which case z will be more negative, or less than x. --> no, this option is not possible
d. y=10, x = -11, z = -13 --> yes, possible
e. y=10 --> y+1.5 = 11.5 and y+2 = 12, so we could have x=11.75, z=13 --> yes, possible

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