.

[sssss]

.

[NL]

Is |x| <1?
(1) x/|x|< x
(2) |x| > x

How do we get x > x/-x or x > -1 from the first statement?
(1) x/|x|< x

The statement is little abstract to me too, so I pick possible numbers for x to see what (1) means:

I used simple reasoning first: (to know what kind of number I should pick)
x/|x| always gives 2 possible values: 1 or -1 (because |x| >0 and x#0)

--> so, I should pick numbers bigger or smaller than 1 and -1.

* x = 2 → 2/|2| <2? Yes, satisfy the condition
* x = -2 → -2/|-2| < -2? Not satisfy.
* x = -1/2 → -1/2/|-1/2| <-1/2? Yes, satisfy.
* x = 1/2 → 1/2/|-1/2| < 1/2? Not satisfy

--> I conclude that 0>x >-1 or x >1 (not sure? drawing a range line)

[NL]

- What I don’t really understand about the explanation for the statement (1) is: why it considers 2 cases dealing with 0, not 1 & -1?

- Do you have any quicker way to clarify the statement if you get confused with reasoning?

[RonPurewal]

- What I don’t really understand about the explanation for the statement (1) is: why it considers 2 cases dealing with 0, not 1 & -1?

- Do you have any quicker way to clarify the statement if you get confused with reasoning?

Those aren't really "cases dealing with 0". Those are just positives and negatives"”the same kinds of numbers you tried yourself.

The point is that the expression |x| has differentbehavior for the different signs of x.

It appears you already understand this fact, since you are already testing numbers of both possible signs. And, you also understand that |x|/x is either -1 or 1, depending on the sign of x.
So, no worries.

[sssss]

.

[NL]

The point is that the expression |x| has differentbehavior for the different signs of x.

I don't really get this. |x| >0 always when x #0
so when x is put in | | , it's in a prison already (in term of positive and negative). But looking at x when it's in the prison can't help assure how it behaves when it's free.

Those aren't really "cases dealing with 0". Those are just positives and negatives"”the same kinds of numbers you tried yourself.

This is great, but I still don’t understand :) This case is not only dealing with positive or negative, and that makes it harder.

Through picking number , I see the statement (1) tells us:
* x takes value within 0 to -1, exclusive,
* or, x take value bigger than 1.
Here is the value line for x:

////-2//////////-1----------0////////////1-----------2-----

//// means values don’t work
---- means values work

The question asks: is |x| <1?
--> means whether x is between -1 and 1? --> Yes and No. Insufficient.


(2) |x| > x
--> means x< 0
Is x between 1 and -1? the answer is Y and N, as we see the value of x here:
------------1------------0


Combine (1) and (2):
(1) ////-2//////////-1----------0////////////1-----------2-----
(2) -----------------1-----------0

--> value of x is this: 1----------0
Is x is between -1 and 1? YES.

(It's so easy to make mistake with this problem. I picked E at the first time solving it)

[RonPurewal]

This last post contains essentially everything that can be understood about the problem, so, you're fine.

I like the "in and out of prison" analogy.

[NL]

x/|x| always gives 2 possible values: 1 or -1 (because |x| >0 and x#0) --> so, I should pick numbers bigger or smaller than 1 and -1.

This is wrong. The reason for picking integers and decimals is that x/|x| will "behaves" differently with those types of number.
And the reason to pick 1/2 & -1/2, but not 3/2 & -3/2 (with which the question cannot be solved) is that the question asks |x| <1?
--> So, there are 2 reasons to pick 1/2 & -1/2 for x.
But I didn't have any explicit thought when I did it. Maybe it was just a luck.

- My question is:
Determine a good number to pick is time consuming sometimes. Does it require specific skills that we can practice or just "open your eyes and recognize"?

[RonPurewal]

Plan A: Open your eyes and recognize.

Plan B: Just guess numbers, throw them in there, and see what happens.

Often, you'll discover Plan A in the middle of doing Plan B.

[NL]

* x = 2 → 2/|2| <2? Yes, satisfy the condition
* x = -2 → -2/|-2| < -2? Not satisfy.
* x = -1/2 → -1/2/|-1/2| <-1/2? Yes, satisfy.
* x = 1/2 → 1/2/|-1/2| < 1/2? Not satisfy

--> This is ridiculous!
I just reviewed this problem and didn’t know why I made it so complicated. I can simply use algebra:

(1) x/|x| = 1 or = -1, depending on x is negative or positive.
(from this angle, yes, the problem just deals with positive and negative as Ron said)
So, x>1 or x>-1 → x >-1 Insufficient.
(2) |x| >x means x<0 Insufficient.
(1) & (2): -1<x<0: sufficient.

(A life lesson for me: Looking at right "place", things become simple)

[RonPurewal]

Ok.

I'm going to lock this thread, since it has a strange non-title and half of the posts are gone for some reason.