If x is different from 0, is |x| < 1 ?

If x is different from 0, is |x| < 1 ?

(1) x*|x|<x --> if \(x>0\) then we can divide both parts by it and we'l get \(|x|<1\) (so we would have an YES answer. Complete range for this case would be: \(0<x<1\)) BUT if \(x<0\) then we can also divide both parts by it though this time we should switch the sign of the inequality as we are dividing by negative value: \(|x|>1\) (so we would have a NO answer. Complete range for this case would be: \(x<-1\)). Not sufficient. (Basically we should know whether \(x\) is positive or negative).

(2) |x| > x --> well this basically tells that \(x\) is negative, as if \(x\) were positive (or zero) then \(|x|\) would be equal to \(x\). Still insufficient to say whether \(|x|<1\).

(1)+(2) From (2) \(x<0\) thus from (1) \(|x|>1\) and the answer to the question is NO (solution based on both statements is \(x<-1\)). Sufficient.

Answer: C.

Similar question: if-x-is-not-equal-to-0-is-x-less-than-1-1-x-x-x-86140.html

Hope it helps.
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OK. First of all: I marked "|x|<1" in red for (1), because you don't get that |x| < 1 from this statement. If it were so then then you'd have an YES answer right away (as the question asks exactly about the same thing "is |x|<1")

Next, "is \(|x|<1\)?" means "is \(-1<x<1\)?"

(1) x*|x|<x holds true in two cases for \(x<-1\) and \(0<x<1\):
-----(-1)----(0)----(1)----
So as you can see this one is not sufficient to answer the question.

(2) |x| > x. Absolute value property: when \(x\geq{0}\) then \(|x|=x\) and when \(x<{0}\) (so when \(x\) is negative) then \(|x|=-x\). For the second case \(|x|=positive>x=negative\). That' why \(|x|>x\) means that \(x<0\) (or as I wrote if \(x\) were positive we would have \(|x|=x\) not \(|x|>x\)). You can also consider a simple example: \(|-3|>-3\) --> \(-3<0\). Thus the range for (2) is:
-------------(0)--------
Also insufficient to answer the question.

(1)+(2) Intersection of the ranges from (1) and (2) is:
----(-1)----(0)----(1)---- --> \(x<-1\). Which makes the answer to the question "is \(-1<x<1\)?" NO.

Check Absolute Value chapter of Math Book for more on this topic: math-absolute-value-modulus-86462.html

Hope it's clear.
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I suppose that you have put in red my logic on stat 1 because not so clear.

Stat (1) : x > 1 or -x > 1.........x < -1

Stat (2) : x < x or -x > x.........x < -x

This is the logic that conduct me to say that the 2 stats are insufficient. I don't understand completely why in the 2 one |x| > x ........ x is < 0 ??'


I also read the topic that you suggested; your third reply is very helpful but the doubt remain on stat ( 2)

Thanks
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Thank you a lot.

Now right away I'll read absolute value chapter.
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If x is different from 0, is |x| < 1 ?
x≠0, is -1<x<1?

(1) x |x| < x
For this to hold true: 0<x<1 or x<-1

0<x<1
1/2 * |1/2| < 1/2
1/4 < 1/2 Valid

x<-1
-2 * |-2| < -2
-4 < -2 Valid

x could be between 0 and 1 or it could be less than -1.
INSUFFICIENT

(2) |x| > x
|x| is by definition ≥0 so for |x| to be greater than x, x must be negative. Depending on the value of -x, it may or may not satisfy the inequality.
INSUFFICIENT

1+2) 0<x<1 or x<-1 and x is negative. The only intersection is when x<-1 and "x is negative" meaning x<-1 and the inequality in the stem holds true.
SUFFICIENT

(C)

i solved it in the following way.. what am i doing wrong ?

for x|x|<x

a. x>0: xx<x -> x<1 i.e. 0<x<1
b. x<0: -xx<x -> -x<1 -> x>-1 which is wrong.

ln another problem i solved x/|x| similarly, and got the correct answer .

for x/|x|< x

x>0: x/x<x -> 1<x
x<0: x/-x<x ->-1<x<0


i am confused ..

For (b) when x<0 you have: -x*x < x --> divie by negative x and flip the sign: -x>1 --> x<-1.
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If x is different from 0, is |x| < 1 ?

(1) x*|x| < x

(2) |x| > x
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1) x*|x| < x : on first look it seems to be sufficient, Divide by X from both side and we will get the answer, But this is the catch here.
We only know that X != 0 . X can e negative as well and if X is negative then it will reverse the sign. So we get two conditions here. X +ve and x-ve

For X +ve = divide by X = |X| <1
For X -ve = divide by -X = |X| >1
So not sufficient.

2) |x| > x : It says that X is -ve. Only in this scenerio |X| can be greater than X. This is also not sufficient.

Now Lets take ! and 2 both
We know from 2 that X is -ve so from 1 we get |X| >1. Answer is no |X| is not less than 1.
So C is sufficient

Hi Bunuel, can x also be x>-1 for Statement 1? Coz when I sub in the values of (-1/2) or (-1/4), x satisfies the equation. But on your number line it is in black, which indicates x>-1 does not hold true. Please kindly enlighten me.

Neither of these values satisfies x*|x| < x.

If x = -1/2, then (x*|x| = -1/4) > -1/2 NOT <.
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Hi,

I like to use smart numbers for this type of questions otherwise I feel completely lost.

Let x be : -2 or -1/2 or 1/2 or 2

1) cases where x*Ixl<x :

With x=-2
-2*l2l<-2 ==> -4<-2 and here l2l>1, is lxl<1? NO

With x= 1/2
1/2*1/2<1/2 ==> 1/4<1/2 and here l1/2l<1, is lxl<1? YES

we have a yes and no so it's insufficient

2) Cases where lxl>x

When x=-2
l2l>-2 ==> is lxl<1? NO

When x=-1/2
l1/2l>-1/2 ==> is lxl<1? YES

We have a yes and no so it's insufficient

1)+2) We have only one value that is consistent with the two statements and it's -2. Therefore, is lxl<1? the answer is NO.

Answer C)

Statement 1:
CRITICAL POINTS occur when the two sides of an inequality are EQUAL or when the inequality is UNDEFINED.
\(x|x|=x\) when x=-1, x=0 or x=1

The critical points are x=-1, x=0 and x=1, implying the following number line:
..........-1..........0..........1..........
To determine which ranges for x are valid, test one value to the left and one value to the right of each critical point.
If we test x=-2, x=-1/2, x=1/2 and x=2, only x=-2 and x=1/2 satisfy \(x|x|<x\), implying that the valid ranges are x<-1 and 0<x<1.

Case 1: x<-1
In this case, |x|>1, so the answer to the question stem is NO.
Case 2: 0<x<1
In this case, |x|<1, so the answer to the question stem is YES.
INSUFFICIENT.

Statement 2:
|x|>x for all negative values of x.
If x=-1/2, then |x|<1, so the answer to the question stem is YES.
If x=-2, then |x|>1, so the answer to the question stem is NO.
INSUFFICIENT.

Statements combined:
Only Case 1 satisfies both statements.
In Case 1, the answer to the question stem is NO.
SUFFICIENT.


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Got to learn so much from this question! I was making mistake while defining the range. This brushes up your concept of inequality!