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Which of the following inequalities is equal to |x-3|x||<8?

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Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 00:17
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Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

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Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post Updated on: 25 Aug 2017, 02:38
3
4
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


Case 1: if \(x >0\) then \(|x| = x\), hence the equation can be written as
\(|x-3x|<8\), or \(|-2x|<8\) or \(2x<8\)
therefore \(x<4\)-----(1)

Case 2: if \(x<0\), then \(|x| = -x\), hence the equation can be written as
\(|x -3(-x)|<8\), or \(|x+3x|<8\), or \(|4x|<8\) or \(|x|<2\)
therefore \(-x<2\) or \(x>-2\)----(2)

Case 3: if \(x = 0\), then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
\(-2<x<4\)

Option \(C\)

Originally posted by niks18 on 25 Aug 2017, 02:32.
Last edited by niks18 on 25 Aug 2017, 02:38, edited 1 time in total.
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 00:44
1
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


I believe there is a TYPO in Q and Q means |x-3|*|x|<8

best way is working on the choices
substiute x as 0..
\(|0-3||0|<8....0<8\).. True
so choice A and B, which exclude 0 can be eliminated
Now substitute x as 3
\(|3-3||3|=0<8\)... true
eliminate B and D as they exclude 3

ans C
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 00:52
chetan2u wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


I believe there is a TYPO in Q and Q means |x-3|*|x|<8

best way is working on the choices
substiute x as 0..
\(|0-3||0|<8....0<8\).. True
so choice A and B, which exclude 0 can be eliminated
Now substitute x as 3
\(|3-3||3|=0<8\)... true
eliminate B and D as they exclude 3

ans C


No, it's correct. \(|x-3|x||<8\) is equivalent to \(-2<x<4\).
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 02:11
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Bunuel can show the process that how can we get the range of x in an absolute inequality ?
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 02:34
3
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


\(|x-3|x||<8 \iff -8 < x-3|x| < 8\)

Case 1: \(x \geq 0\)

We have \(-8 < x - 3x < 8 \implies -8 < -2x < 8 \implies 4 > x > -4\)
Thus \(0 \leq x < 4\)

Case 2: \(x < 0\)

We have \(-8 < x + 3x < 8 \implies -8 < 4x < 8 \implies -2 < x < 2\)
Thus \(-2 < x < 0\)

Combine both 2 cases, we have \(-2 < x < 4\). Answer C
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Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 09:34
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


Answered this question using plugin method. Plug -1 and 3, if both of the numbers satisfy the equation, then we can choose C.

Absolute in absolute makes me confused :-D :-D
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 09:39
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


Case 1: if \(x >0\) then \(|x| = x\), hence the equation can be written as
\(|x-3x|<8\), or \(|-2x|<8\) or \(2x<8\)
therefore \(x<4\)-----(1)

Case 2: if \(x<0\), then \(|x| = -x\), hence the equation can be written as
\(|x -3(-x)|<8\), or \(|x+3x|<8\), or \(|4x|<8\) or \(|x|<2\)
therefore \(-x<2\) or \(x>-2\)----(2)

Case 3: if \(x = 0\), then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
\(-2<x<4\)

Option \(C\)


Dear niks18 , can you please explain more the one that I highlighted?
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 09:49
2
septwibowo wrote:
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


Case 1: if \(x >0\) then \(|x| = x\), hence the equation can be written as
\(|x-3x|<8\), or \(|-2x|<8\) or \(2x<8\)
therefore \(x<4\)-----(1)

Case 2: if \(x<0\), then \(|x| = -x\), hence the equation can be written as
\(|x -3(-x)|<8\), or \(|x+3x|<8\), or \(|4x|<8\) or \(|x|<2\)
therefore \(-x<2\) or \(x>-2\)----(2)

Case 3: if \(x = 0\), then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
\(-2<x<4\)

Option \(C\)


Dear niks18 , can you please explain more the one that I highlighted?


Hiseptwibowo
its a property of mod function which can be explained as follows -

let \(x = -3\), then
\(|-3| = 3\) this is same as \(-(-3)\)

hence if \(x<0\) (i.e negative), then \(|x| = -x\)
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 25 Aug 2017, 09:57
niks18 wrote:
Hiseptwibowo
its a property of mod function which can be explained as follows -

let \(x = -3\), then
\(|-3| = 3\) this is same as \(-(-3)\)

hence if \(x<0\) (i.e negative), then \(|x| = -x\)


Ah!! thanks niks18 , at first I thought that how can an absolute results a negative number. Now I got it, (-X) is not negative because X itself is negative.

Many thanks! :thumbup: :thumbup:
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 27 Aug 2017, 18:11
1
=> -8 < x-3|x| <8

1) x >= 0
-8 < x-3x < 8
-8 < -2x < 8
-4 < x < 4
0 <= x < 4


2) x < 0
-8 < x-3|x| < 8
-8 < x+3x < 8
-8 < 4x < 8
-2 < x < 2
-2 < x < 0

Thus, -2 < x < 4
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 12 Sep 2017, 10:48
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


Case 1: if \(x >0\) then \(|x| = x\), hence the equation can be written as
\(|x-3x|<8\), or \(|-2x|<8\) or \(2x<8\)
therefore \(x<4\)-----(1)

Case 2: if \(x<0\), then \(|x| = -x\), hence the equation can be written as
\(|x -3(-x)|<8\), or \(|x+3x|<8\), or \(|4x|<8\) or \(|x|<2\)
therefore \(-x<2\) or \(x>-2\)----(2)

Case 3: if \(x = 0\), then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
\(-2<x<4\)

Option \(C\)




Hey
Have I interpreted it correctly?

1) If x>=0
|x|= x


2) If x<0

|x|=-x because -(-(-x))


3) If x<0 and the sign is also given, then

|-x|=x
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Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 12 Sep 2017, 11:24
Shiv2016 wrote:
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0


Case 1: if \(x >0\) then \(|x| = x\), hence the equation can be written as
\(|x-3x|<8\), or \(|-2x|<8\) or \(2x<8\)
therefore \(x<4\)-----(1)

Case 2: if \(x<0\), then \(|x| = -x\), hence the equation can be written as
\(|x -3(-x)|<8\), or \(|x+3x|<8\), or \(|4x|<8\) or \(|x|<2\)
therefore \(-x<2\) or \(x>-2\)----(2)

Case 3: if \(x = 0\), then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
\(-2<x<4\)

Option \(C\)




Hey
Have I interpreted it correctly?

1) If x>=0
|x|= x


2) If x<0

|x|=-x because -(-(-x))


3) If x<0 and the sign is also given, then

|-x|=x


Hi Shiv2016

The highlighted part is not correct.
As you have already mentioned that \(x<0\) i.e \(x\) is negative and we know that mod function is always positive
so as per your equation \(LHS=RHS\) implies that \(positive = negative\) which is incorrect.
assume that x = -2
so per statement 3: |-(-2)| = -2, or |2|=-2 Not possible

Also you have added one extra "-" in statement 2
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 12 Sep 2017, 11:38
So does that mean we have to always take |-x|= x when x<0 for the simple reason that || always gives positive value?

I am actually confused because it took me some time to understand absolute values and I started solving questions this way only.
I always took || as positive but in some questions, || gives negative value e.g. if x<0, then |x|= -x which is said to be positive in some solutions.
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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New post 12 Sep 2017, 11:49
Shiv2016 wrote:
So does that mean we have to always take |-x|= x when x<0 for the simple reason that || always gives positive value?

I am actually confused because it took me some time to understand absolute values and I started solving questions this way only.
I always took || as positive but in some questions, || gives negative value e.g. if x<0, then |x|= -x which is said to be positive in some solutions.


Hi Shiv2016

1. Mod is always positive
2. Negative sign inside mod function can be converted to positive i.e. \(|-x|\) is same as \(|x|\)
3. In an Equality \(LHS = RHS\)

when we are saying that \(x<0\), then \(|-x|\) cannot be equal to \(x\) but we can say that \(|-x|=-x\)
assume \(x=-2; |-(-2)|=-(-2)\), or \(|2|=2\), which is perfectly fine
Hence whenever \(x<0\), we can write mod function as \(|x|=-x\)
for \(x>0; |x|=x\)
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Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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