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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

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

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 \(-2<x<4\).
_________________

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

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25 Aug 2017, 03:32

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 \(C\)

Last edited by niks18 on 25 Aug 2017, 03:38, edited 1 time in total.

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

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25 Aug 2017, 10:39

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?
_________________

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

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25 Aug 2017, 10:49

2

This post received KUDOS

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 \(|x| = -x\)

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

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25 Aug 2017, 10:57

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.

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

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12 Sep 2017, 11:48

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

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

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12 Sep 2017, 12:24

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

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

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

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12 Sep 2017, 12:38

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.
_________________

Help me make my explanation better by providing a logical feedback.

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

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12 Sep 2017, 12:49

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.

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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