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

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

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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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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" ##### Most Helpful Community Reply PS Forum Moderator Joined: 25 Feb 2013 Posts: 1217 Location: India GPA: 3.82 Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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. ##### General Discussion Math Expert Joined: 02 Aug 2009 Posts: 7036 Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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 _________________ 1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html GMAT online Tutor Math Expert Joined: 02 Sep 2009 Posts: 50627 Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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 $$-2<x<4$$. _________________ Manager Joined: 14 Sep 2016 Posts: 135 Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 25 Aug 2017, 02:11 1 Bunuel can show the process that how can we get the range of x in an absolute inequality ? Senior CR Moderator Status: Long way to go! Joined: 10 Oct 2016 Posts: 1378 Location: Viet Nam Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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 _________________ Manager Joined: 27 Dec 2016 Posts: 239 Concentration: Marketing, Social Entrepreneurship GPA: 3.65 WE: Marketing (Education) Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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 _________________ There's an app for that - Steve Jobs. Manager Joined: 27 Dec 2016 Posts: 239 Concentration: Marketing, Social Entrepreneurship GPA: 3.65 WE: Marketing (Education) Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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? _________________ There's an app for that - Steve Jobs. PS Forum Moderator Joined: 25 Feb 2013 Posts: 1217 Location: India GPA: 3.82 Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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 $$|x| = -x$$ Manager Joined: 27 Dec 2016 Posts: 239 Concentration: Marketing, Social Entrepreneurship GPA: 3.65 WE: Marketing (Education) Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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! _________________ There's an app for that - Steve Jobs. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6517 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink] ### Show Tags 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Joined: 02 Sep 2016
Posts: 688
Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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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
PS Forum Moderator
Joined: 25 Feb 2013
Posts: 1217
Location: India
GPA: 3.82
Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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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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Posts: 688
Re: Which of the following inequalities is equal to |x-3|x||<8?  [#permalink]

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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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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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13 Sep 2018, 05:48
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Re: Which of the following inequalities is equal to |x-3|x||<8? &nbs [#permalink] 13 Sep 2018, 05:48
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# Which of the following inequalities is equal to |x-3|x||<8?

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