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if (10-x)/3 <-2x, which of the following must be true?

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if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 02:50
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if (10-x)/3 < -2x, which of the following must be true?

I) 2 < x
II) |x-5| >= 7
III) (|x-1|/|x|) > 1

A. I Only
B. II Only
C. III Only
D. II and III
E. I, II, and III
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if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post Updated on: 11 May 2019, 06:18
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Hi eabhgoy

in the "must be true questions", you have to deal with question the other way around.
the inequality : (10-x)/3 < -2x , is the TRUE ground that you will start from. by simplifying it, it will be x< -2.

so the question says that:
if the value of x is less than -2 ... (-2.5, -3, ........)
these values are covered by which of the following I, II , III?

I) 2 < x ---> for sure false, because it is not covering any of the values of x< -2

II) |x-5| >= 7 : so the range is x>=12 and x<= -2 ---> it covers ALL the values of "x<-2" and MORE
(ALL is a must, MORE doesn't affect the judgement because if you try ANY value of "x<-2", it will fit into "|x-5| >= 7"

III) (|x-1|/|x|) > 1 : so the range is x <1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

however, if in the choices, there is (for example):
x<-4 -----> this will be false because it is not covering ALL values of "x<-2" (if x = -3 for instance)
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Originally posted by MahmoudFawzy on 31 Mar 2019, 08:42.
Last edited by MahmoudFawzy on 11 May 2019, 06:18, edited 1 time in total.
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 03:20
IMO D

after simplifying the given equation we get

x+2<0

x<-2

now only 2nd and 3rd hold true for this

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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 04:52
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chakrads wrote:
if (10-x)/3 < -2x, which of the following must be true?

I) 2 < x
II) |x-5| >= 7
III) (|x-1|/|x|) > 1

A. I Only
B. II Only
C. III Only
D. II and III
E. I, II, and III


If X=13, then |x-5|>=7 is true but (10-x)/3<-2x is not true, then how is the answer D?
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 11:27
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chakrads wrote:
if (10-x)/3 < -2x, which of the following must be true?

I) 2 < x
II) |x-5| >= 7
III) (|x-1|/|x|) > 1

A. I Only
B. II Only
C. III Only
D. II and III
E. I, II, and III


Given \(\frac{10-x}{3}< -2x\)
10-x<-6x
5x<-10
x<-2

Take x=-2.1
I)) 2<-2.1 No

II) \(|-2.1-5|\geq 7\)or \(7.1\geq7\) Yes


III)\(\frac{|-2.1-1|}{|-2.1|}\) >1 \(\Rightarrow\) \(\frac{3.1}{2.1}\)>1 Yes

Hence only II and III must be true .
Hope this helps.
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if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 13:14
Mahmoudfawzy83 wrote:
Hi eabhgoy

in the "must be true questions", you have to deal with question the other way around.
the inequality : (10-x)/3 < -2x , is the TRUE ground that you will start from. by simplifying it, it will be x< -2.

so the question says that:
if the value of x is less than -2 ... (-2.5, -3, ........)
these values are covered by which of the following I, II , III?

I) 2 < x ---> for sure false, because it is not covering any of the values of x< -2

II) |x-5| >= 7 : so the range is x>=12 and x<= -2 ---> it covers ALL the values of "x<-2" and MORE
(ALL is a must, MORE doesn't affect the judgement because if you try ANY value of "x<-2", it will fit into "|x-5| >= 7"


III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

however, if in the choices, there is (for example):
x<-4 -----> this will be false because it is not covering ALL values of "x<-2" (if x = -3 for instance)


Hello Mahmoudfawzy83 !

Could you please explain to me the red statement?

If x must be less than -2, how come could II holds true?

What happens if x is -2 as the II statement says that x could be equal or less than -2 if it's equal to -2 then it can't be true.

Kind regards!
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 13:58
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Hi jfranciscocuencag

first, you must know that it was confusing for me at the beginning.

I will try explaining through a simplified question:

if x = 1, which of the following must be true

1) x >1
2) x >-1
3) \(x\geq{1}\)

what the question is really asking about is:
is 1 > 1 ? ------> no ..... why? .... because 1 is not in the range of >1
is 1 > -1? -----> yes ...... why? ..... because 1 is in the range of >-1
is \(1\geq{1}\)? -------> yes ..... why? ..... because 1 is in the range of \(\geq{1}\)

I may haven't answered your question yet,
but do you agree with the simplified question so far?
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 14:32
Mahmoudfawzy83 wrote:
Hi jfranciscocuencag

first, you must know that it was confusing for me at the beginning.

I will try explaining through a simplified question:

if x = 1, which of the following must be true

1) x >1
2) x >-1
3) \(x\geq{1}\)

what the question is really asking about is:
is 1 > 1 ? ------> no ..... why? .... because 1 is not in the range of >1
is 1 > -1? -----> yes ...... why? ..... because 1 is in the range of >-1
is \(1\geq{1}\)? -------> yes ..... why? ..... because 1 is in the range of \(\geq{1}\)

I may haven't answered your question yet,
but do you agree with the simplified question so far?


Mahmoudfawzy83, yes, it actually helped.

But now, for example in statement two we got two values when x is equal or more than 12 and equal or less than -2.

So, in this kind of questions, is it ok if the statement just satisfies one of the two inequalities?

Kind regards!
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 15:01
Mahmoudfawzy83 wrote:
is 1 > -1? -----> yes ...... why? ..... because 1 is in the range of >-1

Now I am going to attack the answer I just said myself:
What happens if x is 0 as the II statement says that x could be any value more than -1?
if it's equal to 0 then it can't be true


the counter argument (in red) I used is irrelevant, because we am not testing whether x = 1 or not in the first place (it is just a fact I am using to test the other statements)
we am testing the fitting of x in the ranges stated in I, II, and III

it is similar to the question you just wrote:
jfranciscocuencag wrote:
If x must be less than -2, how come could II holds true?
What happens if x is -2 as the II statement says that x could be equal or less than -2 if it's equal to -2 then it can't be true.
Kind regards!


In summary,
"must be true" questions are different, and has a special mindset to deal with it.
as long as ALL the values are covered by the proposed range (even though there are extra irrelevant values), then the statement is true,
but if the proposed range is not covering ALL the values, then false (as in the example in green below)
Mahmoudfawzy83 wrote:
Hi eabhgoy

in the "must be true questions", you have to deal with question the other way around.
the inequality : (10-x)/3 < -2x , is the TRUE ground that you will start from. by simplifying it, it will be x< -2.

so the question says that:
if the value of x is less than -2 ... (-2.5, -3, ........)
these values are covered by which of the following I, II , III?

I) 2 < x ---> for sure false, because it is not covering any of the values of x< -2

II) |x-5| >= 7 : so the range is x>=12 and x<= -2 ---> it covers ALL the values of "x<-2" and MORE
(ALL is a must, MORE doesn't affect the judgement because if you try ANY value of "x<-2", it will fit into "|x-5| >= 7"

III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

however, if in the choices, there is (for example):
x<-4 -----> this will be false because it is not covering ALL values of "x<-2" (if x = -3 for instance)


Try practicing on the "Must or Could be True Questions" under the problem solving part of the question bank:
https://gmatclub.com/forum/search.php?view=search_tags
start with the easy level to get familiar with this type of questions.
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 31 Mar 2019, 21:24
Mahmoudfawzy83 wrote:
Hi eabhgoy

in the "must be true questions", you have to deal with question the other way around.
the inequality : (10-x)/3 < -2x , is the TRUE ground that you will start from. by simplifying it, it will be x< -2.

so the question says that:
if the value of x is less than -2 ... (-2.5, -3, ........)
these values are covered by which of the following I, II , III?

I) 2 < x ---> for sure false, because it is not covering any of the values of x< -2

II) |x-5| >= 7 : so the range is x>=12 and x<= -2 ---> it covers ALL the values of "x<-2" and MORE
(ALL is a must, MORE doesn't affect the judgement because if you try ANY value of "x<-2", it will fit into "|x-5| >= 7"

III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

however, if in the choices, there is (for example):
x<-4 -----> this will be false because it is not covering ALL values of "x<-2" (if x = -3 for instance)


Thanks buddy, makes sense!
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if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 11 May 2019, 01:02
III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

Hi Mahmoudfawzy83, Could you please help me with the steps to solve the 3rd inequality? I'm lost.
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if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post Updated on: 11 May 2019, 02:18
We know that x<-2

Assume that 3rd statement is true
|x-1|/|x| > 1
Or |x-1|>|x| (We can cross-multiply here, because |x| is always positive under given constraint)
or |x-1| - |x| >0
If x is less than -2, Then x-1 is negative
Hence |x-1|= -(x-1)
and |x|= -x
-(x-1) - (-x)>0
-x+1+x>0
1>0
Hence our assumption was true, 3rd statement will always true under given constraint

99ramanmehta wrote:
III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

Hi Mahmoudfawzy83, Could you please help me with the steps to solve the 3rd inequality? I'm lost.

Originally posted by nick1816 on 11 May 2019, 01:36.
Last edited by nick1816 on 11 May 2019, 02:18, edited 2 times in total.
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 11 May 2019, 02:00
nick1816 wrote:
We know that x<-2 which means x is always negative

Assume that 3rd statement is true
|x-1|/|x| > 1
Or |x-1|>|x| (We can cross-multiply here, because |x| is always positive under given constraint)
or |x-1| - |x| >0
If x is negative Then x-1 is also negative
Hence |x-1|= -(x-1)
and |x|= -x
-(x-1) - (-x)>0
-x+1+x>0
1>0
Hence our assumption was true, 3rd statement will always true under given constraint

99ramanmehta wrote:
III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

Hi Mahmoudfawzy83, Could you please help me with the steps to solve the 3rd inequality? I'm lost.


Thanks for the explanation Nick. I, too, was able reach to the conclusion that |x-1| - |x| >0 and hence 1>0. What I want to understand is that how did Mahmoudfawzy83 solve this equation to find the range is x <-1/2. Can you please help on this?
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if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 11 May 2019, 02:01
99ramanmehta wrote:
III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

Hi Mahmoudfawzy83, Could you please help me with the steps to solve the 3rd inequality? I'm lost.


Hi 99ramanmehta

\(\frac{|x-1|}{|x|} > 1\)

because |x| is positive (we are sure of the sign), multiplying, both sides by |x| is valid, so:
\(|x-1| > |x|\)

and because both sides are not negative , squaring both sides is valid (so we can get rid of the "||")

\(x^2 - 2x + 1 > x^2\)
\(1 > 2x\)
\(\frac{1}{2}> x\)

an important step is to check the outcome by trying it in the original equation --> choose \(\frac{1}{4}\) for example:
\(|\frac{\frac{1}{4}-1}{\frac{1}{4}}|\) = \(|-3| = 3 > 1\) ---> so confirmed

(note that the range is all values less than \(\frac{1}{2}\), except 0 because \(|x|\) is a denominator)

Thanks nick1816 for sharing your idea with me and 99ramanmehta ,

I would like to comment on your approach
nick1816 wrote:
If x is negative Then x-1 is also negative

..... But x-1 can be negative \(\frac{-1}{2}\), while x is \(\frac{1}{2}\) (positive) , so your assumption is not perfect.

to confirm, try \(x = \frac{3}{4}\) (which lies in your range), and you will find it invalid.
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 11 May 2019, 02:17
I've corrected it now. Thanks!!!

I would like to comment on your approach
nick1816 wrote:
If x is negative Then x-1 is also negative

..... But x-1 can be negative \(\frac{-1}{2}\), while x is \(\frac{1}{2}\) (positive) , so your assumption is not perfect.

to confirm, try \(x = \frac{3}{4}\) (which lies in your range), and you will find it invalid.[/quote]
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 11 May 2019, 06:15
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Mahmoudfawzy83 wrote:
99ramanmehta wrote:
III) (|x-1|/|x|) > 1 : so the range is x <-1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE

Hi Mahmoudfawzy83, Could you please help me with the steps to solve the 3rd inequality? I'm lost.


Hi 99ramanmehta

\(\frac{|x-1|}{|x|} > 1\)

because |x| is positive (we are sure of the sign), multiplying, both sides by |x| is valid, so:
\(|x-1| > |x|\)

and because both sides are not negative , squaring both sides is valid (so we can get rid of the "||")

\(x^2 - 2x + 1 > x^2\)
\(1 > 2x\)
\(\frac{1}{2}> x\)

an important step is to check the outcome by trying it in the original equation --> choose \(\frac{1}{4}\) for example:
\(|\frac{\frac{1}{4}-1}{\frac{1}{4}}|\) = \(|-3| = 3 > 1\) ---> so confirmed

(note that the range is all values less than \(\frac{1}{2}\), except 0 because \(|x|\) is a denominator)

Thanks nick1816 for sharing your idea with me and 99ramanmehta ,

I would like to comment on your approach
nick1816 wrote:
If x is negative Then x-1 is also negative

..... But x-1 can be negative \(\frac{-1}{2}\), while x is \(\frac{1}{2}\) (positive) , so your assumption is not perfect.

to confirm, try \(x = \frac{3}{4}\) (which lies in your range), and you will find it invalid.



Thank you so much Mahmoudfawzy83 for the clarification. I followed the below mentioned approach:

Critical points 0,1

1. For x<0; 1>0 => true

2. For x>=1; -1>0 => false

3. For 0<x<1;
-(x-1)>x
-x+1>x
1>2x
x<1/2

in your earlier answer, you'd specified the range as x<-1/2 instead of x<1/2, hence the confusion. :)
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Re: if (10-x)/3 <-2x, which of the following must be true?  [#permalink]

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New post 13 May 2019, 12:12
Mahmoudfawzy83 wrote:
Hi eabhgoy

in the "must be true questions", you have to deal with question the other way around.
the inequality : (10-x)/3 < -2x , is the TRUE ground that you will start from. by simplifying it, it will be x< -2.

so the question says that:
if the value of x is less than -2 ... (-2.5, -3, ........)
these values are covered by which of the following I, II , III?

I) 2 < x ---> for sure false, because it is not covering any of the values of x< -2

II) |x-5| >= 7 : so the range is x>=12 and x<= -2 ---> it covers ALL the values of "x<-2" and MORE
(ALL is a must, MORE doesn't affect the judgement because if you try ANY value of "x<-2", it will fit into "|x-5| >= 7"

III) (|x-1|/|x|) > 1 : so the range is x <1/2 ----> the same as II, it covers ALL values of "x<-2" and MORE


Can you please explain how x <1/2 is same as x<-2? What if x is greater than -2 and less than 1/2.


however, if in the choices, there is (for example):
x<-4 -----> this will be false because it is not covering ALL values of "x<-2" (if x = -3 for instance)
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Re: if (10-x)/3 <-2x, which of the following must be true?   [#permalink] 13 May 2019, 12:12
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