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# If – 1 < x < 1 and x is not equal to 0, then which of the following mu

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EMPOWERgmat Instructor
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GMAT 1: 800 Q51 V49
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If – 1 < x < 1 and x is not equal to 0, then which of the following mu  [#permalink]

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Updated on: 27 Oct 2015, 21:39
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QUANT 4-PACK SERIES Problem Solving Pack 1 Question 4 IF -1 < x < 1 and ...

If – 1 < x < 1 and x is not equal to 0, then which of the following must be true?

I. $$|x| > x^2$$

II. $$x – x^2 > x^3$$

III. $$|1 – x| = |x – 1|$$

A) I only
B) II only
C) I and II only
D) I and III only
E) II and III only

48 Hour Window Answer & Explanation Window
OA, and explanation will be posted after the 48 hour window closes.

This question is part of the Quant 4-Pack series

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# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Originally posted by EMPOWERgmatRichC on 22 Oct 2015, 20:08. Last edited by EMPOWERgmatRichC on 27 Oct 2015, 21:39, edited 1 time in total.  EMPOWERgmat Discount Codes Magoosh Discount Codes Manhattan GMAT Discount Codes Manager Joined: 01 Jan 2015 Posts: 63 Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 22 Oct 2015, 21:02 1 EMPOWERgmatRichC wrote: RANDOM 4-PACK SERIES Pack 1 Question 4 IF -1 < x < 1 and ... If – 1 < x < 1 and x is not equal to 0, then which of the following must be true? I. |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ III. |1 – x| = |x – 1| A) I only B) II only C) I and II only D) I and III only E) II and III only 48 Hour Window Answer & Explanation Window Earn KUDOS! Post your answer and explanation. OA, and explanation will be posted after the 48 hour window closes. This question is part of the Random 4-Pack series After glancing at the answer choices, I observed that there is no answer choice that includes all 3 statements. Statement 3 is clearly true because the absolute distance from x to 1 will certainly be equal to the absolute distance from 1 to x. Then I noticed that there is no answer choice for Statement 3 only. This eliminates answer choices A,B, and C. Now it is between D and E. Now either statement 1 is true or statement 2 is true, but both can't be true. It is easier to disprove something with a counterexample then proving something must be true. If you plug in $$x=-\frac{1}{2}$$ into statement 2, you will find it is not true. This eliminates answer choice E and leaves me with answer choice D. VP Status: It's near - I can see. Joined: 13 Apr 2013 Posts: 1257 Location: India Concentration: International Business, Operations GMAT 1: 480 Q38 V22 GPA: 3.01 WE: Engineering (Consulting) Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 22 Oct 2015, 21:55 1 1 EMPOWERgmatRichC wrote: RANDOM 4-PACK SERIES Pack 1 Question 4 IF -1 < x < 1 and ... If – 1 < x < 1 and x is not equal to 0, then which of the following must be true? I. |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ III. |1 – x| = |x – 1| A) I only B) II only C) I and II only D) I and III only E) II and III only 48 Hour Window Answer & Explanation Window Earn KUDOS! Post your answer and explanation. OA, and explanation will be posted after the 48 hour window closes. This question is part of the Random 4-Pack series My solution: Let's take two values of x as -0.5 and 0.5, then I) Any value inside the mode is positive then whether we take 0.5 or -0.5 this will remain positive. We also know that squaring a fraction gives us a value less then the original fraction(if the original fraction is positve) and always positive. So statement (I) is always true as 0.5 > 0.125. II) Not always true: If x is -0.5 then the expression becomes 0.625 = 0.625 III) Always holds true as anything inside the mode is positive. If x is -0.5 then mode 1-(-0.5) = mode -0.5-1 becomes 1.5 =1.5 (Don't know how to use mode symbol) Option D _________________ "Do not watch clock; Do what it does. KEEP GOING." Manager Joined: 11 Sep 2013 Posts: 110 Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 23 Oct 2015, 00:12 EMPOWERgmatRichC wrote: RANDOM 4-PACK SERIES Pack 1 Question 4 IF -1 < x < 1 and ... If – 1 < x < 1 and x is not equal to 0, then which of the following must be true? I. |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ III. |1 – x| = |x – 1| A) I only B) II only C) I and II only D) I and III only E) II and III only 48 Hour Window Answer & Explanation Window Earn KUDOS! Post your answer and explanation. OA, and explanation will be posted after the 48 hour window closes. This question is part of the Random 4-Pack series When -1<x<1: |x| > $$x^{2}$$ => I must be true x – $$x^{2}$$ > $$x^{3}$$ => Not sure it is true with all values of x |1 – x| = |x – 1| true with all the value of x => it is also true with all x: -1<x<1: III must be true Ans: D Current Student Joined: 20 Mar 2014 Posts: 2639 Concentration: Finance, Strategy Schools: Kellogg '18 (M) GMAT 1: 750 Q49 V44 GPA: 3.7 WE: Engineering (Aerospace and Defense) Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 23 Oct 2015, 04:21 1 EMPOWERgmatRichC wrote: RANDOM 4-PACK SERIES Pack 1 Question 4 IF -1 < x < 1 and ... If – 1 < x < 1 and x is not equal to 0, then which of the following must be true? I. |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ III. |1 – x| = |x – 1| A) I only B) II only C) I and II only D) I and III only E) II and III only 48 Hour Window Answer & Explanation Window Earn KUDOS! Post your answer and explanation. OA, and explanation will be posted after the 48 hour window closes. This question is part of the Random 4-Pack series Given, $$- 1 < x < 1$$ and $$x \neq 0$$, For a Must be true question, ALL cases posisble must satisfy the given conditions. i) $$|x| > x^2$$ ---> 1 positive quantity > another positive squared quantity , can only happen for fractions between -1 and 1. As this is the given range of x, this has to be true for all cases. Additionally, algebraically, when $$x<0$$ --> $$|x|=-x$$ ---> $$-x>x^2$$ --->$$x^2+x<0$$ --->$$-1<x<0$$ ...(1) when $$x\geq 0$$-->$$|x|=x$$---> $$x>x^2$$ ---> $$x^2-x<0$$ ---> $$0<x<1$$ ...(2) Both (1) and (2) lie in the given ranges and hence must be true for all cases. Eliminate B and E. ii) $$x – x^2 > x^3$$ ---> $$x^3+x^2-x <0$$ ---> $$x(x^2+x-1)<0$$ ---> $$x(x-a)(x-b)<0$$ ---> where $$a<0$$ and $$b>0$$--->$$x<a$$ and $$0<x<b$$. As $$a=\frac{-1-\sqrt{5}}{2}$$ (= a quantity <-1) and $$b=\frac{-1+\sqrt{5}}{2}$$, clearly see that $$x<a$$ is out of the given range of -$$1<x<$$1, hence making this statement NOT a must be true statement. Eliminate C. iii) $$|1 – x| = |x – 1|$$, $$|x-1| > 0$$ for all values $$-1<x<1$$ and as $$|x-1| > 0$$ for $$-1<x<0$$ and as this range lies within the given range, this is a MUST BE TRUE statement, making D as the correct answer. EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12411 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 26 Oct 2015, 16:42 Learning4mU wrote: My solution: Let's take two values of x as -0.5 and 0.5, then I) Any value inside the mode is positive then whether we take 0.5 or -0.5 this will remain positive. We also know that squaring a fraction gives us a value less then the original fraction(if the original fraction is positve) and always positive. So statement (I) is always true as 0.5 > 0.125. Hi Learning4mU, Your logic IS correct, but you have to be careful about your calculations. $$(.5)^{2}$$ = .25 (NOT .125). Working through this prompt, your logic is sound, so the miscalculation wouldn't hurt you. In other types of questions though (especially DS), a miscalculation could cost you the points. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12411
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu  [#permalink]

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26 Oct 2015, 17:06
1
1
Hi All,

In Roman Numeral questions, the arrangement of the answer choices often provides a clue as to how you could deal with the prompt in the most efficient way possible (and avoid some of the work). From these answer choices, we know that only 1 or 2 of the 3 Roman Numerals MUST be true, so we can deal with the Roman Numerals 'out of order' if it will save us some effort.

We're given a range of possible values (-1 < x < 1) and that X CANNOT be 0. We can prove/disprove the various Roman Numerals by TESTing VALUES and looking for patterns.

I. I. |x| > $$x^{2}$$

|x| will be positive for any value of X that we can choose. Since we'll be SQUARING a fraction (regardless of whether it's a positive fraction or a negative fraction), the result WILL be smaller than the |x|.

eg.
IF....
X = 1/2
$$(1/2)^{2}$$ = 1/4

IF...
X = -1/3
$$(-1/3)^{2}$$ = 1/9

Thus, Roman Numeral 1 is ALWAYS TRUE.

II. x – $$x^{2}$$ > $$x^{3}$$

This Roman Numeral is probably the 'scariest looking', but it can also be dealt with by TESTing VALUES.

While....
(1/2) - $$(1/2)^{2}$$ > $$(1/2)^{3}$$
1/2 - 1/4 > 1/8
1/4 > 1/8
Is TRUE

(-1/2) - $$(-1/2)^{2}$$ > $$(-1/2)^{3}$$
-1/2 - 1/4 > -1/8
-3/4 > -1/8
is NOT TRUE

Thus, Roman Numeral 2 is NOT always true.

III. |1 – x| = |x – 1|

This Roman Numeral is actually a math 'truism'; it's always true for any rational value of X. You can use as many different values as you like, the resulting calculations are ALWAYS equal.

Thus, Roman Numeral 3 is ALWAYS TRUE.

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** VP Status: It's near - I can see. Joined: 13 Apr 2013 Posts: 1257 Location: India Concentration: International Business, Operations GMAT 1: 480 Q38 V22 GPA: 3.01 WE: Engineering (Consulting) Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 26 Oct 2015, 21:12 EMPOWERgmatRichC wrote: Learning4mU wrote: My solution: Let's take two values of x as -0.5 and 0.5, then I) Any value inside the mode is positive then whether we take 0.5 or -0.5 this will remain positive. We also know that squaring a fraction gives us a value less then the original fraction(if the original fraction is positve) and always positive. So statement (I) is always true as 0.5 > 0.125. Hi Learning4mU, Your logic IS correct, but you have to be careful about your calculations. $$(.5)^{2}$$ = .25 (NOT .125). Working through this prompt, your logic is sound, so the miscalculation wouldn't hurt you. In other types of questions though (especially DS), a miscalculation could cost you the points. GMAT assassins aren't born, they're made, Rich Thanks a lot Rich for pointing out this silly error of mine. Surely it has worked here but some other day it can be a blunder and cost me heavy points. I ll certainly take care of this. _________________ "Do not watch clock; Do what it does. KEEP GOING." Intern Joined: 01 Feb 2015 Posts: 5 Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 10 Jun 2016, 07:16 Engr2012 wrote: EMPOWERgmatRichC wrote: bunuel wrote: RANDOM 4-PACK SERIES Pack 1 Question 4 IF -1 < x < 1 and ... If – 1 < x < 1 and x is not equal to 0, then which of the following must be true? I. |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ III. |1 – x| = |x – 1| A) I only B) II only C) I and II only D) I and III only E) II and III only 48 Hour Window Answer & Explanation Window Earn KUDOS! Post your answer and explanation. OA, and explanation will be posted after the 48 hour window closes. Given, $$- 1 < x < 1$$ and $$x \neq 0$$, For a Must be true question, ALL cases posisble must satisfy the given conditions. i) $$|x| > x^2$$ ---> 1 positive quantity > another positive squared quantity , can only happen for fractions between -1 and 1. As this is the given range of x, this has to be true for all cases. Additionally, algebraically, when $$x<0$$ --> $$|x|=-x$$ ---> $$-x>x^2$$ --->$$x^2+x<0$$ --->$$-1<x<0$$ ...(1) when $$x\geq 0$$-->$$|x|=x$$---> $$x>x^2$$ ---> $$x^2-x<0$$ ---> $$0<x<1$$ ...(2) Both (1) and (2) lie in the given ranges and hence must be true for all cases. Eliminate B and E. ii) $$x – x^2 > x^3$$ ---> $$x^3+x^2-x <0$$ ---> $$x(x^2+x-1)<0$$ ---> $$x(x-a)(x-b)<0$$ ---> where $$a<0$$ and $$b>0$$--->$$x<a$$ and $$0<x<b$$. As $$a=\frac{-1-\sqrt{5}}{2}$$ (= a quantity <-1) and $$b=\frac{-1+\sqrt{5}}{2}$$, clearly see that $$x<a$$ is out of the given range of -$$1<x<$$1, hence making this statement NOT a must be true statement. Eliminate C. iii) $$|1 – x| = |x – 1|$$, $$|x-1| > 0$$ for all values $$-1<x<1$$ and as $$|x-1| > 0$$ for $$-1<x<0$$ and as this range lies within the given range, this is a MUST BE TRUE statement, making D as the correct answer. I have been making this mistake for a lot of modulus problems...please help me... when x<0 we have x^2+x<0 that is x(x+1)<0 so x<0 and x+1<0 hence X<0 and x<-1 how is it that x>-1 according to your solution? You said when x<0 the range is -1<x<0 please explain Current Student Joined: 18 Oct 2014 Posts: 874 Location: United States GMAT 1: 660 Q49 V31 GPA: 3.98 Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 10 Jun 2016, 09:20 EMPOWERgmatRichC wrote: QUANT 4-PACK SERIES Problem Solving Pack 1 Question 4 IF -1 < x < 1 and ... If – 1 < x < 1 and x is not equal to 0, then which of the following must be true? I. |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ III. |1 – x| = |x – 1| A) I only B) II only C) I and II only D) I and III only E) II and III only 48 Hour Window Answer & Explanation Window Earn KUDOS! Post your answer and explanation. OA, and explanation will be posted after the 48 hour window closes. This question is part of the Quant 4-Pack series x is a +ve or -ve fraction <1 but not equal to 0 I. |x| > $$x^{2}$$ x^2 and |x| will always be +ve and squaring +ve fraction always decrease the value. hence |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ If we take x= 1/2 or -1/2 and put the value in the given equation, it might or might not be right. III. |1 – x| = |x – 1| If we put 1/2 or -1/2 in the given equation, the result will be same. True D) I and III only is the answer. _________________ I welcome critical analysis of my post!! That will help me reach 700+ Intern Joined: 01 Feb 2015 Posts: 5 Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 10 Jun 2016, 10:02 Divyadisha wrote: EMPOWERgmatRichC wrote: QUANT 4-PACK SERIES Problem Solving Pack 1 Question 4 IF -1 < x < 1 and ... If – 1 < x < 1 and x is not equal to 0, then which of the following must be true? I. |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ III. |1 – x| = |x – 1| A) I only B) II only C) I and II only D) I and III only E) II and III only 48 Hour Window Answer & Explanation Window Earn KUDOS! Post your answer and explanation. OA, and explanation will be posted after the 48 hour window closes. x is a +ve or -ve fraction <1 but not equal to 0 I. |x| > $$x^{2}$$ x^2 and |x| will always be +ve and squaring +ve fraction always decrease the value. hence |x| > $$x^{2}$$ II. x – $$x^{2}$$ > $$x^{3}$$ If we take x= 1/2 or -1/2 and put the value in the given equation, it might or might not be right. III. |1 – x| = |x – 1| If we put 1/2 or -1/2 in the given equation, the result will be same. True D) I and III only is the answer. hello disha can u answer my question above? please tell me what is the mistake in that approach... EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12411 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu [#permalink] ### Show Tags 10 Jun 2016, 10:22 mup05ro wrote: I have been making this mistake for a lot of modulus problems...please help me... when x<0 we have x^2+x<0 that is x(x+1)<0 so x<0 and x+1<0 hence X<0 and x<-1 how is it that x>-1 according to your solution? You said when x<0 the range is -1<x<0 please explain Hi mup05ro, When dealing with that inequality, you CANNOT "break it into pieces" the way that you did. When you wrote that X < 0, you have to consider how that impacts the other part of the calculation. For example, If X = -2, how does THAT value impact the overall inequality? IF X = -2... X(X+1) = (-2)(-2 + 1) = (-2)(-1) = +2.... but that is NOT less than 0, so X CANNOT be -2. In that same way, while X can be negative, X cannot be less than -1. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Intern
Joined: 01 Feb 2015
Posts: 5
Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu  [#permalink]

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10 Jun 2016, 10:43
EMPOWERgmatRichC wrote:
mup05ro wrote:

when x<0
we have x^2+x<0
that is x(x+1)<0
so x<0 and x+1<0
hence X<0 and x<-1
how is it that x>-1 according to your solution? You said when x<0 the range is -1<x<0

Hi mup05ro,

When dealing with that inequality, you CANNOT "break it into pieces" the way that you did.

When you wrote that X < 0, you have to consider how that impacts the other part of the calculation. For example, If X = -2, how does THAT value impact the overall inequality?

IF X = -2...

X(X+1) =
(-2)(-2 + 1) =
(-2)(-1) = +2.... but that is NOT less than 0, so X CANNOT be -2. In that same way, while X can be negative, X cannot be less than -1.

GMAT assassins aren't born, they're made,
Rich

thank you

how do determine this for every such problem? I would do the steps that I did above and simply say x<-1, so should i always verify with the range I got and conclude? I have not done that for many problems that i got right
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12411
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu  [#permalink]

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10 Jun 2016, 17:09
Hi mup05ro,

Some of the questions that you're going to face on Test Day will test the "thoroughness" of your thinking (Data Sufficiency does this often). If there are certain question types that you find challenging (such as absolute value/modulus questions), then you should consider doing a little more work to prove that you're correct. By TESTing VALUES, you would have PROOF of whether your deductions are correct or not.

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Manager
Joined: 30 Mar 2017
Posts: 136
GMAT 1: 200 Q1 V1
Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu  [#permalink]

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16 Jun 2018, 12:29
We're given that -1<x<0 or 0<x<1. To see if each statement must be true, we should test these ranges.

I. $$x^2<|x|$$
If x is negative, then |x|=-x, so expression becomes
$$x^2<|x|$$
$$x^2<-x$$
$$x>-1$$ (dividing by x and reversing the inequality since x is negative)
So if x is negative, -1<x<0, which falls inside the negative part of the given range for x.

If x is positive, then |x|=x, so expression becomes
$$x^2<|x|$$
$$x^2<x$$
$$x<1$$
So if x is positive, 0<x<1, which falls inside the positive part of the given range for x.

Thus, this statement must be true.

II. $$x^3<x-x^2$$
$$x^3+x^2-x<0$$
If -1<x<0, then we have
$$x^3+x^2-x<0$$
(small negative) + (larger positive) + (even larger positive)
which cannot be less than 0

Thus, this statement must not be true.

III. |1-x|=|-(x-1)|=|x-1|
identity property. always true

Thus, this statement must be true.

Re: If – 1 < x < 1 and x is not equal to 0, then which of the following mu &nbs [#permalink] 16 Jun 2018, 12:29
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