BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

GMAT Prep Absolute question

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Source: — Problem Solving |
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Mon Sep 12, 2016 3:29 am
Mo2men wrote:If x≠0, is |x| < 1 ?

(1) x^2 < 1
(2) |x| < 1/x
Statement 1:
Since x is nonzero, only a positive or negative fraction between -1 and 1 will satisfy x²<1.
The absolute value of a positive or negative fraction must be less than 1.
SUFFICIENT.

Statement 2:
Only a positive fraction between 0 and 1 will satisfy |x| < 1/x.
The absolute value of a positive fraction must be less than 1.
SUFFICIENT.

The correct answer is D.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
User avatar
Legendary Member
Posts: 2135
Joined: Mon Feb 03, 2014 9:26 am
Location: https://martymurraycoaching.com/
Thanked: 955 times
Followed by:140 members
GMAT Score:800

by MartyMurray » Mon Sep 12, 2016 3:32 am
Question: If x ≠ 0, is |x| < 1?

Rephrase: Is -1 < x < 1

Statement 1: x² < 1

If x² < 1, then -1 < x < 1.

Sufficient.

Statement 2: |x| < 1/x

Try some numbers. There are six categories that could be tried.

x < -1, x = -1, -1 < x < 0, 0 < x < 1, x = 1, 1 < x

-2: |-2| < 1/-2 This is impossible as the left side is positive and the right side is negative. A positive number is not less than a negative number.

No negative numbers work. Try the positive categories

1/2: |1/2| < 1/(1/2) This is true.

1: |1| < 1/1 This is not true.

2: |2| < 1/2 This is not true, and illustrates the fact that for any x such that x > 1, Statement 2 will not be true.

The only one that works is 1/2.

|1/2| < 1

Sufficient.

The correct answer is D.

There is a trick to Statement 2.

One may think that one can multiply both sides by x and get x|x| < 1.

However that idea is not entirely correct.

In working with the inequality have to consider the possibility that x could be negative, in which case when we multiply both sides by x, the sign of the inequality would have to be reversed.

In considering the case in which x is negative, when we multiply both sides by x, we get x|x| > 1.

Since a negative number multiplied by an absolute value must generate a negative product, if x is negative, x|x| > 1 is not true.

So once again we have showed that x cannot be negative.

We also have to consider the case in which x is positive. If x is positive when we multiply we get x|x| < 1.

That statement is true when x is a fraction.

So 0 < x < 1, and once again we have showed that Statement 2 is sufficient.
Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.
Top
Legendary Member
Posts: 712
Joined: Fri Sep 25, 2015 4:39 am
Thanked: 14 times
Followed by:5 members

by Mo2men » Mon Sep 12, 2016 3:48 am
GMATGuruNY wrote:
Mo2men wrote:If x≠0, is |x| < 1 ?

(1) x^2 < 1
(2) |x| < 1/x
Statement 1:
Since x is nonzero, only a positive or negative fraction between -1 and 1 will satisfy x²<1.
The absolute value of a positive or negative fraction must be less than 1.
SUFFICIENT.

Statement 2:
Only a positive fraction between 0 and 1 will satisfy |x| < 1/x.
The absolute value of a positive fraction must be less than 1.
SUFFICIENT.

The correct answer is D.
Dear GMATGuru,

In statement 2,

if I want to use critical point. I struggle when x<0

is it -x<1/x or -x<1/-x?

Should replace every 'x' with '-x' or only 'x' inside modulus??
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Mon Sep 12, 2016 7:01 am
Mo2men wrote: Dear GMATGuru,

In statement 2,

if I want to use critical point. I struggle when x<0

is it -x<1/x or -x<1/-x?

Should replace every 'x' with '-x' or only 'x' inside modulus??
Since an absolute value cannot be less than a negative, |x| < 1/x is valid only if x is nonnegative.
Thus, we can simplify as follows:
x < 1/x
x² < 1
x < 1.
Since x≠0, 0 < x < 1.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
User avatar
GMAT Instructor
Posts: 1462
Joined: Thu Apr 09, 2015 9:34 am
Location: New York, NY
Thanked: 39 times
Followed by:22 members

by Jeff@TargetTestPrep » Mon Sep 12, 2016 4:48 pm
Mo2men wrote:If x≠0, is |x| < 1 ?

(1) x^2 < 1
(2) |x| < 1/x
We are given that x is not equal to 0 and we need to determine whether the absolute value of x is less than 1.

Statement One Alone:

x^2 < 1

We can take the square root of both sides of the inequality x^2 < 1:

√(x^2) < √1.

Since √(x^2) = |x|, |x| < 1.

Statement one alone is sufficient. Eliminate answer choices B, C and E.

Statement Two Alone:

|x| < 1/x

Since |x| will always be positive, in order for 1/x to be greater than |x|, x must be positive. Furthermore, the only way |x| < 1/x holds true is if x is a proper fraction. Thus, x must be a positive proper fraction. Since a positive proper fraction is between zero and one, the absolute value of x is less than one. Statement two alone is also sufficient.

Answer: D

Jeffrey Miller
Head of GMAT Instruction
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews
Top
GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Thu Sep 15, 2016 7:35 pm
We could start by rephrasing the prompt:

Is -1 < x < 1?

S1:

x*x < 1

It's probably easiest to see if we plug in numbers. If x ≥ 1, this won't work. If -1 ≥ x it won't work either. So we must have -1 < x < 1, and this answers our question! Sufficient.

S2:

|x| < 1/x

If x is positive, this is the same as statement one! We multiply both sides by x, and get |x| * x < 1.

If x is negative, this is impossible, since the left side is positive and the right side is negative.

So we must have x positive, and each statement is sufficient.
Top
Post new topic Post Reply
• Page 1 of 1