Last visit was: 23 Jul 2024, 23:17 It is currently 23 Jul 2024, 23:17
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
Tags:
Show Tags
Hide Tags
GMAT Club Legend
GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6027
Own Kudos [?]: 13823 [1]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Send PM
Tutor
Joined: 04 Aug 2010
Posts: 1328
Own Kudos [?]: 3234 [3]
Given Kudos: 9
Schools:Dartmouth College
Send PM
Re: If a ≠ 0, is a + a−1 > 2? (1) a > 0 (2) a < 1 [#permalink]
GMATGuruNY wrote:
Asad wrote:
If \(a ≠ 0\), is \(a + a^{−1} > 2\)?

(1) \(a > 0\)
(2) \(a < 1\)


CRITICAL POINTS occur when the two sides of the inequality are EQUAL or when the inequality is UNDEFINED.
Given \(a + a^{−1} > 2\):
The two sides are equal when a=1, since \(1 + 1^{−1} = 1 + \frac{1}{1^1} = 2\).
The inequality is undefined when a=0, since \(0^{-1} = \frac{1}{0^1} =\) undefined.

To determine which ranges satisfy the inequality, test one value to the left and one value to the right of each critical point.
Here, we must test a<0, 0<a<1 and a>1.
If we test a=-1, a=1/2 and a=2, only a=1/2 and a=2 satisfy \(a + a^{−1} > 2\), implying that the valid ranges are 0<a<1 and a>1.

Question stem, rephrased:
Is \(a\) a positive value other than 1?

Statement 1:
If a=2, the answer to the rephrased question stem is YES.
If a=1, the answer to the rephrased question stem is NO.
INSUFFICIENT.

Statement 2:
If a=1/2, the answer to the rephrased question stem is YES.
If a=-1, the answer to the rephrased question stem is NO.
INSUFFICIENT.

Statements combined:
Since 0<a<1, the answer to the rephrased question stem is YES.
SUFFICIENT.


GMATGuruNY
Thank you so much for your response. Sir, could explain a bit the highlighted part? i mean: how do someone convinced that we must test those values?
Also, why it is other than 1
Thanks__
Tutor
Joined: 04 Aug 2010
Posts: 1328
Own Kudos [?]: 3234 [0]
Given Kudos: 9
Schools:Dartmouth College
Send PM
Re: If a ≠ 0, is a + a−1 > 2? (1) a > 0 (2) a < 1 [#permalink]
Expert Reply
Asad wrote:
GMATGuruNY wrote:
Asad wrote:
If \(a ≠ 0\), is \(a + a^{−1} > 2\)?

(1) \(a > 0\)
(2) \(a < 1\)


CRITICAL POINTS occur when the two sides of the inequality are EQUAL or when the inequality is UNDEFINED.
Given \(a + a^{−1} > 2\):
The two sides are equal when a=1, since \(1 + 1^{−1} = 1 + \frac{1}{1^1} = 2\).
The inequality is undefined when a=0, since \(0^{-1} = \frac{1}{0^1} =\) undefined.

To determine which ranges satisfy the inequality, test one value to the left and one value to the right of each critical point.
Here, we must test a<0, 0<a<1 and a>1.
If we test a=-1, a=1/2 and a=2, only a=1/2 and a=2 satisfy \(a + a^{−1} > 2\), implying that the valid ranges are 0<a<1 and a>1.

Question stem, rephrased:
Is \(a\) a positive value other than 1?

Statement 1:
If a=2, the answer to the rephrased question stem is YES.
If a=1, the answer to the rephrased question stem is NO.
INSUFFICIENT.

Statement 2:
If a=1/2, the answer to the rephrased question stem is YES.
If a=-1, the answer to the rephrased question stem is NO.
INSUFFICIENT.

Statements combined:
Since 0<a<1, the answer to the rephrased question stem is YES.
SUFFICIENT.


GMATGuruNY
Thank you so much for your response. Sir, could explain a bit the highlighted part? i mean: how do someone convinced that we must test those values?
Also, why it is other than 1
Thanks__



The critical points, plotted on a number line:
<------0-------1------>
0 and 1 are the only values where the \(a + a^{−1}\) is undefined or where \(a + a^{−1} = 2\).
Implication:
In each of the colored ranges, \(a + a^{−1} < 2\) or \(a + a^{−1} > 2\).
To determine in which ranges \(a + a^{−1} > 2\), we must test only one value in each of the colored ranges.

a=-1 is in the red range.
When a=-1, \(a + a^{−1} < 2\).
Thus, the red range (a<0) is not valid.

a=1/2 is in the green range.
When a=1/2, \(a + a^{−1} > 2\).
Thus, the green range (0<a<1) is valid.

a=2 is in the blue range.
When a=2, \(a + a^{−1} > 2\).
Thus, the blue range (a>1) is valid.

Result:
\(a + a^{−1} > 2\) when \(a\) is in the green range (0<a<1) or in the blue range (a>1).
Together, the green and blue ranges include every positive value except for 1.
Thus, the answer to the question stem is YES if \(a\) is ANY POSITIVE VALUE OTHER THAN 1.

Question stem. rephrased:
Is \(a\) a positive value other than 1?
Current Student
Joined: 17 Jun 2020
Status:Stuck with 99 points. Call 911!
Posts: 30
Own Kudos [?]: 23 [1]
Given Kudos: 254
Location: Bangladesh
Concentration: Finance, General Management
GMAT 1: 630 Q45 V32
WE:General Management (Transportation)
Send PM
Re: If a ≠ 0, is a + a−1 > 2? (1) a > 0 (2) a < 1 [#permalink]
1
Bookmarks
Bunuel
I tried to solve it using algebra. Here's my process. Is it right?
a+a^-1>2
or a+1/a-2>0
or (a-1)^2. a^-1>0
So, the question becomes is 1/a>0 when a≠1?
1. a>0, It works in all the cases except when a=1. so, Not sufficient
2. a<1, a can be + or -. So, Not sufficient.
(1) + (2) = 0<a<1 , So, Clearly a is + & a≠1. So, Sufficent . Answer. (C)
User avatar
Non-Human User
Joined: 09 Sep 2013
Posts: 34052
Own Kudos [?]: 853 [0]
Given Kudos: 0
Send PM
Re: If a 0, is a + a1 > 2? (1) a > 0 (2) a < 1 [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
GMAT Club Bot
Re: If a 0, is a + a1 > 2? (1) a > 0 (2) a < 1 [#permalink]
Moderator:
Math Expert
94589 posts