Last visit was: 21 May 2024, 04:58 It is currently 21 May 2024, 04:58
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
GRE Forum Moderator
Joined: 02 Nov 2016
Posts: 14033
Own Kudos [?]: 33962 [14]
Given Kudos: 5804
GPA: 3.62
Send PM
Retired Moderator
Joined: 19 Oct 2018
Posts: 1877
Own Kudos [?]: 6382 [0]
Given Kudos: 704
Location: India
Send PM
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 18874
Own Kudos [?]: 22278 [0]
Given Kudos: 285
Location: United States (CA)
Send PM
Director
Director
Joined: 29 Oct 2015
Posts: 726
Own Kudos [?]: 332 [0]
Given Kudos: 475
Send PM
Which of the following inequalities, if true, is sufficient alone to [#permalink]
X^5 - x^3 < 0 
x^3(x+1) (x-1) <0


Attached is the wave line diagram ; only valid range is x < -1 and x is between 0 and 1 because in these two ranges only the inequality becomes valid i.e negative. egmat

E works and is the answer.­
Attachments

wavy line.png
wavy line.png [ 14.37 KiB | Viewed 455 times ]

Tutor
Joined: 11 Aug 2023
Posts: 925
Own Kudos [?]: 1616 [0]
Given Kudos: 83
GMAT 1: 800 Q51 V51
Send PM
Re: Which of the following inequalities, if true, is sufficient alone to [#permalink]
Expert Reply
Which of the following inequalities, if true, is sufficient alone to show that \(\sqrt[3]{x} < \sqrt[5]{x}\)?

We could just check the answer choices.

(A) \(-1 < x < 0\)

When \(x\) is a fraction less than \(1\), \(\sqrt[5]{x}\) is further from \(0\) than \(\sqrt[3]{x}\).

So, if \(-1 < x < 0\), then \(\sqrt[3]{x} > \sqrt[5]{x}\).

Eliminate.

(B) \(x > 1\)

Clearly incorrect since, if \(x > 1\), then \(\sqrt[3]{x} > \sqrt[5]{x}\).

Eliminate.

(C) \(|x| < - 1\)

Impossible.

Eliminate.

(D) \(|x| > 1\)

As we saw for (A), if \(x\) is between \(-1\) and \(0\), \(\sqrt[3]{x} > \sqrt[5]{x}\).

Eliminate.

(E) \(x < -1\)

For all values of \(x\) such that \(x < -1\), \(\sqrt[5]{x}\) is closer to \(0\), in other words, less negative, than \(\sqrt[3]{x}\).
­
So, if \(x < -1\), then \(\sqrt[3]{x} < \sqrt[5]{x}\).

Keep.

Correct answer: E­
Tutor
Joined: 16 Oct 2010
Posts: 14891
Own Kudos [?]: 65400 [1]
Given Kudos: 431
Location: Pune, India
Send PM
Re: Which of the following inequalities, if true, is sufficient alone to [#permalink]
1
Kudos
Expert Reply
 
Sajjad1994 wrote:
Which of the following inequalities, if true, is sufficient alone to show that \(\sqrt[3]{x} < \sqrt[5]{x}\) ?

(A) \(-1 < x < 0\)

(B) \(x > 1\)

(C) \(|x| < - 1\)

(D) \(|x| > 1\)

(E) \(x < -1\)

­
When we raise an inequality to an odd power, the inequality sign does not change. It stays the same. 

Think about it. If both are positiive, both remain positive and the smaller magnitude remains smaller.
If both are negative, both remain negative and the smaller magnitude remains smaller. So what was less negative stays less negative.
If one is positive and the other negative then their signs remain as such. So negative one remains smaller. 

\(\sqrt[3]{x} < \sqrt[5]{x}\)
When we raise it to the power 15, we get
\(x^5 < x^3\)

Now you can use inequalities or focus on the number line. Try cases from x > 1 and 0 < x < 1 
If x = 2, this obviously does hold true. 
But if x = 1/2, then it does.
Since it works in the range 0 < x < 1, it will be valid in the range x < -1 too since we are dealing with odd powers. 

Hence 0 < x < 1 or x < -1 

(A) \(-1 < x < 0\)
Not in our range

(B) \(x > 1\)
Not in our range

(C) \(|x| < - 1\)
Absolute value cannot be negative. Ignore.

(D) \(|x| > 1\)
x > 1 is not in our range so not acceptable

(E) \(x < -1\)
This is within our range so if it is true, x^5 is less than x^3. 

Answer (E)

Method 2:
Alternatively, if you do not want to raise both sides to the power 15, simply use test cases of 2^15 and 1/2^15
GMAT Club Bot
Re: Which of the following inequalities, if true, is sufficient alone to [#permalink]
Moderator:
Math Expert
93365 posts