GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 Sep 2018, 23:35

### 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

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

# Find the range of values of x such that (x-5)^3 (2-4x) < 0

Author Message
TAGS:

### Hide Tags

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1994
Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

Updated on: 23 Aug 2018, 21:25
00:00

Difficulty:

65% (hard)

Question Stats:

53% (01:10) correct 47% (00:58) wrong based on 155 sessions

### HideShow timer Statistics

Solving inequalities- Number Line Method - Practice Question #1

Find the range of values of x such that $$(x-5)^3 (2-4x) < 0$$

A. $$\frac{1}{2} < x < 5$$
B. $$0 < x < \frac{1}{2}$$
C. $$x >5$$
D. $$\frac{1}{2} < x < 5$$
E. $$(x < \frac{1}{2}$$) and ($$x > 5$$)

Next Question

To read the article: Solving inequalities- Number Line Method

_________________

Number Properties | Algebra |Quant Workshop

Success Stories
Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Originally posted by EgmatQuantExpert on 23 Aug 2018, 04:23.
Last edited by EgmatQuantExpert on 23 Aug 2018, 21:25, edited 1 time in total.
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 1994
Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

Updated on: 27 Aug 2018, 02:05
1

Solution

Given:
• An inequality, $$(x-5)^3 (2-4x) < 0$$

To find:
• The range of x, that satisfies the above inequality

Approach and Working:
• So, if we observe carefully, the inequality given to us can be written as,
o $$(x-5)^2 * (x-5) * (2-4x) < 0$$
• We know that, any number of the form N2 is always positive, expect for N = 0.
o So, we can say that, $$(x - 5)^2$$ is always positive, expect for x = 5,
o Thus, the inequality can be written as (x - 5) * (2 - 4x) < 0
• Now, let’s multiply the inequality by -1 to make the coefficient of x, positive
o And, note that the inequality sign must be changed as we are multiplying it by a negative number, -1
o Thus, the inequality becomes, (x - 5) * (4x - 2) > 0
• The zero points of this inequality are x = 5 and $$x = \frac{1}{2}$$
• Plot these points on the number line.
o Since, both (x – 5) and (4x – 2) > 0, the inequality will be positive, for all the points to the right of 5
o And, it is negative, in the region, between $$\frac{1}{2}$$ and 5, since (x – 5) is negative and (4x – 2) is positive in this region
o It is again positive for all the values less than $$\frac{1}{2}$$, since both (x – 5) and (4x – 2) are negative in this region

Therefore, the range of x is $$x < \frac{1}{2}$$ and x > 5

Hence, the correct answer is option E.

_________________

Number Properties | Algebra |Quant Workshop

Success Stories
Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Originally posted by EgmatQuantExpert on 23 Aug 2018, 04:33.
Last edited by EgmatQuantExpert on 27 Aug 2018, 02:05, edited 3 times in total.
Math Expert
Joined: 02 Aug 2009
Posts: 6795
Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

23 Aug 2018, 05:38
2
EgmatQuantExpert wrote:

Find the range of values of x such that $$(x-5)^3 (2-4x) < 0$$

A. $$\frac{1}{2} < x < 5$$
B. $$0 < x < \frac{1}{2}$$
C. $$x >5$$
D. $$\frac{1}{2} < x < 5$$
E. $$(x < \frac{1}{2}$$) and ($$x > 5$$)

$$(x-5)^3 (2-4x) < 0...........(x-5)^3(1-2x)<0$$

two cases-
1) $$x-5<0$$ or $$x<5$$, then $$1-2x>0$$ or $$2x<1$$ or $$x<\frac{1}{2}$$.... thus $$x<\frac{1}{2}$$
2) $$x-5>0$$ or $$x>5$$, then $$1-2x<0$$ or $$2x>1$$ or $$x>\frac{1}{2}$$.... thus $$x>5$$

combined $$x<\frac{1}{2}$$ and $$x>5$$

E
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

GMAT online Tutor

Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 851
WE: Supply Chain Management (Energy and Utilities)
Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

23 Aug 2018, 06:35
EgmatQuantExpert wrote:
Solving inequalities- Number Line Method - Practice Question #1

Find the range of values of x such that $$(x-5)^3 (2-4x) < 0$$

A. $$\frac{1}{2} < x < 5$$
B. $$0 < x < \frac{1}{2}$$
C. $$x >5$$
D. $$\frac{1}{2} < x < 5$$
E. $$(x < \frac{1}{2}$$) and ($$x > 5$$)

$$(x-5)^3 (2-4x) < 0$$
Or, $$-2\left(x-5\right)^3\left(2x-1\right)<0$$
Or, $$2\left(x-5\right)^3\left(2x-1\right)>0$$

Critical points:- 5, 1/2
Using wavy curve method:-
range of x:-
$$x<\frac{1}{2}$$ or $$x>5$$

Ans. (E)

P.S:- In option (E) , OR sounds logical instead of AND.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Manager
Joined: 18 Apr 2018
Posts: 59
Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

05 Sep 2018, 02:29
EgmatQuantExpert wrote:

Solution

Given:
• An inequality, $$(x-5)^3 (2-4x) < 0$$

To find:
• The range of x, that satisfies the above inequality[/list

Approach and Working:
• So, if we observe carefully, the inequality given to us can be written as,
o $$(x-5)^2 * (x-5) * (2-4x) < 0$$
• We know that, any number of the form N2 is always positive, expect for N = 0.
o So, we can say that, $$(x - 5)^2$$ is always positive, expect for x = 5,
o Thus, the inequality can be written as (x - 5) * (2 - 4x) < 0
• Now, let’s multiply the inequality by -1 to make the coefficient of x, positive
o And, note that the inequality sign must be changed as we are multiplying it by a negative number, -1
o Thus, the inequality becomes, (x - 5) * (4x - 2) > 0
• The zero points of this inequality are x = 5 and $$x = \frac{1}{2}$$
• Plot these points on the number line.
o Since, both (x – 5) and (4x – 2) > 0, the inequality will be positive, for all the points to the right of 5
o And, it is negative, in the region, between $$\frac{1}{2}$$ and 5, since (x – 5) is negative and (4x – 2) is positive in this region
o It is again positive for all the values less than $$\frac{1}{2}$$, since both (x – 5) and (4x – 2) are negative in this region

Therefore, the range of x is $$x < \frac{1}{2}$$ and x > 5

Hence, the correct answer is option E.

Hello EgmatQuantExpert, thanks for the wholesome explanation. I would like to know why after multiplying the whole inequality by -1, the sign in (x-5) did no change only that of 2-4x changed. Thanks.

Posted from my mobile device
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 186
Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

06 Sep 2018, 19:23
EgmatQuantExpert wrote:

Find the range of values of x such that $$(x-5)^3 (2-4x) < 0$$

A. $$\frac{1}{2} < x < 5$$
B. $$0 < x < \frac{1}{2}$$
C. $$x >5$$
D. $$\frac{1}{2} < x < 5$$
E. $$(x < \frac{1}{2}$$) and ($$x > 5$$)

$$?\,\,\,:\,\,\,{\left( {x - 5} \right)^3}\left( {2 - 4x} \right) < 0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{\left( {x - 5} \right)\left( {1 - 2x} \right) < 0}$$

$$\left( {x - 5} \right)\left( {1 - 2x} \right) = 0\,\,\,\,\, \Leftrightarrow \,\,\,\,x = 5\,\,{\text{or}}\,\,x = \frac{1}{2}$$

$$?\,\,:\,\,\,\left\{ {x < \frac{1}{2}} \right\}\,\,\, \cup \,\,\,\left\{ {x > 5} \right\}$$

The solution above follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.
_________________

Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!

Intern
Joined: 10 Feb 2017
Posts: 36
Location: Viet Nam
GPA: 3.5
WE: General Management (Education)
Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

06 Sep 2018, 21:09
I thought the answer must be written in form like x < 1/2 or x > 5 as there is no way x can satisfy both the condition. The "And" here made me confused.
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 186
Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

06 Sep 2018, 21:34
Hungluu92vn wrote:
I thought the answer must be written in form like x < 1/2 or x > 5 as there is no way x can satisfy both the condition. The "And" here made me confused.

Hi, Hungluu92vn!

Excellent observation. I have avoided to use the term "and", because EACH x cannot be simultaneously in ]-infinity, 1/2[ and in ]5, +infinity[.
That´s why, for the solution set of a inequation such as the one presented in the question stem, the proper word is OR , the proper symbol is U (reunion).

Other experts used the term AND, specifically in this question, because the word "range" was presented in the question stem!
Their interpretation: there are real values "x" that belong to ]-infinity, 1/2[ AND there are (OTHER) real values "x" that belong to ]5, +infinity[.
(It is as if "x" is a dummy variable: different "roles" each time it appears! This is not the classical way of looking into this "theme" but... you got the point!)

Regards,
fskilnik.
_________________

Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!

Intern
Joined: 07 Aug 2018
Posts: 37
Location: Croatia (Hrvatska)
GMAT 1: 560 Q39 V28
Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

09 Sep 2018, 05:21
Under 30sec approach

You directly see that x can be bigger than 5. Eliminate A,B and D.
Left with C or E. --> x can also be negative therefore eliminate C. Hence, E
Intern
Joined: 15 Aug 2018
Posts: 4
Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

### Show Tags

09 Sep 2018, 06:13
Hello Egmatquantexpert why are multiplying by -1? What is the need for it?

Posted from my mobile device
Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0 &nbs [#permalink] 09 Sep 2018, 06:13
Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.