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# Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-

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Joined: 04 Jan 2015
Posts: 2593
Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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Updated on: 23 Aug 2018, 20:25
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Difficulty:

35% (medium)

Question Stats:

73% (01:49) correct 27% (01:59) wrong based on 140 sessions

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Solving inequalities- Number Line Method - Practice Question #2

Find the range of values of x such that $$(x+1) (12-3x)^5 (x+3) (5-x) < 0.$$
A. $$x > 5$$
B. $$4 < x < 5$$
C. $$-3 < x < -2$$
D. $$(-3< x < -1) and (4 < x < 5)$$
E. $$(x < -3) and (x > 5)$$

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Originally posted by EgmatQuantExpert on 23 Aug 2018, 03:26.
Last edited by EgmatQuantExpert on 23 Aug 2018, 20:25, edited 1 time in total.
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Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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Updated on: 27 Aug 2018, 01:07

Solution

Given:
• An inequality, $$(x+1) (12-3x)^5 (x+3) (5-x) < 0$$

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 $$(x+1) (12-3x) (12-3x)^4 (x+3) (5-x) < 0$$
• We know that, any number of the form $$N^2$$ is always positive, expect for N = 0.
o So, we can say that, $$(12-3x)^4$$ is always positive, expect for x = 4,
o Thus, the inequality can be written as (x+1) (12-3x) (x+3) (5-x) < 0
• Now, let’s multiply the inequality by -1, twice, to make the coefficients of x, in 12-3x and 5-x, positive
o And, note that the inequality sign does not change as we are multiplying it by -1, twice
• Thus, the inequality becomes, (x+1) (3x - 12) (x+3) (x - 5) < 0
• The zero points of this inequality are x = -1, x = 4, x = -3 and x = 5
• Plot these points on the number line.
o Since, 5 is the greatest among all, the inequality will be positive, for all the points to the right of 5, on the number line
o And, it is negative, in the region, between 4 and 5
o It is again positive in the region, between -1 and 4
o And, it is again negative in the region, between -3 and -1
o And, it is positive for all the values of x, less than -3

Therefore, the range of x is 5 < x < 4 and -3 < x < -1

Hence, the correct answer is option D.

_________________

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

Originally posted by EgmatQuantExpert on 23 Aug 2018, 03:31.
Last edited by EgmatQuantExpert on 27 Aug 2018, 01:07, edited 3 times in total.
Director
Status: Learning stage
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Posts: 958
WE: Supply Chain Management (Energy and Utilities)
Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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23 Aug 2018, 05:08
EgmatQuantExpert wrote:
Solving inequalities- Number Line Method - Practice Question #2

Find the range of values of x such that $$(x+1) (12-3x)^5 (x+3) (5-x) < 0.$$
A. $$x > 5$$
B. $$4 < x < 5$$
C. $$-3 < x < -2$$
D. $$(-3< x < -1) and (4 < x < 5)$$
E. $$(x < -3) and (x > 5)$$

$$(x+1) (12-3x)^5 (x+3) (5-x) < 0$$
Or, $$243\left(x-4\right)^5\left(x-5\right)\left(x+1\right)\left(x+3\right)<0$$

using wavy curve method:-
$$-3<x<-1\quad \mathrm{or}\quad \:4<x<5$$

Ans. (D)
_________________

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PKN

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Joined: 18 Apr 2018
Posts: 97
Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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05 Sep 2018, 04:38
PKN wrote:
EgmatQuantExpert wrote:
Solving inequalities- Number Line Method - Practice Question #2

Find the range of values of x such that $$(x+1) (12-3x)^5 (x+3) (5-x) < 0.$$
A. $$x > 5$$
B. $$4 < x < 5$$
C. $$-3 < x < -2$$
D. $$(-3< x < -1) and (4 < x < 5)$$
E. $$(x < -3) and (x > 5)$$

$$(x+1) (12-3x)^5 (x+3) (5-x) < 0$$
Or, $$243\left(x-4\right)^5\left(x-5\right)\left(x+1\right)\left(x+3\right)<0$$

using wavy curve method:-
$$-3<x<-1\quad \mathrm{or}\quad \:4<x<5$$

Ans. (D)

Hello PKN hope you are having a good day. I attempted this question using the wavy line method as well but didn't answer correctly. To rearrange the expressions in the (x-a)(x-b) form and i multiplied through by -1 and this changed the inequality sign to >0 but I didn't arrive at the right answer anyway. Could you please elaborate on how you rearranged 5-x to x-5 without changing the sign of the inequality. Thanks. I'm not a Math guru, just pushing hard for a 700 score.

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Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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05 Sep 2018, 10:21
1
Kem12 wrote:
PKN wrote:
EgmatQuantExpert wrote:
Solving inequalities- Number Line Method - Practice Question #2

Find the range of values of x such that $$(x+1) (12-3x)^5 (x+3) (5-x) < 0.$$
A. $$x > 5$$
B. $$4 < x < 5$$
C. $$-3 < x < -2$$
D. $$(-3< x < -1) and (4 < x < 5)$$
E. $$(x < -3) and (x > 5)$$

$$(x+1) (12-3x)^5 (x+3) (5-x) < 0$$
Or, $$243\left(x-4\right)^5\left(x-5\right)\left(x+1\right)\left(x+3\right)<0$$

using wavy curve method:-
$$-3<x<-1\quad \mathrm{or}\quad \:4<x<5$$

Ans. (D)

Hello PKN hope you are having a good day. I attempted this question using the wavy line method as well but didn't answer correctly. To rearrange the expressions in the (x-a)(x-b) form and i multiplied through by -1 and this changed the inequality sign to >0 but I didn't arrive at the right answer anyway. Could you please elaborate on how you rearranged 5-x to x-5 without changing the sign of the inequality. Thanks. I'm not a Math guru, just pushing hard for a 700 score.

Posted from my mobile device

Hi Kem12,
Greetings of the day!!
Given f(x)<0.
Notice that two of the factors of f(x) are not in the form of $$(x-a)^n$$ rather they are in the form of $$(a-x)^n$$.
Our requirement:- All factors must present in the form $$(x-a)^n$$, If not, then we have to convert to $$(a-x)^n$$.
So, odd forms are: $$(12-3x)^5$$ & (5-x)
1) $$(12-3x)^5$$ can be written as:$$(-(3x-12))^5$$ or, $$(-1)^5*(3x-12)^5$$ or, $$(-3)^5*(x-4)^5$$ Or, $$-243(x-4)^5$$
2) (5-x) can be written as : -(x-5)

Now f(x)= $$(x+1) (12-3x)^5 (x+3) (5-x)$$=$$(x-(-1))*{-243(x-4)^5}*(x-(-3))*{-(x-5)}=243(x-(-1))*(x-4)^5*(x-(-3))*(x-5)$$
Now simplified expression as per desired form:
$$243(x-(-1))*(x-4)^5*(x-(-3))*(x-5)<0$$ Or, $$(x-(-1))*(x-4)^5*(x-(-3))*(x-5)<0$$

Step-1
critical points: -1, 4,-3,5
Step-2
Arrange the critical points in ascending fashion:-3,-1,4,5
Step-3
Draw the curve.

Please revert in case of any difficulty in step-3.

All the best. I am no expert, I am at learning stage.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Manager
Joined: 18 Apr 2018
Posts: 97
Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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07 Sep 2018, 00:39
Hi PKN, thanks for the explanation. But I just want to confirm whether in the 3rd step, the negative in -(x-5) and -243(x-4)^5 canceled because of even number of negatives, right?

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Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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07 Sep 2018, 01:17
1
Kem12 wrote:
Hi PKN, thanks for the explanation. But I just want to confirm whether in the 3rd step, the negative in -(x-5) and -243(x-4)^5 canceled because of even number of negatives, right?

Posted from my mobile device

Hi Kem12,
You know , multiplying negative sign an even number of times results positive polarity.

1) (-)*(-)=(+)
2) (-)*(+)=(-)
3) (-)*(-)*(-)=(-)

1. Whenever you multiply or divide an inequality by a positive number, you must keep the inequality sign.
2. Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.

Refer note(2) above, '
(x+1) (x-4) {-(x-5)} <0
Or, (-1)(x+1) (x-4) (x-5)<0
Or, (-1)*(-1)(x+1) (x-4) (x-5)>0*(-1) (flip of sign occurs , multiplying both side of inequality by (-1))
Or, (x+1) (x-4) (x-5)>0 ((-1)*(-1)=+1 and 0*(-1)=0)

Hope it helps.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Manager
Joined: 18 Apr 2018
Posts: 97
Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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07 Sep 2018, 04:07
Totally clear now. Thanks PKN

Posted from my mobile device
Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 958
WE: Supply Chain Management (Energy and Utilities)
Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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07 Sep 2018, 04:15
Kem12 wrote:
Totally clear now. Thanks PKN

Posted from my mobile device

Need not to mention. (blue color) depicts everything.

You are always welcome.
_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-   [#permalink] 07 Sep 2018, 04:15
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