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

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New post Updated on: 23 Aug 2018, 21:25
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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, 04:26.
Last edited by EgmatQuantExpert on 23 Aug 2018, 21:25, 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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New post Updated on: 27 Aug 2018, 02:07

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

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

Hence, the correct answer is option D.

Answer: D

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Originally posted by EgmatQuantExpert on 23 Aug 2018, 04:31.
Last edited by EgmatQuantExpert on 27 Aug 2018, 02:07, edited 3 times in total.
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Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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New post 23 Aug 2018, 06:08
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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Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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New post 05 Sep 2018, 05:38
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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New post 05 Sep 2018, 11:21
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.

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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. :-)
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PKN

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

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New post 07 Sep 2018, 01:39
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?

If the inequality were just (x+1) (x-4) -(x-5) <0 how will we go about this? Thanks.

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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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New post 07 Sep 2018, 02:17
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?

If the inequality were just (x+1) (x-4) -(x-5) <0 how will we go about this? Thanks.

Posted from my mobile device

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

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

flip of sign in inequalities:-
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.
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Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5-  [#permalink]

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New post 07 Sep 2018, 05:07
Totally clear now. Thanks PKN

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

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New post 07 Sep 2018, 05:15
Kem12 wrote:
Totally clear now. Thanks PKN

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Need not to mention. :thumbup: (blue color) depicts everything. :-)

You are always welcome.
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Re: Find the range of values of x such that [m](x+1) (12-3x)^5 (x+3) (5- &nbs [#permalink] 07 Sep 2018, 05:15
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