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

Question

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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Official Answer

D

EgmatQuantExpert · Answer

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

https://e-gmat.com/blogs/wp-content/uploads/2018/08/PQ2-e1535358164366.png

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

Hence, the correct answer is option D.

Answer: D

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Quant-workshop_Webinar-e1533628475738.jpg

PKN · Answer

(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)

Kem12 · Answer

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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PKN · Answer

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.

Kem12 · Answer

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

PKN · Answer

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.

Kem12 · Answer

Totally clear now. Thanks  PKN

Posted from my mobile device

PKN · Answer

Need not to mention.

(blue color) depicts everything.

You are always welcome.

MHIKER · Answer

Why wasn't the sign of equality flipped after mulplying both side by -1?

Doer01 · Answer

Hi  Baten80  ,

It was. It was flipped twice. So it became what it was initially.

armaankumar · Answer

(x+1) (12-3x)^5 (x+3) (5-x) < 0. => (x+1) 3^5(4-x)^5 (x+3) (5-x) < 0. => (x+1)(4-x)^5 (x+3) (5-x) < 0. => (x+1)(4-x)^4 (4-x)(x+3) (5-x) < 0. => (x+1)(4-x)(x+3) (5-x) < 0. We can test each option, A. x > 5 Doesn't Work B. 4 < x < 5 Works C. -3 < x < -2 Works D. (-3< x < -1) and (4 < x < 5) Works E. (x < -3) and (x > 5) Doesn't Work Since, both B. and C. work, We check for D as it also includes the inequality of B. D satisfies the equation

bv8562 · Answer

I got the part that when we multiply the given inequality twice by -1 the inequality would not change but I did not understand how 5-x became x-5 after multiplying the given inequality twice by -1 because this what I get when I do the same: 5-x < 0 multiplying it by -1 -1*(5-x) < -1*0 x-5 > 0 again multiplying it by -1 -1*(x-5) > -1*0 5-x < 0 I arrive at the same expression What am I doing wrong please help me understand? EgmatQuantExpert

Dhwanii · Answer

EgmatQuantExpert , gone through your wavy line method, really helpful but just wanted to understand what if on the other side of expression we don't have a 0 then we will still use wavy line method but by bringing that articular number (with sign changed) on other side of inequality sign and put a 0 and then solve right ? Basically for us to use wavy line method we need to have 0 on one side of inequality sign right?

bumpbot · Answer

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

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

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