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

Question

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 (\(x > 5\))

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Official Answer

E

chetan2u · Accepted Answer

(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

EgmatQuantExpert · Answer

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 https://e-gmat.com/blogs/wp-content/uploads/2018/08/PQ1-e1535353974796.png Therefore, the range of x is x < \frac{1}{2} and x > 5 Hence, the correct answer is option E. Answer: E https://e-gmat.com/blogs/wp-content/uploads/2018/08/Quant-workshop_Webinar-e1533628475738.jpg

PKN · Answer

(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.

Kem12 · Answer

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.

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fskilnik · Answer

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

\left( {x - 5} ight)\left( {1 - 2x} ight) = 0\,\,\,\,\, \Leftrightarrow \,\,\,\,x = 5\,\,{	ext{or}}\,\,x = \frac{1}{2}

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

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

Regards,
fskilnik.

Hungluu92vn · Answer

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.

fskilnik · Answer

Hi, Hungluu92vn!

Excellent observation. I have avoided to use the term "and", because EACH x cannot be simultaneously in ]-infinity, 1/2 5, +infinity -infinity, 1/2 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.

T1101 · Answer

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

Abhinav93 · Answer

Hello Egmatquantexpert why are multiplying by -1? What is the need for it?

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bansalgaurav · Answer

Note: While solving inequalities, remember the following simple rules :
-->If the product of two numbers is less than zero, then one of them is less than zero and the other one is greater than zero. ==> if ab <0 ==> a<0 & b>0 OR a>0 & b<0
-->If the product of two numbers is greater than zero, then either both the numbers are lesser than zero or both the numbers are greater than zero ==> if ab > 0 ==> a>0 & b>0 OR a<0 & b<0
--> Multiplying both sides of an inequality reverses the sign of inequality. Example If a<0 ==> -a>0

Now the given equation can be simplified as : 2[(x-5)^3][(1-2x)] < 0

Thus we can say either of the following has to hold true::
1: x-5 < 0 and 1-2x >0 2: x-5 > 0 and 1-2x <0
The above two conditions can be further simplified in terms of X as:
1: x < 5 and x < 1/2 { note if 1-2x > 0 ==>2x-1 <0 . multiplying by -1 on both sides reverses the inequality sign}
Both the conditions can hold true only if x<1/2
2: x> 1/2 and x>5
Both the conditions can hold true if x>5
Thus the answer is E.

bansalgaurav · Answer

Multiplying by -1 eases the equation. If 1-2x<0 ==> -2x < -1 ==> -x < -1/2 Now since we have to find answer in terms of X not in terms of -X hence we multiply by -1 here which yields :: (-x)(-1) > (-1/2)(-1) ==> x>1/2 Note:: Multiplication of any negative number on both sides of an inequality reverses the sign of inequality. Note, whenever you see "-x" in inequality and if the answer is in therms of "x", first convert that "x" into positive by multiplying the inequality by -1. In this question my first step would be 1-2x<0 ==> 2x-1>0 ==>x>1/2 . this helps me solve the ineq. in a smoother manner

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

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