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

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

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Updated on: 23 Aug 2018, 21:25
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45% (medium)

Question Stats:

63% (01:34) correct 37% (01:23) wrong based on 236 sessions

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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 ($$x > 5$$)

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To read the article: Solving inequalities- Number Line Method

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

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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.

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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.
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Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

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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
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Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

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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.
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Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

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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.

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Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

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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.
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Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

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

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

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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
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Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0  [#permalink]

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

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Re: Find the range of values of x such that (x-5)^3 (2-4x) < 0   [#permalink] 09 Sep 2018, 06:13
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