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# Which of the following describes all the values of x for whi

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Senior Manager
Joined: 21 Oct 2013
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Which of the following describes all the values of x for whi  [#permalink]

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30 Jul 2014, 02:53
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35% (medium)

Question Stats:

69% (01:36) correct 31% (01:54) wrong based on 279 sessions

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Which of the following describes all the values of x for which x(x−1)(x−2)≥0?

A. 0 ≤ x ≤ 1 or x ≥ 2
B. x ≤ 0 or x ≥ 2
C. 0 ≤ x ≤ 1
D. 1 ≤ x ≤ 2 or x ≥ 2
E. 0 ≤ x ≤ 2
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Joined: 20 Dec 2013
Posts: 73
Which of the following describes all the values of x for whi  [#permalink]

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30 Jul 2014, 03:18
2
goodyear2013 wrote:
Which of the following describes all the values of x for which x(x−1)(x−2)≥0?

A. 0 ≤ x ≤ 1 or x ≥ 2
B. x ≤ 0 or x ≥ 2
C. 0 ≤ x ≤ 1
D. 1 ≤ x ≤ 2 or x ≥ 2
E. 0 ≤ x ≤ 2

In terms of graphs, Cubic equation has 3 roots : x=0, 1 , 2.
for, X >=2 all three terms positive => graph will hit x axis at x=2 from downward to upward and keep on going up as x increases from 2.
=> x<2 graph is in negative zone : till when ? => till next root point i.e. x=1 ..
And so on so forth ..
Voila It's A. Graphs made it really easy !
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Joined: 02 Sep 2009
Posts: 58443
Re: Which of the following describes all the values of x for whi  [#permalink]

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30 Jul 2014, 03:40
1
3
goodyear2013 wrote:
Which of the following describes all the values of x for which x(x−1)(x−2)≥0?

A. 0 ≤ x ≤ 1 or x ≥ 2
B. x ≤ 0 or x ≥ 2
C. 0 ≤ x ≤ 1
D. 1 ≤ x ≤ 2 or x ≥ 2
E. 0 ≤ x ≤ 2

x(x−1)(x−2)≥0 --> the "roots" are 0, 1 and 2, this gives us 4 ranges:

x<0;
0≤x≤1;
1<x<2;
x≥2.

Next, test some extreme value for x: if x is some large enough number, say 10, then all three multiples will be positive which gives the positive result for the whole expression, so when x≥2, the expression is positive. Now the trick: as in the 4th range expression is positive, then in the 3rd it'll be negative, in the 2nd it'll ne positive and finally in the 1st it'll be negative: - + - + . So, the ranges when the expression is positive or 0 are: 0≤x≤1 and x≥2.

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Re: Which of the following describes all the values of x for whi  [#permalink]

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05 Aug 2014, 00:40
I'm not getting this - could anybody please elaborate a bit more.

Is there any way els around this than presented by Bunuel? Thx
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Joined: 30 Jul 2008
Posts: 19
Re: Which of the following describes all the values of x for whi  [#permalink]

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08 Jul 2015, 07:34
Bunuel wrote:
goodyear2013 wrote:
Which of the following describes all the values of x for which x(x−1)(x−2)≥0?

A. 0 ≤ x ≤ 1 or x ≥ 2
B. x ≤ 0 or x ≥ 2
C. 0 ≤ x ≤ 1
D. 1 ≤ x ≤ 2 or x ≥ 2
E. 0 ≤ x ≤ 2

x(x−1)(x−2)≥0 --> the "roots" are 0, 1 and 2, this gives us 4 ranges:

x<0;
0≤x≤1;
1<x<2;
x≥2.

Next, test some extreme value for x: if x is some large enough number, say 10, then all three multiples will be positive which gives the positive result for the whole expression, so when x≥2, the expression is positive. Now the trick: as in the 4th range expression is positive, then in the 3rd it'll be negative, in the 2nd it'll ne positive and finally in the 1st it'll be negative: - + - + . So, the ranges when the expression is positive or 0 are: 0≤x≤1 and x≥2.

Hi Bunuel,

I am able to understand 99% of it and got the answer right, but I have one question.

Why <= is used for some sets of roots only.

x<0;
0≤x≤1;
1<x<2;
x≥2.
Math Expert
Joined: 02 Sep 2009
Posts: 58443
Re: Which of the following describes all the values of x for whi  [#permalink]

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08 Jul 2015, 08:52
robinpallickal wrote:
Bunuel wrote:
goodyear2013 wrote:
Which of the following describes all the values of x for which x(x−1)(x−2)≥0?

A. 0 ≤ x ≤ 1 or x ≥ 2
B. x ≤ 0 or x ≥ 2
C. 0 ≤ x ≤ 1
D. 1 ≤ x ≤ 2 or x ≥ 2
E. 0 ≤ x ≤ 2

x(x−1)(x−2)≥0 --> the "roots" are 0, 1 and 2, this gives us 4 ranges:

x<0;
0≤x≤1;
1<x<2;
x≥2.

Next, test some extreme value for x: if x is some large enough number, say 10, then all three multiples will be positive which gives the positive result for the whole expression, so when x≥2, the expression is positive. Now the trick: as in the 4th range expression is positive, then in the 3rd it'll be negative, in the 2nd it'll ne positive and finally in the 1st it'll be negative: - + - + . So, the ranges when the expression is positive or 0 are: 0≤x≤1 and x≥2.

Hi Bunuel,

I am able to understand 99% of it and got the answer right, but I have one question.

Why <= is used for some sets of roots only.

x<0;
0≤x≤1;
1<x<2;
x≥2.

The question asks for the ranges of x for which x(x−1)(x−2)0. So, the transition points, "roots" (0, 1, and 2) must be included in ranges we consider (x<0; 0≤x≤1; 1<x<2; x≥2.). Actually it does not matter to which number you apply = sign as long as you apply it to all of them (0, 1, and 2).
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Joined: 30 Jul 2008
Posts: 19
Which of the following describes all the values of x for whi  [#permalink]

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09 Jul 2015, 03:17
Thanks a lot Bunuel.
Intern
Joined: 25 May 2017
Posts: 4
Re: Which of the following describes all the values of x for whi  [#permalink]

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31 Oct 2018, 12:24
Bunuel wrote:
goodyear2013 wrote:
Which of the following describes all the values of x for which x(x−1)(x−2)≥0?

A. 0 ≤ x ≤ 1 or x ≥ 2
B. x ≤ 0 or x ≥ 2
C. 0 ≤ x ≤ 1
D. 1 ≤ x ≤ 2 or x ≥ 2
E. 0 ≤ x ≤ 2

x(x−1)(x−2)≥0 --> the "roots" are 0, 1 and 2, this gives us 4 ranges:

x<0;
0≤x≤1;
1<x<2;
x≥2.

Next, test some extreme value for x: if x is some large enough number, say 10, then all three multiples will be positive which gives the positive result for the whole expression, so when x≥2, the expression is positive. Now the trick: as in the 4th range expression is positive, then in the 3rd it'll be negative, in the 2nd it'll ne positive and finally in the 1st it'll be negative: - + - + . So, the ranges when the expression is positive or 0 are: 0≤x≤1 and x≥2.

Hi Bunuel,

The graph inequalities strategy appears good for 2 solutions, can you kindly explain how you are assigning signs here and if the approach is always applicable.
Re: Which of the following describes all the values of x for whi   [#permalink] 31 Oct 2018, 12:24
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