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If x < 0, which of the following must be true?

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Joined: 02 Sep 2009
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If x < 0, which of the following must be true? [#permalink]

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03 Dec 2014, 07:19
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Tough and Tricky questions: Must or Could be True Questions.

If x < 0, which of the following must be true?

I. x^2 > 0
II. x − 2x > 0
III. x^3 + x^2 < 0

A) I only
B) I & II
C) II & III
D) All of the above
E) None of the above

Kudos for a correct solution.

Source: Chili Hot GMAT
[Reveal] Spoiler: OA

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Posts: 167
GMAT 1: 740 Q49 V41
GPA: 3.94
Re: If x < 0, which of the following must be true? [#permalink]

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03 Dec 2014, 09:28
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We're told that x is negative and that is all. Let's evaluate.

Since x is negative, x^2 is always positive (>0). True.
Since x is negative then x-2x>0. True as we're essentially adding a negative number to a larger (effectively positive number). For example, if x=-.5, then -0.5 + 1>0 or if x = -1, then -1+2>0.
However, the third doesn't have to be true. If x is a negative fraction, then x^2 is positive and larger than x^3. If x=-1/2, then x^3+x^2 = -1/8 + 1/4>0. However, if x=-1, then -1+1=0. Insufficient, not always true.

Choice B
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Re: If x < 0, which of the following must be true? [#permalink]

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03 Dec 2014, 19:35
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I. x^2 > 0 -TRUE- Irrespective of x sign, it will always be positive because of even power and since x is less than 0 x^2 can't be 0.
II. x − 2x > 0 -TRUE- Since x<0 , -x will be positive
III. x^3 + x^2 < 0 -Can not say-
x^2(x+1)<0 , since x^2 will always be positive => x+1 < 0 => x < -1. But we know that x<0, so x could be -1/2 etc.

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Re: If x < 0, which of the following must be true? [#permalink]

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03 Dec 2014, 19:44
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Expert's post
Bunuel wrote:

Tough and Tricky questions: Must or Could be True Questions.

If x < 0, which of the following must be true?

I. x^2 > 0
II. x − 2x > 0
III. x^3 + x^2 < 0

A) I only
B) I & II
C) II & III
D) All of the above
E) None of the above

Kudos for a correct solution.

Source: Chili Hot GMAT

Given: x is negative.
Which of the following must be true?

I. $$x^2 > 0$$
This is always true for real numbers except if x is 0. Since x is negative, this must be true.

II. $$x - 2x > 0$$
$$-x > 0$$
$$x < 0$$ (multiplied both sides by -1). This is given so it must be true.

III. $$x^3 + x^2 < 0$$
$$x^2*(x + 1) < 0$$
x^2 is positive so for x^2 * (x+1) to be negative, x+1 should be negative i.e. x < -1. We know that x < 0 but it could very well lie between 0 and -1 and hence this needn't be true. It may or may not be.

So only (I) and (II) must be true.

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Re: If x < 0, which of the following must be true? [#permalink]

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03 Dec 2014, 19:53
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Answer = B) I & II

I. x^2 > 0

Square of any number is always positive. Always true

II. x - 2x > 0

-x > 0
x < 0 (This is the given condition in the problem, which has to be obviously true)

III. x^3 + x^2 < 0

$$For x = \frac{-1}{2}$$

$$\frac{1}{4} - \frac{1}{8} > 0$$ (Not necessarily true)

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Re: If x < 0, which of the following must be true? [#permalink]

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03 Dec 2014, 23:04
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Given: x < 0,

I. x^2 > 0 --> squaring is never negative --> so always true
II. x − 2x > 0 --> -x>0, since x is given as -ve no. so -ve of -ve no. is always +ve --> so always true
III. x^3 + x^2 < 0 -->x^2(x+1)<0 and given is x<0 so it can be -1, in that case (-1)^2(-1+1)<0 = 0<0 --> so not always true

Ans. B) I & II
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Re: If x < 0, which of the following must be true? [#permalink]

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21 Feb 2017, 12:03
Bunuel wrote:

Tough and Tricky questions: Must or Could be True Questions.

If x < 0, which of the following must be true?

I. x^2 > 0
II. x − 2x > 0
III. x^3 + x^2 < 0

A) I only
B) I & II
C) II & III
D) All of the above
E) None of the above

Kudos for a correct solution.

Source: Chili Hot GMAT

We can invalid III by putting the fraction value of x. So, the correct choice is B.
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Re: If x < 0, which of the following must be true? [#permalink]

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21 Feb 2017, 21:54
Bunuel wrote:

Tough and Tricky questions: Must or Could be True Questions.

If x < 0, which of the following must be true?

I. x^2 > 0
II. x − 2x > 0
III. x^3 + x^2 < 0

A) I only
B) I & II
C) II & III
D) All of the above
E) None of the above

Kudos for a correct solution.

Source: Chili Hot GMAT

Given: x<0
I. x^2 > 0 Since x is Non-Zero so x^2 is bound to be Positive hence CORRECT

II. x − 2x > 0
i.e. -x > 0
i.e. x <0 hence CORRECT

III. x^3 + x^2 < 0
i.e. x^2(x+1) < 0
x^2 is always positive but (x+1) may or may not be negative for values of x less than or greater than -1 hence this is not always True

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Re: If x < 0, which of the following must be true?   [#permalink] 21 Feb 2017, 21:54
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