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Is x < 0 ? (1) -4x < 0 (2) -4x^2 < 0

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Intern
Joined: 26 Jun 2012
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Is x < 0 ? (1) -4x < 0 (2) -4x^2 < 0 [#permalink]  03 Jul 2012, 22:30
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83% (01:27) correct 17% (00:59) wrong based on 36 sessions
Is x < 0 ?

(1) -4x < 0
(2) -4x^2 < 0

What I want to know is

1) -4x < 0 ; either 4 is negative or x is negative. Hence cant say if x <0.
But explanation for this is given that
dividing both sides by -4, we get x > 0. Hence we know that x > 0.
Can anyone please help me understand why -4x is not 4 * (-x)? and in case it is 4*(-x) then why cant x<0 ?

2) Is clear to me. Hence no questions
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Kudos [?]: 257 [1] , given: 23

Re: Help clear concept [#permalink]  03 Jul 2012, 22:59
1
KUDOS
jayoptimist wrote:
Q) Is x < 0 ?
1) -4x < 0
2) -4x^2 < 0

What I want to know is

1) -4x < 0 ; either 4 is negative or x is negative. Hence cant say if x <0.
But explanation for this is given that
dividing both sides by -4, we get x > 0. Hence we know that x > 0.
Can anyone please help me understand why -4x is not 4 * (-x)? and in case it is 4*(-x) then why cant x<0 ?

2) Is clear to me. Hence no questions

Hi,

-4x<0,
if x < 0, let say, x=-1, then (-4)(-1)>0
which contrary to the given inequality.

again, if x>0, let say, x=2, then (-4)(2)<0
Thus, we can say x>0

Let me know if you need any further clarification,

Regards,
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My posts: Solving Inequalities, Solving Simultaneous equations, Divisibility Rules

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Math Expert
Joined: 02 Sep 2009
Posts: 29855
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Kudos [?]: 53963 [1] , given: 8260

Re: Is x < 0 ? (1) -4x < 0 (2) -4x^2 < 0 [#permalink]  04 Jul 2012, 00:32
1
KUDOS
Expert's post
jayoptimist wrote:
Is x < 0 ?

(1) -4x < 0
(2) -4x^2 < 0

What I want to know is

1) -4x < 0 ; either 4 is negative or x is negative. Hence cant say if x <0.
But explanation for this is given that
dividing both sides by -4, we get x > 0. Hence we know that x > 0.
Can anyone please help me understand why -4x is not 4 * (-x)? and in case it is 4*(-x) then why cant x<0 ?

2) Is clear to me. Hence no questions

Is x < 0 ?

(1) -4x < 0. Divide both parts by -4 and flip the sign of the inequality since we are dividing by negative number: $$x>0$$. Sufficient. (Or just rewrite as $$4x>0$$, which also leads to $$x>0$$)

(2) -4x^2 < 0 --> $$x^2>0$$. This inequality holds true for positive as well as negative values of $$x$$. Not sufficient.

As for your question: we CAN write $$-4x < 0$$ as $$4*(-x) < 0$$. Now, from $$4*(-x) < 0$$ we have that the product of the positive number 4 and $$-x$$ is negative, so $$-x$$ must be negative: $$-x<0$$ --> $$x>0$$.

Hope it's clear.

_________________
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Joined: 26 Jun 2012
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Re: Is x < 0 ? (1) -4x < 0 (2) -4x^2 < 0 [#permalink]  04 Jul 2012, 00:55
Thanks Bunuel. I got it!
Re: Is x < 0 ? (1) -4x < 0 (2) -4x^2 < 0   [#permalink] 04 Jul 2012, 00:55
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