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|3 + 3x| < –2x

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|3 + 3x| < –2x [#permalink] New post 08 Aug 2018, 16:31
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64% (01:07) correct 35% (01:20) wrong based on 92 sessions
\(|3 + 3x| < –2x\)


Quantity A
Quantity B
\(|x|\)
\(4\)


A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

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[Reveal] Spoiler: OA

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Re: |3 + 3x| < –2x [#permalink] New post 11 Aug 2018, 12:29
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We solve the inequality:

|3+3x| < –2x

We have two possible options:

3+3x < -2x

and

3+3x > 2x

In both cases, we move x terms to the left side and constant terms over to the right side:

5x < -3

and

x > -3

So x is constrained between -5/3 and -3. In both cases, |x| < 4. So the answer is B.
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Re: |3 + 3x| < –2x [#permalink] New post 13 Nov 2019, 18:18
Hi,

I'm having trouble with this one. I selected D as the answer. My reasoning:

When we solve for an absolute value of a variable, we are accounting for both scenarios of the bars reversing x and keeping x as it is. So, when we're given any inequality or equation with absolute value and variables, we have to solve for both scenarios. That makes perfect sense.

But it also follows from this reasoning that both scenarios can't be true--it's only one or the other. For this problem in particular, it could be that x > -3 and not less than -3/5 or it could be that x is less than -3/5 but not greater than -3.

Say that it were only true that x > -3. X could well be 4 or more, making D the correct answer.

Why is this incorrect? Could it be that this is a faulty question by Manhattan? Since this is highly unlikely I'm guessing I'm just missing a detail, so I would love it if someone could point it out for me!

Thank you.
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Re: |3 + 3x| < –2x [#permalink] New post 14 Nov 2019, 07:52
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So, you have good reasoning up to the point where you solve for both scenarios:
X < -3/5
AND
X > -3
Now here's where you are getting it a bit twisted: BOTH CONDITIONS MUST BE MET, NOT ONLY ONE.
So, values of X go from -3 to -3/5 (not including). Or in mathematical terms: X = (-3,-3/5)

Please see the image attached.
I hope this helps!

P.S: I have no idea why I can't quote the comment of einalemjs...
Attachments

Possible values of X.jpg
Possible values of X.jpg [ 9.87 KiB | Viewed 958 times ]

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Re: |3 + 3x| < –2x [#permalink] New post 14 Nov 2019, 07:53
[

Please see my response above...
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Re: |3 + 3x| < –2x [#permalink] New post 14 Nov 2019, 09:20
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Why is it that both scenarios must be true? I thought when you take an absolute value, you are either reversing the negative or leaving it as it was... but it can't be both negative and positive at once. Meaning, while it's useful to know both possible scenarios, it's not actually possible that they both occur simultaneously, since it can only be one OR the other. So what is the reasoning behind that?

It's not like finding the roots of a quadratic or accounting for positive/negative roots, where both are correct. It's a way of accounting for uncertainty in absolute value, but in this case it is either one case or the other, because the starting value could not have been both cases at once.

Given all of this, answer choice D makes the most sense, since it's possible that the |x| is less than, equal to, or greater than 4.

I really don't see how this is faulty reasoning. I hope someone can explain!

EDiT: I think I understand now, from doing more questions. It seems there's nothing wrong with my reasoning--the variables can't be both cases at once. But we're not precisely talking about the sign of the variable, but rather the output of the absolute value sign. Further, since we're dealing with inequalities, the values can satisfy both conditions. In the context of absolute value, since the resulting quantity will always be positive for the same inputs, the variable must satisfy the inequalities of both positive and negative values.

Thanks! Resolved.
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Re: |3 + 3x| < –2x   [#permalink] 14 Nov 2019, 09:20
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