Is x<0?

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Tue Jul 17, 2018 1:01 am

Is x<0?

1) x^5 < 0.
2) x^5 + x + 1 = 0.

The OA is the option D.

How can I get an answer from the second statement? Please, I need a clarification here.

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Source: Beat The GMAT — Data Sufficiency | Tue Jul 17, 2018 1:01 am

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Tue Jul 17, 2018 2:16 am

Hello Vjesus12.

Let's take a look at your question.

We have to say if x<0 or not.

First statement

1) x^5 < 0.

Since the power is an odd number, then it implies that x will also have the same sign, so x<0. Therefore, this statement is sufficient.

Second statement

2) x^5 + x + 1 = 0.

This equation will lead us to $$x^5+x+1=0$$ $$x^5+x=-1$$ $$x\left(x^4+1\right)=-1 $$ $$x\left(x^4+1\right)=negative\ number .$$ Now, x^4+1 is always positive, so it implies that x must be negative, that is to say, x<0.

Therefore, this statement is sufficient.

In conclusion, each statement alone is sufficient. The answer is the option D.

I hope it helps you.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Jul 17, 2018 6:50 am

VJesus12 wrote:Is x<0?

1) x^5 < 0.
2) x^5 + x + 1 = 0.

Target question: Is x<0?

Key Concept: ODD exponents preserve the sign of the base.
In other words, POSITIVE^ODD = POSITIVE and NEGATIVE^ODD = NEGATIVE

Statement 1: x^5 < 0
This is telling us that x^5 = NEGATIVE
So, it must be the case that x is NEGATIVE
So, the answer to the target question is YES, it is definitely the case that x < 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x^5 + x + 1 = 0
Subtract 1 from both sides to get: x^5 + x = -1
Applying the above concept, we know that x^5 and x share the same sign.
That is, EITHER x^5 and x are both positive, OR x^5 and x are both negative
However, in order for the equation x^5 + x = -1 to hold true, it must be the case that x^5 and x are both negative
So, the answer to the target question is YES, it is definitely the case that x < 0
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Mon Aug 27, 2018 12:24 pm

VJesus12 wrote:Is x<0?

1) x^5 < 0.
2) x^5 + x + 1 = 0.

How can I get an answer from the second statement? Please, I need a clarification here.

Hi VJesus12,

x is a real number, therefore the trichotomy law applies to it: x is positive, x is zero or x is negative, and only one of these 3 possibilities occurs.

Now letÂ´s consider statement (2) :

x is not zero (because 0^5 + 0 +1 would be equal to 1, not 0)
x is not positive (otherwise x^5 + x +1 would be greater than 1, not 0) , hence...
x is negative!

The above follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Tue Sep 04, 2018 3:16 am

VJesus12 wrote:Is x<0?

1) x^5 < 0.
2) x^5 + x + 1 = 0.

Statement One Alone:

x^5 < 0

When a negative number is raised to an odd integer exponent, the result is negative. Thus, the only way for x^5 to be less than 0 is if x is less than 0. Statement one is sufficient.

Statement Two Alone:

x^5 + x + 1 = 0

Let's re-express the given equation as x^5 + x = x(x^4 + 1) = -1. Since x^4 + 1 is nonnegative, that the only way for this to be true is if x is a negative number. Statement two is also sufficient.

Answer: D

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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