If x

Question

If x < 0 and y is not equal to 0, which of the following must be true?

I.  x^7 < 0 
II.  x^3y^4 < 0 
III. |x| = -x

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

My confusion is with part 3. Of the question. If someone can explain, that'd be fantastic.
Thank You

Bunuel · Accepted Answer

This is a property of an absolute value: When x\leq{0} then |x|=-x , or more generally when some \ expression\leq{0} then |some \ expression|={-(some \ expression)} . For example: |-5|=5=-(-5) ; When x\geq{0} then |x|=x , or more generally when some \ expression\geq{0} then |some \ expression|={some \ expression} . For example: |5|=5 . You should brush-up fundamentals on modulus: Theory on Absolute Values: math-absolute-value-modulus-86462.html The E-GMAT Question Series on ABSOLUTE VALUE: the-e-gmat-question-series-on-absolute-value-198503.html Properties of Absolute Values on the GMAT: properties-of-absolute-values-on-the-gmat-191317.html Absolute Value: Tips and hints: absolute-value-tips-and-hints-175002.html DS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=37 PS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=58 Hard set on Absolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

TheUltimateWinner · Answer

Hi,
1. odd power of negative value always give negative result. So, 1 is ok.
2. even power of negative value or even power of positive value always give positive result. 
So, negative*positive=negative value. So, 2 is ok
3. absolute value of any negative value always give positive result.
So. positive result=(-)*negative value. if the value of x is same in this regard. So, 3 is ok.
So, the correct choice is E

ozzaayy · Answer

I only picked E because statement 1 & 2 will hold true if x < 0, but 3 is not so sure? Like modulus can open to give both +ve and -ve values, so it's not important that the statement 3 will hold true. If  Bunuel , statement 1 & 2 existed as one of the options, would we still pick 1,2, & 3? Or we'll consider that option in that case

BrushMyQuant · Answer

Given that x < 0 and y ≠ 0 1. x^7 < 0 As x < 0 so any odd power of x will also be < 0 => x^7 < 0 is TRUE 2. x^3y^4 < 0 As x < 0 so any odd power of x will also be < 0 => x^3 < 0 y ≠ 0 and we have an even power of y => y^4 > 0 => x^3y^4 = Negative * Positive = Negative => TRUE 3. |x| = -x x < 0 Let's take x = -k where k > 0 => |x| = |-k| = k = -x => TRUE So, Answer will be E Hope it helps! Watch the following video to MASTER Absolute Value Problems https://www.youtube.com/watch?v=rBWfl4AmQDI

BrushMyQuant · Answer

We will pick option E in any case as for any number x which is negative |x| = -x (this is a property of modulus)
Ex: |-5| = 5 = -(-5)

Modulus is also the distance of the number on the number line from the origin, so |-5| is the distance of the number - 5 from origin which is 5 (i.e. -(-5), i.e -x.
Hope it helps!

Kameekazee · Answer

I still don't get it. How can an absolute value of x be equal to negative x?

egmat · Answer

Hi Kameekazee,

I think the whole knot here is one small thing: you're reading  "-x"  as if it must be a  negative  number. It isn't. The minus sign flips whatever x is - so when x is already negative,  -x turns positive .

That's the piece Bunuel's example (|-5| = 5 = -(-5)) was pointing at, so let's slow it down.

Plug in a real negative number

Say  x = -5  (which fits the rule x < 0).

- The  left side , |x|, is |-5| =  5 . Absolute value just measures distance from 0, so it's always non-negative.
- The  right side , -x, is -(-5). Two minus signs cancel, so this is also  5 .

Both sides equal  5 . So |x| = -x holds. The -x did  not  give a negative - because you were negating a number that was  already  negative.

The trap is thinking "-x" reads as "negative." Really, "-x" means "the opposite sign of x." Since x is negative here, its opposite is positive - and that positive is exactly what |x| gives you.

Quick check to lock it in

Let  x = -12 :
- |x| = |-12| =  12 
- -x = -(-12) =  12

Same value every time. Whenever x is negative, -x is the positive twin of |x|, so statement 3 must be true - which is why  E  is correct.

Rule to keep: -x is negative  only when x is positive . Flip that around and the confusion disappears.

Answer: E

Bunuel · Answer

Because x itself is negative -x becomes positive.

For example, if x = -5, then:

|x| = |-5| = 5

and

-x = -(-5) = 5

So |x| = -x.

The minus sign in -x does not mean the result is negative. It means “take the opposite of x.” Since x is already negative, its opposite is positive.

If x < 0 and y is not equal to 0, which of the following must be true?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If x < 0 and y is not equal to 0, which of the following must be true?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards