Which of the following must be equal to zero for all real numbers x ?

Question

I.  -\frac{1}{x}

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and III only
(D) II and III only
(E) I, II , and III

Bunuel · Answer

SOLUTION

Which of the following must be equal to zero for all real numbers x ?

I.  -\frac{1}{x}

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and II I only
(D) II and III only
(E) I, II , and III

I.  -\frac{1}{x}
eq{0}  for any real number.

II. x + (-x) = x - x = 0.

III. x^0=1, for all nonzero numbers.

Answer: B.

Cosmas · Answer

I go for B.

i. -1/x: if divided by any real number, the answer will never be zero. If 'x' =0; the answer is 'undefined'
ii. x+(-x): any real number will yield a result of zero when substituted for x in the expression
iii. x^0: this yields 1 as an answer if x is any real number.

nutshell · Answer

I. -1/x will not be 0 for any real number; The value will be undefined if x is 0;
II. The value will be 0 for any real number. (ex.,) 2 + (-2) = 2-2 = 0;
III. x^0 = 1 for real numbers;

II only.

Ans is (B)

psal · Answer

Once we eliminate i, we realize from the answer choices that we can ignore ii and look at iii. Once iii is eliminated, we know ii is the answer.

harishbiyani · Answer

Lets look at III first - Anything raise to 0 is 1, so x^0 = 1.

Now eliminate any answer choice that has III in options. Options C, D, E are eliminated.

I -1/x will never be equal to zero for any real number, value will be undefined if x = o

Hence option A is eliminated. We are left only with B, so B is the right answer.

II x + (-x) will be zero for any real number = x. for example x= 2.6. 2.6 - 2.6 = 0

thefibonacci · Answer

I would go with B

-1/x can be -1/2,-1/4,... 
(x)+(-x) will be 0 for all real numbers
x^0 will either yield 1 or be indeterminate for 0^0

stne · Answer

Answer is B

1/0 is undefined so option A is never 0
also 0^0 is undefined or 1 so option C is also never 0

so we are left with option B only 
Answer B

Bunuel · Answer

SOLUTION

Which of the following must be equal to zero for all real numbers x ?

I.  -\frac{1}{x}

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and II I only
(D) II and III only
(E) I, II , and III

I.  -\frac{1}{x}
eq{0}  for any real number.

II. x + (-x) = x - x = 0.

III. x^0=1, for all nonzero numbers.

Answer: B.

Blackbox · Answer

Isn't zero a real number? And if it is,  0^0  is 0 isn't it? Or is it undefined in GMAT?

Bunuel · Answer

0 is a real number.

0^0, in some sources equals to 1 ( not 0 ), some mathematicians say it's undefined.  But you won't need this for the GMAT because the case of 0^0 is not tested on the GMAT.

Blackbox · Answer

Thanks! But it is interesting you say that because if 0 is a real number then I am forced to use 0^0 for statement (3), in which case, this question isn't a GMAT type question. Oh well, I guess I will just chalk this one up to "fringe maths".

Bunuel · Answer

In any way 0^0 does NOT equal 0.

kanusha · Answer

1,  \frac{-1}{x}  # 0
2, x + (-x) = x - x = always 0 for any number
3, x^0 = 1 but not 0 
B

totaltestprepNick · Answer

One way to do this question is to choose numbers for x and then eliminate answer choices that don't fit. Let's say x = 5 and see what happens.

I) -1/x --> -1/5, which is not zero, therefore -1/x is not equal to zero for all real numbers x and we can eliminate answer choices A, C, and E because they contain Roman numeral I.

II) x + (-x) --> 5 + (-5) = 0, so we can keep this in for now.

III) x^0 --> 5^0 = 1, which is not zero, therefore x^0 is not equal to zero for all real numbers x and we can eliminate answer choice D because it contains Roman numeral III.

So, B is correct because it is the only choice that remains.

Which of the following must be equal to zero for all real numbers x ?

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

Which of the following must be equal to zero for all real numbers x ?

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