If there is a least integer that satisfies the inequality 9/x ≥ 2 what

Question

If there is a least integer that satisfies the inequality 9/x ≥ 2, what is that least integer?

A. 0
B. 1
C. 4
D. 5
E. There is not a least integer that satisfies the inequality.

PS41471.01

Debo1988 · Accepted Answer

porwal1

9/x >= 2, this inequality tells us that x can't be negative or 0, because in either case inequality is violated (Example : 9/(-ve number) is negative, 9/0 is undefined)
now solving the inequality gives us :
x<= 9/2 
x<= 4.5
So the possible integer values are {1,2,3,4}, of which the minimum is 1 (answer)

Bunuel · Answer

This is an official questions so it must and is correct.

If 9/x ≥ 2, then x cannot be negative, because if x is negative, then 9/x = 9/negative = negative and cannot be more than a positive number.

dcummins · Answer

The question asks for the least value that satisfies the inequality 9/x > 2

The quickest method is to test the answers

A. 9/0 = 0 but 0 is <2 not greater than 2...Incorrect
B. 9/1 = 9 9>2 this works. 
C 9/4 =2.25 although 2.25 is greater than 2, 4 is not the least value for x that would make the inequality work

For this reason B is correct.

RK007 · Answer

IMHO, regarding option A, we can just say that the denominator cannot be 0 at all! So there's actually no need to do any kind of calculation.

sayan640 · Answer

Start with the least number...

option A :- 9 / 0 = undefined So option A is discarded.

Option B :- Put x = 1 and check . 
9 / 1 >= 2 ...So option B satisfies the inequality. 
B is the answer.

Please give me KUDOs if you liked my explanation.

porwal1 · Answer

Guys is there a way to solve this using algebra and not by plugging in?

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Camach700 · Answer

Bunuel  integer x is not specified to be either positive or negative.

Hence if x was negative the expression would become  x>=9/2  --> least integer 4.

However, if we consider x being positive, the expression gives x=1 as least integer.

I think the question stem shoud specify that x is positive.

EMPOWERgmatRichC · Answer

Hi All,

We're given the inequality 9/X ≥ 2. We're asked for the LEAST integer that satisfies the inequality. While this question is relatively straight-forward, it's written in such a way that you might lose track of what you're ultimately solving for. To reiterate, we want the SMALLEST INTEGER that "fits" the inequality.

We need 9/X to be greater than or equal to 2, so we need X to be POSITIVE. We are NOT asked to make 9/X as small as possible.

Thus, the smallest positive integer that 'fits' the inequality is 1.

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

BrentGMATPrepNow · Answer

If 9/x is greater than 2, we know that x is POSITIVE

Since x is POSITIVE, we can eliminate answer choice A.
At this point, we can test values, starting with the smallest possible value.

B) 1
If x = 1, our inequality becomes 9/1 ≥ 2, which is TRUE
Done!

Answer: B

Cheers,
Brent

ScottTargetTestPrep · Answer

We see that x can’t be 0 or negative (otherwise, 9/x will be undefined or negative, respectively). So x must be positive, and if x = 1, we have 9/x = 9/1 = 9 ≥ 2.

Answer: B

mockingjay · Answer

if the question asked for greatest integer then would 4 be the answer then ?

EMPOWERgmatRichC · Answer

Hi mockingjay,

YES - if the question was changed and we were asked for the largest integer-value of X, then the answer would be 4.

GMAT assassins aren't born, they're made,
Rich

Stonely · Answer

Unbelievable, do you really think 9/0 undefined? It’s infinite and infinity is definitely greater than 2. If this is an official question, I question the test are we to understand that the test limits it’s understanding to algebra and so those of us who have a calculus background are at a disadvantage? Too bad. Bad question

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kornn · Answer

Dear IanStewart , Technically, if X were 0 (9/X would be infinity), then the equality is also satisfied right? A bit confused on the concept. Please help sir

IanStewart · Answer

9/0 is not equal to "infinity". 9/0 is "undefined"; it is not a number at all.

To see why it does not make sense to think of 9/0 as being "infinity", you can borrow from calculus the concept of a 'limit' (which of course, is not something explicitly tested on the GMAT, though certain estimation questions allude to limits in a simple enough way that you don't need to know about them to get an answer). If you think of the expression 9/x, and start with x = 1 and then imagine making x smaller and smaller so it gets closer and closer to zero, the value of 9/x gets bigger and bigger. So you might say, because of that, that "9/0 is infinity". But then imagine starting with x = -1, and making x bigger and bigger so it gets closer to zero. Then 9/x gets more and more negative -- it gets smaller and smaller. So you might then think that "9/0 is negative infinity". Those two interpretations are clearly contradictory, which is why it doesn't make sense to think that 9/0 has any single "value" at all, and is one reason that 9/0 is simply undefined (the other reason being that infinity is not a number in the first place).

hk111 · Answer

This is how I approached the question

I simplified the inequality first,

9/x >= 2

multiply by 1/9

1/x >= 2/9

flip both sides, inequality also flips

x <= 9/2
x<= 4.5

Hence I selected E as the minimum value can be minus infinity. Please someone tell me why this approach is incorrect

EMPOWERgmatRichC · Answer

Hi hk111,

We're given the inequality 9/X ≥ 2. We're asked for the LEAST integer that satisfies the inequality. This means that we want the SMALLEST INTEGER that "fits" the inequality.

We need 9/X to be greater than or equal to 2, so we need X to be POSITIVE. Re-writing the inequality in the way that you did was fine, BUT you lost track of what the question was asking for. We were NOT asked to make 9/X as small as possible. We were essentially asked for the smallest positive integer that 'fits' the inequality - and that is 1.

GMAT assassins aren't born, they're made, 
Rich

Devoshish · Answer

the question is saying find the least integer which is equal to or greater than two , while there are integers that will satisfy the greater than term but none of the integer will satisfy that equal to term. can someone please clear my doubt?

EMPOWERgmatRichC · Answer

Hi Devoshish, The prompt actually asks us to find the least INTEGER value (for X) that satisfies the given inequality 9/X ≥ 2. While this question is relatively straight-forward, it's written in such a way that you might lose track of what you're ultimately solving for. We need 9/X to be greater than or equal to 2, so we need X to be POSITIVE. We are NOT asked to make 9/X as small as possible; we are asked for the smallest integer value that X could be and still satisfy the inequality. Thus, the smallest positive integer that 'fits' the inequality is 1. GMAT assassins aren't born, they're made, Rich Contact Rich at: Rich.C@empowergmat.com

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If there is a least integer that satisfies the inequality 9/x ≥ 2 what

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