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Intern  Joined: 01 May 2012
Posts: 3
If [x] denotes the least integer greater than or equal to x  [#permalink]

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Question Stats: 63% (01:14) correct 37% (01:10) wrong based on 2658 sessions

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If [x] denotes the least integer greater than or equal to x, is [x] = 0?

(1) -1< x< 1
(2) x < 0
Math Expert V
Joined: 02 Sep 2009
Posts: 60553
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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27
54
If [x] denotes the least integer greater than or equal to x, is [x] = 0?

Some function [] rounds UP a number to the nearest integer. For example [1.5]=2, =2, [-1.5]=-1, ...

Question: is $$[x]=0$$? --> is $$-1<x\leq{0}$$?

(1) -1< x< 1. Not sufficient.
(2) x < 0. Not sufficient.

(1)+(2) -1<x<0. Sufficient.

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awkward-ds-105416.html
help-data-sufficiency-equation-and-inequalities-99856.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html

Hope it helps.
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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10
1
13
(1) Find out the question.

Solution:
(1) Question: Is [x] = 0?
(2) Possible values of x: [x] = 0 would mean x is greater than -1 but less than 0.

Statement 1: -1 < x < 1
x=0.5 would give [x] = 1
x=-0.5 would give [x] = 0
INSUFFICIENT.

Statement 2: x < 0
x = -0.5 would give [x] = 0
x = -1 would give [x]=-1
INSUFFICIENT

Together: -1 < x < 0 This is the range of x that we know will give [x] = 0. SUFFICIENT.

##### General Discussion
Intern  Joined: 13 Mar 2012
Posts: 12
GMAT 1: 700 Q50 V34
Re: If {x} denotes the least integer greater than or equal to x,  [#permalink]

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6
5
1) -1<x<1
if x=-0.5; {x}=0;
if x=0.5; {x}= 1
not sufficient
2)if x<0
itself not suficient, i.e cant be 0 or -1 or any negative int.

however 1) & 2) together are sufficient with -1<x<0 , {x}=0

therefore c
Intern  Joined: 24 May 2013
Posts: 5
GMAT Date: 06-27-2013
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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1
I got the right answer but the question is tricky for me: "if [x] denotes the least integer greater than or equal to x"... does it concern only numbers like [1.5]=2; [-1.5]=-1;... or [1.4] will also be rounded to 2?

Thank's
Math Expert V
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Posts: 60553
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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Intern  Joined: 01 Jan 2013
Posts: 48
Location: India
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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Bunuel wrote:
If [x] denotes the least integer greater than or equal to x, is [x] = 0?

Some function [] rounds UP a number to the nearest integer. For example [1.5]=2, =2, [-1.5]=-1, ...

Question: is $$[x]=0$$? --> is $$-1<x\leq{0}$$?

(1) -1< x< 1. Not sufficient.
(2) x < 0. Not sufficient.

(1)+(2) -1<x<0. Sufficient.

Similar questions to practice:
how-x-become-fraction-if-its-been-said-an-integer-94687.html
awkward-ds-105416.html
help-data-sufficiency-equation-and-inequalities-99856.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html

Hope it helps.

Hi Bunuel,
Can you post some 650+ - 700 level PS/DS questions on greatest integer function.

Thanks!!
Math Expert V
Joined: 02 Sep 2009
Posts: 60553
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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4
3
abid1986 wrote:
Bunuel wrote:
If [x] denotes the least integer greater than or equal to x, is [x] = 0?

Some function [] rounds UP a number to the nearest integer. For example [1.5]=2, =2, [-1.5]=-1, ...

Question: is $$[x]=0$$? --> is $$-1<x\leq{0}$$?

(1) -1< x< 1. Not sufficient.
(2) x < 0. Not sufficient.

(1)+(2) -1<x<0. Sufficient.

Similar questions to practice:
how-x-become-fraction-if-its-been-said-an-integer-94687.html
awkward-ds-105416.html
help-data-sufficiency-equation-and-inequalities-99856.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html

Hope it helps.

Hi Bunuel,
Can you post some 650+ - 700 level PS/DS questions on greatest integer function.

Thanks!!

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_________________
Intern  Joined: 02 Jan 2014
Posts: 10
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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If [x] denotes the least integer greater than or equal to x, is [x] = 0?

(1) -1< x< 1
[-0.5] = 0 but [0.5] = 1. Not sufficient

(2) x < 0
[-0.5] = 0 but [-1] = -1 and [-2] = -2. Not sufficient

(1)+(2) -1<x<0
[-0.5] = 0. Sufficient

Manager  Joined: 04 May 2015
Posts: 69
Concentration: Strategy, Operations
WE: Operations (Military & Defense)
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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Hi everyone... This may sound strange but this is the first question I have ever come across [] to denote rounding up. I mistakenly treated it as || (absolute value) so obviously got the question wrong... Can I just confirm that this is a standard type of mathematical notation that is used on the GMAT? I have not seen this anywhere....
Intern  Joined: 26 Feb 2016
Posts: 12
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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Bunuel wrote:
If [x] denotes the least integer greater than or equal to x, is [x] = 0?

Some function [] rounds UP a number to the nearest integer. For example [1.5]=2, =2, [-1.5]=-1, ...

Question: is $$[x]=0$$? --> is $$-1<x\leq{0}$$?

(1) -1< x< 1. Not sufficient.
(2) x < 0. Not sufficient.

(1)+(2) -1<x<0. Sufficient.

Similar questions to practice:
how-x-become-fraction-if-its-been-said-an-integer-94687.html
awkward-ds-105416.html
help-data-sufficiency-equation-and-inequalities-99856.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html

Hope it helps.

"[x] denotes to be the least integer no less than x" i can see why this is rounding down

but "If [x] is the greatest integer less than or equal to x"i dont see how thisis rounding down and why

"[x] denotes the least integer greater than or equal to x" why is this rounding up... i mean if theres only 2 of these i guess i could memorise but i'm trying to understand why...

please help. Ive been stuck on this interpretation for ages i dont want to just memorise it.
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Joined: 20 Mar 2014
Posts: 2549
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
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WE: Engineering (Aerospace and Defense)
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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5
3
sabxu1 wrote:
Bunuel wrote:
If [x] denotes the least integer greater than or equal to x, is [x] = 0?

Some function [] rounds UP a number to the nearest integer. For example [1.5]=2, =2, [-1.5]=-1, ...

Question: is $$[x]=0$$? --> is $$-1<x\leq{0}$$?

(1) -1< x< 1. Not sufficient.
(2) x < 0. Not sufficient.

(1)+(2) -1<x<0. Sufficient.

Similar questions to practice:
how-x-become-fraction-if-its-been-said-an-integer-94687.html
awkward-ds-105416.html
help-data-sufficiency-equation-and-inequalities-99856.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html

Hope it helps.

"[x] denotes to be the least integer no less than x" i can see why this is rounding down

but "If [x] is the greatest integer less than or equal to x"i dont see how thisis rounding down and why

"[x] denotes the least integer greater than or equal to x" why is this rounding up... i mean if theres only 2 of these i guess i could memorise but i'm trying to understand why...

please help. Ive been stuck on this interpretation for ages i dont want to just memorise it.

A quick comment. DO NOT memorize these definitions as they are not fixed. GMAT can change the definition of what the [x] does. Just try to understand what is the given definition.

Try to break it down in manageable chunks.

Case 1: "If [x] is the greatest integer less than or equal to x" ---> let x=4.4 what is the greatest integer LESS THAN or EQUAL to x ? Answer is 4 (5 will be GREATER THAN and NOT less than).You will get the equality when x = integer.

Thus, based on the definition of [x] above, [4.4] = 4 (you rounded DOWN ---> you went to a lower number).

Case 2: "[x] denotes the least integer greater than or equal to x" ---> let x =4.4 ---> based on this current definition, what is the LEAST integer GREATER or EQUAL to x? Answer is 5 (4 will not be correct here as 4<4.4). You will get the equality when x = integer.

Thus, based on the definition of [x] above, [4.4] = 5 (you rounded UP ---> you went to a higher number).

Do not memorize functional definitions as these definitions can change. Example, I can give you a question telling you that [x] denotes a function such that [x] =$$x^2$$ etc.

Hope this helps.
Intern  Joined: 26 Feb 2016
Posts: 12
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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Engr2012 wrote:
sabxu1 wrote:
Bunuel wrote:
If [x] denotes the least integer greater than or equal to x, is [x] = 0?

Some function [] rounds UP a number to the nearest integer. For example [1.5]=2, =2, [-1.5]=-1, ...

Question: is $$[x]=0$$? --> is $$-1<x\leq{0}$$?

(1) -1< x< 1. Not sufficient.
(2) x < 0. Not sufficient.

(1)+(2) -1<x<0. Sufficient.

Similar questions to practice:
how-x-become-fraction-if-its-been-said-an-integer-94687.html
awkward-ds-105416.html
help-data-sufficiency-equation-and-inequalities-99856.html
if-denotes-the-greatest-integer-less-than-or-equal-to-x-94687.html
denotes-to-be-the-least-integer-no-less-than-x-is-130841.html

Hope it helps.

"[x] denotes to be the least integer no less than x" i can see why this is rounding down

but "If [x] is the greatest integer less than or equal to x"i dont see how thisis rounding down and why

"[x] denotes the least integer greater than or equal to x" why is this rounding up... i mean if theres only 2 of these i guess i could memorise but i'm trying to understand why...

please help. Ive been stuck on this interpretation for ages i dont want to just memorise it.

A quick comment. DO NOT memorize these definitions as they are not fixed. GMAT can change the definition of what the [x] does. Just try to understand what is the given definition.

Try to break it down in manageable chunks.

Case 1: "If [x] is the greatest integer less than or equal to x" ---> let x=4.4 what is the greatest integer LESS THAN or EQUAL to x ? Answer is 4 (5 will be GREATER THAN and NOT less than).You will get the equality when x = integer.

Thus, based on the definition of [x] above, [4.4] = 4 (you rounded DOWN ---> you went to a lower number).

Case 2: "[x] denotes the least integer greater than or equal to x" ---> let x =4.4 ---> based on this current definition, what is the LEAST integer GREATER or EQUAL to x? Answer is 5 (4 will not be correct here as 4<4.4). You will get the equality when x = integer.

Thus, based on the definition of [x] above, [4.4] = 5 (you rounded UP ---> you went to a higher number).

Do not memorize functional definitions as these definitions can change. Example, I can give you a question telling you that [x] denotes a function such that [x] =$$x^2$$ etc.

Hope this helps.

Thanks for the explanation and time explain it to me. I thnk i understand it. I agree i should not memorise it. Thats why i was looking for an explanation.
Manager  D
Joined: 17 May 2015
Posts: 245
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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Hi,

Please note that greatest integer and least integer have two separate standard symbols, and GMAT is not going to use one symbol for another. Please refer the question no 111(DS) and 178(PS) OG 16th edition.

Two symbols are as follows:

$$\lceil x \rceil$$: least integer greater than or equal to x.

[x]: greatest integer less than or equal to x.

Hope this helps.
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GMAT 1: 700 Q49 V33
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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How to deal with such kind of gmat question in a general way?
Manager  S
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GMAT 1: 640 Q50 V25 GMAT 2: 710 Q50 V35 GPA: 3.48
If [x] denotes the least integer greater than or equal to x  [#permalink]

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1
chesstitans wrote:
How to deal with such kind of gmat question in a general way?

Hi,

I think the point here is to understand the phrase "the least integer greater than or equal to x". It means that if we have a number x, then just round it up to the possible smallest integer and we will get [x]. Remember that x could be anything, not just an integer. For example, if x=-1.2, then [x]=-1. Or if x=5, then [x]=5.

This is a DS question. Then we read the information given in each statement and put it into number line to figure out whether we could answer the question.

For example, (1) -1<x<1
Put -1 and 1 into the number line. We'll see that there are many situations to test. If -1<x=<0, then [x]=0. But that will not be necessarily the case, if 0<x<1. Because if 0<x<1, then [x]=1.
Intern  B
Joined: 09 Jan 2017
Posts: 5
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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Meaning of least value
for positive values
if x = 0.1
then also [x] = 1
if x=0.9
then also [x] = 1

For Negative values
if x= -0.1
then also [x] = 0
if x=-0.9
then also [x] = 0

statement 1 :- given -1 <x < 1
so [x] can be 0 or 1;
Not suffic.

statement 2 :- given x <0 ;

so [x] can be 0 ,-1,-2 all the negative values.

Not suff.

so combining both together for getting the soltn.

We will get the answer C.
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4219
If [x] denotes the least integer greater than or equal to x  [#permalink]

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1
Top Contributor
macwanjason wrote:
If [x] denotes the least integer greater than or equal to x, is [x] = 0?

(1) -1< x< 1
(2) x < 0

Target question: Is [x] = 0?
This is a good candidate for rephrasing the target question.

Given: [x] denotes the least integer greater than or equal to x

Let's make sure we understand what this [] symbol means.
Here are a few examples:
[3.1] = 4
[5.8] = 6
[0.7] = 1
[-0.9] = 0
 = 0
[-4.6] = -4
[-1] = -1

We can see that [x] = 0, when -1 < x ≤ 0
So, we can REPHRASE the target question . . .

REPHRASED target question: Is -1 < x ≤ 0 ?

Aside: My video below has tips on rephrasing the target question

Statement 1: -1< x< 1
Let's TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = -0.5. In this case, the answer to the REPHRASED target question is YES, it IS the case that -1 < x ≤ 0
Case b: x = 0.5. In this case, the answer to the REPHRASED target question is NO, it is NOT the case that -1 < x ≤ 0
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x < 0
Let's TEST some values.
There are several values of x that satisfy statement 2. Here are two:
Case a: x = -0.5. In this case, the answer to the REPHRASED target question is YES, it IS the case that -1 < x ≤ 0
Case b: x = -3. In this case, the answer to the REPHRASED target question is NO, it is NOT the case that -1 < x ≤ 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
When we combine the two statements, we get the inequality -1 < x < 0
So, for ALL possible values of x, the answer to the REPHRASED target question is YES, it IS the case that < x ≤
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent

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