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# If [x] denotes the least integer greater than or equal to x

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

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08 May 2012, 05:34
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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
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Joined: 02 Sep 2009
Posts: 58445
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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09 May 2012, 01:53
26
50
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]=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
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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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18 Dec 2012, 01:03
9
12
(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.

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##### General Discussion
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Joined: 13 Mar 2012
Posts: 14
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Re: If {x} denotes the least integer greater than or equal to x,  [#permalink]

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08 May 2012, 06:56
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
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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09 Jun 2013, 12:53
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
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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09 Jun 2013, 12:57
1
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Intern
Joined: 01 Jan 2013
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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04 Nov 2013, 00:31
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]=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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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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04 Nov 2013, 00:42
3
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]=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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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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24 Apr 2014, 07:38
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

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

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07 Oct 2015, 20:59
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....
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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25 Apr 2016, 05:23
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]=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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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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25 Apr 2016, 05:43
4
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]=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.
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Posts: 12
Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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25 Apr 2016, 05:50
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]=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.
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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13 Jan 2017, 06:26
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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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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15 Jun 2017, 21:26
How to deal with such kind of gmat question in a general way?
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If [x] denotes the least integer greater than or equal to x  [#permalink]

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19 Jun 2017, 01:48
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.
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Re: If [x] denotes the least integer greater than or equal to x  [#permalink]

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19 Jul 2017, 09:36
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.
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If [x] denotes the least integer greater than or equal to x  [#permalink]

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27 Feb 2019, 07:26
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] = 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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If [x] denotes the least integer greater than or equal to x   [#permalink] 27 Feb 2019, 07:26
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