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

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.

Answer: C.

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Hope it helps.
_________________

When answering Data Sufficiency:
(1) Find out the question.
(2) Ponder about the possible values that will lead you to the answer.

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.

Answer: C

Video solution from Quant Reasoning:
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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

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

Yes, [1.4]=2.

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_________________

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

Thanks!!

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_________________

"[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.
_________________

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.
Inconsistent answers mean insufficient statement.

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

Answer: C

Cheers,
Brent

RELATED VIDEO

Hello Bunuel
Thanks for the explanation. May I know WHY we don't consider x< 1 when we combined both statement?
_________________

(1) ----(-1)---------(1)-----
(2) ------------0------------

Overlep:
------(-1)----0----
_________________

If integer is mentioned why do we need to take floating point numbers and round them?
Plus you have written 3.1 = 4, I am sorry but I didnt understand this

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

Let's examine [3.1]
There are infinitely many integers that are greater than or equal to 3.1. They include: 4, 5, 6, 7, 8, 9, .....
The bracket notation tells us to take the LEAST integer among the integers greater than or equal to 3.1.
So, [3.1] = 4

Notice that the question does not state that x must be an integer. All we know is that the OUTPUT will be an integer.

Does that help?

The least integer function takes any real number as the input; the output is the smallest integer greater than/equal to the number provided as the input.
For example, if x = 3.7, [x] = 4 since the smallest integer bigger than 3.7 is 4.

Strategy to solve least integer functions:

1) Plot the given value on the number line
2) The least integer for the given value will be the closest integer to the right, on the number line.

The original question stem, “Is [x] = 0?” can be rephrased as “Is -1<x≤0?”.

Statement I alone is insufficient to answer the question. Answer options A and D can be eliminated. Possible answer options are B, C or E

Statement II alone is insufficient to answer the question. Answer option B can be eliminated. Possible answer options are C or E.

Combining the data given in statements I and II, the common range is -1<x<0. This is sufficient to answer the question with a definite YES.
Statements I and II are sufficient when taken together. Answer option E can be eliminated.

The correct answer option is C.

Hope that helps!
Aravind B T
_________________

Answer: Option C

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