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Math Expert V
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For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 67% (01:45) correct 33% (01:49) wrong based on 1358 sessions

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For all z, [z] denotes the least integer greater than or equal to z. Is [x] = 0 ?

(1) -1< x < -0.1
(2) [x + 0.5] = 1

Data Sufficiency
Question: 96
Category: Algebra Operations with real numbers
Page: 160
Difficulty: 650

The Official Guide For GMAT® Quantitative Review, 2ND Edition

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Math Expert V
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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##### General Discussion
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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3
1
Option A.

From S1:x=any value greater than -1 but less than -0.1.For every value of x,
[x]=0 only.Sufficient.

From S2:x=0.1,0.2,0.3,...,0.5
OR x=-0.1,-0.2,-0.3,-0.4.
Therefore,[x]=0 or 1.Not sufficient.
Math Expert V
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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1
et us do [2.4]
What are the integers greater than or equal to 2.4?
3,4,5...
What is the least among them?
3
So [2.4] = 3

What is ?
Integers greater than or equal to 5 are 5,6,7,...
Leats among them is 5.
so  = 5

[] is called a step function. because, the graph looks like steps.

Hope this helps!!
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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(1) -1 < x < -0.1. It'S Sufficient, as it's within the range - IXI = 0

(2) [x + 0.5] = 1 --> 0<x+0.5≤1 --> −0.5<x≤0.5 Not sufficient, if x=-0,2 -> IXI = 0, if x=0,3 -> IXI = 1
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Bunuel wrote:

(2) [x + 0.5] = 1 --> 0<x+0.5≤1
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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sairam95 wrote:
Bunuel wrote:

(2) [x + 0.5] = 1 --> 0<x+0.5≤1

hi

[b]For all z, [z] denotes the least integer greater than or equal to z.
so if [x+0.5] =1, x+0.5 has to be between 0 and 1 including 1..
if x+0.5 is between -0.999999 and 0, inclusive, then [x+0.5] =0..
basically what ever is betwen [] takes the higher integer value or same value if it is integer...
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Could you please explain the logic for the statement 2 .I understood [z] denotes the least integer greater than or equal to z is meant by −1<x≤0.
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Ishanvs wrote:
Could you please explain the logic for the statement 2 .I understood [z] denotes the least integer greater than or equal to z is meant by −1<x≤0.

"Least integer greater than or equal to z implies that [z] >= z but [z] takes the least value that it can.

So if z = 0.3, [z] = 1
If z = 1.98, [z] = 2

Ques: Is [x] = 0?
When will [x] be 0? When -1 < x <= 0
So we need to know whether -1 < x <= 0?

Stmnt 2: [x + 0.5] = 1

Say, x + 0.5 = z
Given: [z] = 1
If [z] = 1, then we know that 0 < z <= 1. Note that if z is 0, [z] = 0. If z > 1, then [z] > 1 too.

This implies that 0 < x + 0.5 <= 1
Subtracting 0.5 from the inequality, we get
-0.5 < x <= 0.5
We know that x lies between -0.5 and 0.5. Some of these values lie in the -1 to 0 range and some do not. Hence we can't say whether x will lie in the -1 to 0 range.
Not sufficient alone.
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Okay has anyone actually seen a question like this on the actual GMAT? Maybe I'm crazy, but with Kaplan for example, there were lots of practice questions that did not prepare me for the actual GMAT. On the QA of the actual GMAT I took, there were much more questions like those on Veritas' prep CAT's- lots of exponents, lots of algebra, lots of functions
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Nunuboy1994 wrote:
Okay has anyone actually seen a question like this on the actual GMAT? Maybe I'm crazy, but with Kaplan for example, there were lots of practice questions that did not prepare me for the actual GMAT. On the QA of the actual GMAT I took, there were much more questions like those on Veritas' prep CAT's- lots of exponents, lots of algebra, lots of functions

This question is from the Official Quant guide. So you could definitely see something like this in the actual GMAT.
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Sorry does this 2nd statement show modulus??
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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longhaul123 wrote:
Sorry does this 2nd statement show modulus??

No. The stem introduces a function [], which rounds UP a number to the nearest integer. For example [1.5]=2, =2, [-1.5]=-1, ... The second statement also has the same function.
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Bunuel - For statement 1, do we not consider 0.05?

Thanks
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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eishan123 wrote:
Bunuel - For statement 1, do we not consider 0.05?

Thanks

(1) says: -1 < x < -0.1. Is 0.05 in that range?
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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My mistake, I did not see the minus. Sorry about that.
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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1) -1 < x < -0.1
[x] = 0 when -1 < x ≤ 0
Sufficient.

(2) [x+0.5] = 1
Range of x is -0.5 < x < 0.5
If x is -0.4 then it rounds up to 0
If x is 0.4 then it rounds up to 1
Not sufficient.

A number line can be helpful to visualize.
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For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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[x] = 0 iff -1<x<=0

Statement 1 - Falls in the above range. Hence, Sufficient. Get rid of B,C,E
Statement 2 says [x+0.5]=1. Now here x may be -0.1 i.e [x]=0 or x may be +0.1 i.e. [x]=1. Hence this is clearly insufficient. Get rid of optionB too

Clearly, A is the winner
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Re: For all z, [z] denotes the least integer greater than or equal to z.  [#permalink]

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Bunuel wrote:
SOLUTION

For all z, [z] denotes the least integer greater than or equal to z. 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 < -0.1. Sufficient.

(2) [x + 0.5] = 1 --> $$0<x+0.5\leq{1}$$ --> $$-0.5<x\leq{0.5}$$. Not sufficient.

Bunuel, I read all the explanations in this post but still unable to understand how [x + 0.5] transforms into an inequality... Can you please elaborate? Thank you. Re: For all z, [z] denotes the least integer greater than or equal to z.   [#permalink] 05 Jul 2019, 09:36

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