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

Question

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

(1) -1< x < -0.1
(2)   = 1

Bunuel · Accepted Answer

KarishmaB · Answer

"Least integer greater than or equal to z implies that >= z but takes the least value that it can. So if z = 0.3, = 1 If z = 1.98, = 2 Ques: Is = 0? When will be 0? When -1 < x <= 0 So we need to know whether -1 < x <= 0? Stmnt 2: = 1 Say, x + 0.5 = z Given: = 1 If = 1, then we know that 0 < z <= 1. Note that if z is 0, = 0. If z > 1, then > 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.

AKG1593 · Answer

Option A.

From S1:x=any value greater than -1 but less than -0.1.For every value of 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, =0 or 1.Not sufficient.

gmatkum · Answer

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

BrainLab · Answer

(1) -1 < x < -0.1. It'S Sufficient, as it's within the range - IXI = 0 (2) = 1 --> 0 −0.5 IXI = 0, if x=0,3 -> IXI = 1

sairam595 · Answer

Please advise how below statement can be formed :

(2)   = 1 --> 0<x+0.5≤1

chetan2u · Answer

hi

For all z,   denotes the least integer greater than or equal to z.
so if   =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   =0..
basically what ever is betwen   takes the higher integer value or same value if it is integer...

Ishanvs · Answer

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

longhaul123 · Answer

Sorry does this 2nd statement show modulus??

Bunuel · Answer

No. The stem introduces a function  , which rounds UP a number to the nearest integer. For example  =2,  =2,  =-1, ... The second statement also has the same function.

energetics · Answer

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

(2)   = 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.

Basshead · Answer

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

The least integer  greater than or equal  to z means we round up.   = 6,   = 5.

(1) -1< x < -0.1

= 0. SUFFICIENT.

(2)   = 1

If x = 0 then,   = 0
If x = 0.5 then   = 1

INSUFFICIENT.

Answer is A.

ScottTargetTestPrep · Answer

Solution:

We need to determine whether   = 0 where   denotes the least integer greater than or equal to x.

Statement One Alone:

Since the least integer greater than any number whose value is between -1 and -0.1 is 0,   = 0. Statement one alone is sufficient.

Statement Two Alone:

If   = 1, then we have:

0 < x + 0.5 ≤ 1

-0.5 < x ≤ 0.5

We see that if x = -0.4, then   = 0. However, if x = 0.4, then   = 1. Statement two alone is not sufficient.

Answer: A

GMATinsight · Answer

Question: Is   = 0 ?

This will be true if x is between -1 and 0 i.e.

Question REPHRASED: Is -1 < x ≤ 0?

Statement 1   :  -1< x < -0.1
 SUFFICIENT

Statement 2  :   = 1
i.e. -0.5 < x ≤ 0.5
 NOT SUFFICIENT

Answer: Option A

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

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GMAT Data Sufficiency (DS) Questions

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

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