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

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]=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.

Answer: A.

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"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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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.

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

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

Hope this helps!!

(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

Please advise how below statement can be formed :

(2) [x + 0.5] = 1 --> 0<x+0.5≤1
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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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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.

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]=2, [-1.5]=-1, ... The second statement also has the same function.
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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.

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

The least integer greater than or equal to z means we round up. [5.5] = 6, [5] = 5.

(1) -1< x < -0.1

[x] = 0. SUFFICIENT.

(2) [x + 0.5] = 1

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

INSUFFICIENT.

Answer is A.
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Solution:

We need to determine whether [x] = 0 where [x] 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, [x] = 0. Statement one alone is sufficient.

Statement Two Alone:

If [x + 0.5] = 1, then we have:

0 < x + 0.5 ≤ 1

-0.5 < x ≤ 0.5

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

Answer: A
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