If [x] denotes the greatest integer less than or equal to x

Hi all,

this is from OG 12 q.100/p.281

If [x] denotes the greatest integer less than or equal to x, is [x] = 0 ?
(1) 5x + 1 = 3 + 2x
(2) 0 < x < 1

The answer mentions that this [x] = 0 is equivalent to 0 ≤ x < 1.

I have hard time understanding it.

What would be [x] = 5 ?

5 <= x < 6 ???

Merging similar topics. Please refer to the solution above.

As for your question:
If \([x]=5\) then yes, \(5\leq{x}<6\), as ANY \(x\) from this range if round down to the integer gives 5. The same way if \([x]=0\) then \(0\leq{x}<1\), as ANY \(x\) from this range if round down to the integer gives 0.

Hope it's clear.

Thanks Bunuel for the answer & merging my question with other questions.
I understand the the lower limit but find it hard to grasp the upper limit.
can the upper limit always be x+1 for [x] ?

No upper limit is not x+1 for [x]. For example if [x]=5 then x<6 because if it's more than or equal to 6 then 5 won't be the greatest integer less than or equal to x.

More examples:

[5]=5;
[5.3]=5;
[5.9]=5;
[6]=6;
[-0.1]=-1;
...

Hope it's clear.

(1)

3x = 2

=> x = 2/3 ~ 0.67

So [x] = 0

Sufficient

(2)

x = 0.5, 0.9

So [x] = 0

Sufficient

Answer - D

I've seen this exact formula used before. Basically, they are just telling you that the brackets mean to round down to the nearest integer. The solution above is correct. Both statements indicate that x is a fraction between 0 and 1, and would therefore round down to 0.

What value of x would give [x]=0?
According to the rule given in the problem: [x] denotes the greatest integer less than or equal to x.
\(0<= x < 1\) will give us [x] = 0.

(1) 5x + 1 = 3 + 2x
3x = 2
x = 2/3 which satisfies 0 <= x < 1 Thus SUFFICIENT.

(2) 0 < x < 1 SUFFICIENT.

Answer: D

Why bother solve statement-1? From st-1 we can tell that we will get the exact value of 'x' for sure and that's all we need to answer the question one way or the other. So sufficient. Is my logic right mods?

Solution:

We need to determine whether [x] = 0 where [x] denotes the greatest integer less than or equal to x. If the rule is hard to follow, let’s use a few examples.

[12.53] = 12, since 12 is the greatest integer less than or equal to 12.53.

[-1/2] = -1, since -1 is the greatest integer less than or equal to -1/2.

[1/2] = 0, since 0 is the greatest integer less than or equal to 1/2.

Using this last example, we know that if x is a positive fraction between zero and one, or is zero itself, the answer will be yes.

Statement One Alone:

5x + 1 = 3 + 2x

Let’s first simplify the equation.

5x + 1 = 3 + 2x

3x = 2

x = 2/3

Since 0 is the greatest integer less than or equal to 2/3, we see that [2/3] = 0.

Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

0 < x < 1

Because 0 < x < 1, x is a number between 0 and 1 (but, of course, not including 0 nor 1). Thus, the greatest integer less than or equal to such a number will always be zero. Statement two alone is also sufficient to answer the question.

The answer is D.

Great explanation.

For [x]=0, the inequality is -1<x<=0 but what if the question states [x] = 5.. What will the inequality be in that case?

will it still be -1<x<=0 ?? Please explain

[x] = 5 would mean that \(5\leq{x}<6\). Any number from that range when rounded down to the integer gives 5.

Hello.. I wanted to ask why is 2/3 rounded to zero and not to 1 ?? O.667 so since 6 is greater than 4, shouldn't it be rounded up?
Thanks for your help..

0.667 rounded to the nearest integer is indeed 1. But the function we have does not simply rounds number. Given function, represented by the symbol \([]\), rounds down any number to an integer value:

\([3.4]=3\), \([2]=2\), \([-7.5]=-8\), ... So, \([\frac{2}{3}]=0\).

Check other Rounding Functions Questions in our Special Questions Directory.

Hello Bunuel
Thanks for the nice explanation.
Here i need to add one more thing. As we still don't know what is going on in both statements (Can we refrain our eyes from statements for some moments?) we should rephrase the question stem ( \([x]=0\) ?) as \(-1 <x≤ 0\) too. Am I missing anything?

No. If \(-1 <x< 0\), then the greatest integer less than or equal to x, will be -1, not 0. For example, if x = -1/2, then the greatest integer less than or equal to -1/2 is -1.

Thanks for the response Bunuel
I am talking when the scenario is -1 <x≤ 0 (not -1 <x< 0). So, what if the value of x=0? Shouldn't it be the greatest integer less than or equal to x, is [x] = 0?
Thanks_-

If x = 0, then then the greatest integer less than or equal to 0 is 0 itself. So, \(0\leq{x}<1\) in the solution above is correct.