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[x] denotes the largest integer less than or equal to x. If [x] = -1,

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[x] denotes the largest integer less than or equal to x. If [x] = -1, [#permalink] New post   10 Apr 2020, 08:24
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[x] denotes the largest integer less than or equal to x. If [x] = -1, which of the following must be true?


A. \(x = 1\)

B. \(-2 \leq x < -1\)

C. \(-2 < x \leq -1\)

D. \(-1 \leq x < 0\)

E. \(-1<x \leq 0\)


M37-09

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D
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Re: [x] denotes the largest integer less than or equal to x. If [x] = -1, [#permalink] New post   18 Nov 2021, 05:01
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Official Solution:

\([x]\) denotes the largest integer less than or equal to \(x\). If \([x] = -1\), which of the following must be true?

A. \(x = 1\)
B. \(-2 \leq x < -1\)
C. \(-2 < x \leq -1\)
D. \(-1 \leq x < 0\)
E. \(-1< x \leq 0\)


Some function [] rounds DOWN a number to the nearest integer. For example:

\([2.7] = 2\) because 2 the largest integer less than or equal to 2.7;

\([3] = 3\) because 3 the largest integer less than or equal to 3;

\([-1.7] = -2\) because -2 the largest integer less than or equal to -1.7.

So, \([x] = -1\) means that -1 is the largest integer less than or equal to \(x\), which implies that \(-1 \leq x < 0\). For any number from that range, the largest integer less than or equal that number will be -1.


Answer: D
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Re: [x] denotes the largest integer less than or equal to x. If [x] = -1, [#permalink] New post   10 Apr 2020, 08:45
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Bunuel wrote:
[x] denotes the largest integer less than or equal to x. If [x] = -1, which of the following must be true?


A. \(x = 1\)

B. \(-2 \leq x < -1\)

C. \(-2 < x \leq -1\)

D. \(-1 \leq x < 0\)

E. \(-1<x \leq 0\)



[x] = -1

Now [x] can be x, so [x] = -1=x.
So our solution has to include -1. Only C and D include -1.
OR
[x] can be the largest integer less than or equal to x,
so \([x]\leq{x}...-1\leq{x}..\)

Also x<0, otherwise [x]=0..

combined
\(-1\leq{x}<0\)..

D
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Re: [x] denotes the largest integer less than or equal to x. If [x] = -1, [#permalink] New post   10 Apr 2020, 09:03
[x] denotes the largest integer less than or equal to x. If [x] = -1, which of the following must be true?
example :[3.4]=3 , [2]=2,
[−7.5]=−8, (pattern -7 -1 =-8 )
here, [x] = -1 so,
x= must be some fractions with negative signs with no integral parts. ( 0 -1 = -1)
so , -1<x<0
now , considering critical points,
when x=0, 0, -1, -2, ... all are less than or equal to x. but greatest of them is 0 . so, [x] = 0 in this case. so x cannot be 0.
when x=-1, -1, -1.5 , -2 ,-3 ... all are less than or equal to x. but greatest of them is -1 . so, [x] = -1 in this case. so x can be -1.

so final range −1≤x<0 (must be true)

correct answer D
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Re: [x] denotes the largest integer less than or equal to x. If [x] = -1, [#permalink]
10 Apr 2020, 09:03
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