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18 Nov 2021, 05:01
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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. \(-1< x \leq 0\)
A. \(x = 1\)
B. \(-2 \leq x < -1\)
C. \(-2 < x \leq -1\)
D. \(-1 \leq x < 0\)
E. \(-1< x \leq 0\)
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
\([x]\) 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
18 Nov 2025, 03:59
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I like the solution - it’s helpful. [x] = -1.
Take the scenarios,
+ve values
x=1, then
f[x]= 1
x=2.5,
f[x] = 2. The largest integer less than or equal to 2.5 is 2.
x=0, then
f[x] = 0
if x=.5,then
f[x] = 0 again. The largest integer less than or equal to .5 is 0. So x cant be greater than 0.
-ve values:
x=(-.5), then
f[x]= -1. The largest integer less than or equal to -0.5 is -1. Possible for x to exist.
Lets check the extremes now.
if x=(-1), then
f[x]= -1. Possible for x to exist.
x=-1.5, then
f[x]= -2 The largest integer less than or equal to -1.5 is -2.
Now this number is not possible. because as per the given info f[x] = -1.
So x can be -1 or less than 0. The range becomes 1≤ x < 0
−
Take the scenarios,
+ve values
x=1, then
f[x]= 1
x=2.5,
f[x] = 2. The largest integer less than or equal to 2.5 is 2.
x=0, then
f[x] = 0
if x=.5,then
f[x] = 0 again. The largest integer less than or equal to .5 is 0. So x cant be greater than 0.
-ve values:
x=(-.5), then
f[x]= -1. The largest integer less than or equal to -0.5 is -1. Possible for x to exist.
Lets check the extremes now.
if x=(-1), then
f[x]= -1. Possible for x to exist.
x=-1.5, then
f[x]= -2 The largest integer less than or equal to -1.5 is -2.
Now this number is not possible. because as per the given info f[x] = -1.
So x can be -1 or less than 0. The range becomes 1≤ x < 0
−
