If [x] denotes the largest integer smaller than x, is [x]>[x]?

s111

Using the highlighted part above we need to answer the question is [x]>[−x]?

so x = 3 then [x]=2 i.e. x=[x]+1 is done to check if the value x = 3 is acceptable as per the 1st statement or not since it's acceptable because it satisfies the first statement so now we need to know what kind of answer does this value of x gives for the primary question which is is [x]>[−x]?

Never forget the primary question in DS which in this question is is [x]>[−x]?

we find all values of x using statement to check whether there is consistency in the answers or not in order to call the statement sufficient or insufficient

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What can we say right away?
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[x] denotes the largest integer smaller than x. I'm going to assume 'smaller' here means 'less than' and not 'of lesser magnitude.'

This is effectively applying a floor function (analogous to a ceiling function

Let x be some real number. There exists some integer k such that k < x <= k+1; we're just saying [x] = k.

Let's see some examples (see any difference when x < 0 vs. x >= 0?)

--> Suppose x = 0; then [x] = -1
--> Suppose x = -.2.5, then [x] = -3
--> Suppose x = 4.5, then [x] = 4


We want to know if [x] > [-x]

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What does statement A tell us?
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x = [x] + 1

So this is telling us that x is an integer; recall we called [x] that integer k. This is just saying x = k + 1, which must be an integer.

Let's try two integers. I like 2 and -2; note these don't have to be of equivalent magnitude. It's just helpful.
--> What happens when x = 2? [x] = 1.
--> What happens when x = -2? [x] = -3.

Remember we wanted to know if [x] > [-x]. Let's plug for each of these cases.
--> if x = 2, we need to evaluate if [2] > [-2]. Is 1 > -3? Yes.
--> if x = -2, we need to evaluate if [-2] > [2]. Is -3 > 1? No.

So Statement A is insufficient.

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What does statement B tell us?
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x + 1 > 0. This implies x > -1 (Note that x doesn't have to be an integer, though!)

Let's try two values. I want to use -0.5 and 2 as my preferred ones to test.
--> We already know x = 2 shows us that [x] > [-x] from the above.

What happens if x = -0.5? We get [x] = -1. To check the inequality, we also have to know [-x] = [0.5] = 0.
--> So is [x] > [-x]? This is the same as asking if -1 > 0 is true. It's clearly false.

So we can tell the Statement B is insufficient.


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What if we take the statements together.
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Remember A told us that x has to be an integer. Statement B tells us that x has to be greater than -1. So we know x has to be either 0 or a positive integer.

Let's try two values for this one. I want to use 0 and 2.
--> We already know x = 2 implies [x] > [-x]

What happens if x = 0? Well [x] = -1. and [-x] = -1. Is it true that -1 > -1? No, because -1 = -1 .

So The statements in combination are insufficient.

Official Solution:


If \([x]\) denotes the largest integer smaller than \(x\), is \([x] > [-x]\)?

We are given a function, [ ], which rounds DOWN a number to the nearest integer. For example:

\([2.7] = 2\), because 2 is the largest integer less than 2.7;

\([3] = 2\), because 2 is the largest integer less than 3;

\([-1.7] = -2\), because -2 is the largest integer less than -1.7.

(1) \(x = [x] + 1\).

The above is true for all integers. For example, if x = 3, then [3] = 2 (the largest integer smaller than 3), so 3 = 2 + 1. This statement simply implies that \(x\) is an integer. However, this is not sufficient to answer the question. For example, if \(x = 1\), then \([x] = [1] = 0\) and \([-x] = [-1] = -2\), and the answer would be YES. But, if \(x = -1\), then \([x] = [-1] = -2\) and \([-x] = [1] = 0\), and the answer would be NO. Not sufficient.

(2) \(x + 1 > 0\).

This statement implies that \(x > -1\), which is clearly insufficient to answer the question.

(1)+(2) Combining the statements, we can conclude that \(x\) is an integer greater than -1: 0, 1, 2, 3, and so on. If \(x\) is a positive integer, the answer will be YES. However, if \(x = 0\), then \([x] = [0] = 0\) and \([-x] = [0] = 0\), and the answer will be NO. Not sufficient.


Answer: E

Can we not simply translate the question stem as asking is x>0? This is the simplified form in my opinion

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Yes, for \([x] > [-x]\) to hold true, \(x\) must be greater than 0. This is because if \(x < 0\), \([x]\) will be the greatest integer less than a negative value, which is negative, while \([-x]\) will be the greatest integer less than a positive value, which is positive (or 0 if \(-1 \leq x < 0\)). If \(x = 0\), both \([x]\) and \([-x]\) will be -1, making them equal to each other. However, when \(x > 0\), \([x]\) will be the greatest integer less than a positive value, which is positive (or 0 if \(0 < x \leq 1\)), while \([-x]\) will be the greatest integer less than a negative value, which is negative. Therefore, the question essentially asks whether \(x > 0\).