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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
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So The statements in combination are insufficient.