M04-17

Official Solution:


Is \(x\) a positive number?

(1) \(x^3=x\).

Rearrange: \(x^3 - x = 0\). Factor out \(x\): \(x(x^2 - 1) = 0\). Factor further: \(x(x - 1)(x + 1) = 0\). Thus, \(x = 0\), \(x = 1\), or \(x = -1\). Not sufficient.

(2) \(|x|=x\).

This statement implies that \(x\) is a non-negative number, so \(x\) can be 0 or any positive number. Not sufficient.

(1) + (2) Combining the statements, we find that \(x\) can be 0, which is not a positive number, or 1, which is a positive number. Since we have two different answers, the combined statements are still not sufficient.


Answer: E
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. In statement 1, if I divide both sides by X it leaves me with X squared = 1. This eliminates the possibility of 0. Is it illegal to divide both sides by X in this case?

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.
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Are you suggesting that 0 is not a positive number?

Please help.

0 is NOT a positive integer.

ZERO:

1. 0 is an integer.

2.. 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

3. 0 is neither positive nor negative integer (the only one of this kind).

4. 0 is divisible by EVERY integer except 0 itself.

Check more here: https://gmatclub.com/forum/number-proper ... 74996.html
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Is the absolute value of zero defined ? I got this question Incorrect because I read on the internet the absolute value of zero is undefined.

Please let me know.

Thank You
Bunuel

|0| = 0.

You can think of an absolute value as a distance from 0. So, absolute value of 0, is 0, because the distance between 0 and 0 is 0.
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I think this is a high-quality question and I agree with explanation. Quick question, if the second statement said that the absolute value of x was -x, would the answer be C?

If (2) were |x| = -x, then the answer would have been B. |x| = -x means that x is negative or 0, so not a positive number.
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When you have a question like “Is x a positive number?”, remember that the question is probably giving you a clue that ‘x’ could possibly be non-positive as well. If you take this subtle clue, you will end up solving the question with a more open mind rather than trying to prove that x is a positive number.

From statement I alone, \(x^3\) = x. A very common mistake that some students make on such statements is to cancel off variables and solve an entirely different equation. There are a couple of things you could do instead.

#1 – Take all terms on to the LHS, keeping the RHS as 0. Express the LHS as a product or a quotient by taking terms common, which will help you decide the signs of the terms.

#2 – If you end up cancelling variables, remember that every cancellation corresponds to a root of ZERO.

In general, when you have a cubic equation, it’s a good idea to try the values of 0,1 and -1.

\(x^3\) = x can be re-written as \(x^3\) -x = 0. Taking x as common, we have x(\(x^2\) – 1) = 0. This means x=0 or \(x^2\)-1 =0 which means \(x^2\) = 1which in turn gives us x = 1 or x = - 1.
-1 is negative, 0 is neither negative nor positive and 1 is positive. Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, |x| = x. This equation can only be satisfied by positive values OR ZERO, since the LHS will always be non-negative.
Statement II alone is insufficient. Answer option B can be eliminated, possible answer options are C or E.

Combining the data from statements I and II, we have the following:
From statement I alone, x = 0 or -1 or 1.
From statement II alone, x = 0 or 1.
Clearly, the common values of x are 0 and 1. Even after combining the statements, we do not know if x is definitely positive.
The combination of statements is insufficient. Answer option C can be eliminated.

The correct answer option is E.

Hope that helps!
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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