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

(2) \(|x|=x\). This statement tells that \(x\) is some non-negative number, so \(x\) can be 0 or any positive number. Not sufficient.

(1)+(2) \(x\) can be 0, so not a positive number as well as 1, so a positive number, two different answers, hence not sufficient.

Greatttt many thanks I did not get zero for both as I did not factor it in statement 1 , Similar for statement 2 as I underestimate the value of zero many thanks for the question and steps

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?

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.
_________________

I did the correct steps and understand that 1 and 2 are not sufficient alone. I thought the answer is D, but why is it E instead? How is E a better answer than D?

I did the correct steps and understand that 1 and 2 are not sufficient alone. I thought the answer is D, but why is it E instead? How is E a better answer than D?

The answer is E because Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked. C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked. D. EACH statement ALONE is sufficient to answer the question asked. E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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.