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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