Problem Solving

If x = ± |x| , then which one of the following statements could be true?
I. x = 0
II. x < 0
III. x > 0
(A) None (B) I only (C) III only (D) I and II (E) II and III

How to solve this?

btg760 wrote:If x = ± |x| , then which one of the following statements could be true?
I. x = 0
II. x < 0
III. x > 0
(A) None (B) I only (C) III only (D) I and II (E) II and III

How to solve this?

The definition of |x| is:

X, if x is >_ 0

-x, if x is < 0


If x is >_0, then X=+- (X). This values includes 0, positive and negative values. So I, II and III are possible

if x is < 0, then x =+- (-X) = +- X. Here only 0 is not possible.

Since the question says which could be true, I, II and III could be true as shown above in both situations.

From the looks of it, "x = ± |x|" means two things at once:
a. x = -|x|. Since |x| is always positive or equal to zero, -|x| will always be negative or equal to zero. So from case a you get that x <=0.
b. x = +|x|, which is true only when x is positive or equal to zero. In this second case, x >= 0.

As you can see, there's only one value that fits both cases, x = 0. The answer will be B.

anybody else?