AAPL wrote:$$If\ \ 5\ge|x|\ge0$$ which of the following must be true?
$$I.\ x\ge0$$
$$II.\ \ x>-5$$
$$III.\ \ 25\ge x^2\ge-25$$
$$A.\ None$$
$$B.\ II\ only$$
$$C.\ III\ only$$
$$D.\ I\ and\ III\ only$$
$$E.\ II\ and\ III\ only$$
|x| = the distance between x and 0.
|x| ≥ 0 implies that the distance between x and 0 must be greater than or equal to 0.
This will be true for any value of x.
5 ≥ |x| implies that the distance between x and 0 must be less than or equal to 5, as follows:
-5<---------->5.
Thus, 5 ≥ |x| ≥ 0 implies that x must be within the green range above.
Since it's possible that x=-5, Statements I and II do not have to be true.
Eliminate B, D and E.
The square of any value in the green range above will be between -25 and 25, inclusive.
Thus, Statement III must be true.
The correct answer is
C.
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