Problem Solving

$$If\ \ 5\ge|x|\ge0$$ which of the following must be true?

$$I.\ x\ge0$$
$$II.\ \ x>-5$$
$$III.\ \ 25\ge x^2\ge-25$$

$$A.\ None$$
$$B.\ II\ only$$
$$C.\ III\ only$$
$$D.\ I\ and\ III\ only$$
$$E.\ II\ and\ III\ only$$

The OA is C.

Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.

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.