Is m < 0? (1) -m = |-m| (2) m^2 = 9

Official Solution:

Is \(m \lt 0\)?

(1) \(-m=|-m|\).

Firstly, \(|-m|=|m|\), (for example: \(|-3|=|3|=3\)), so we can rewrite the statement as \(-m=|m|\). Since \(|x| = -x\) when \(x \leq 0\), then \(-m=|m|\) implies \(m \leq 0\). Therefore, \(m\) could be either negative, giving a YES answer to the question, or zero, giving a NO answer to the question. This statement is not sufficient.

(2) \(m^2=9\).

The above implies that either \(m=3\) (positive) or \(m=-3\) (negative). This information alone is not sufficient to answer the question.

(1)+(2) We know from (1) that \(m \leq 0\), and from (2) we have two possibilities: \(m=3\) or \(m=-3\). Since \(m \leq 0\), we can eliminate \(m=3\), leaving us with \(m=-3\), which is negative. Thus, the answer to the question "is \(m \lt 0\)" is YES. Sufficient.

Answer: C.

Hope it's clear.