Hi, there. I'm happy to help with this.
Prompt: √(x-3)^2 = 3-x?
Notice first of all that √(x-3)^2 = |x-3|.
Notice also that 3-x = -(x-3)
When does |x-3| = -(x-3)? It might be easier to express: when does |u| = -u? That's true when u is negative, when u < 0. Similarly, |x-3| = -(x-3) when x-3 < 0, or when x < 3. So, really, the question reduces to: is x < 3?
Statement #1: x is not equal to 3
Useless. Statement #1, by itself, is
insufficient.
Statement #2: -x|x|>0
Well, first of all, if x = 0, this would not be true, so we know x = 0 is not a possibility. Then, we know that |x| must be a positive number, so we can divide the inequality by |x| and not change the order of the inequality. That leaves us with -x>0. Multiply by a negative 1 (remembering to reverse the order of inequality), and we get x<0. Statement #2 is telling us that x is less than zero.
Well, if x is less than zero, then it's certainly true that x < 3. Therefore, this statement provide enough information to give a definitive answer to the prompt question. Statement #2 is
sufficient.
Answer =
A
Does that make sense? Please let me know if you have any further questions.
Mike
