Question Banks -Number Properties #11

Below is the question and explanation. My question is about the explanation for choice D. It says if we rewrite the expresssion, we get ac-bc>0. I don't follow how this is done? Why is the inequality signed flipped? Thanks


If (a – b)c < 0, which of the following cannot be true?



a < b


c < 0


|c| < 1


ac > bc


a2 – b2 > 0




If (a – b)c < 0, the expression (a – b) and the variable c must have opposite signs.
Let's check each answer choice:
(A) UNCERTAIN: If a < b, a – b would be negative. It is possible for a – b to be negative according to the question.

(B) UNCERTAIN: It is possible for c to be negative according to the question.

(C) UNCERTAIN: This means that -1 < c < 1, which is possible according to the question.

(D) FALSE: If we rewrite this expression, we get ac – bc > 0. Then, if we factor this, we get: (a – b)c > 0. This directly contradicts the information given in the question, which states that (a – b)c < 0.

(E) UNCERTAIN: If we factor this expression, we get (a + b)(a – b) < 0. This tells us that the expressions a + b and a – b have opposite signs, which is possible according to the question.
The correct answer is D.

Can you explain choice E? I don't see why when we factor choice E the inequality sign flips as it does in the explanation.

Why does a+b and a-b have opposite signs, I don't see this?

This is problem #10, not 11, from the Number Properties Question Bank. :)

First question (about choice D)

ac > bc
subtract bc from both sides
ac - bc > bc - bc
simplify
ac - bc > 0

You ask why the sign flips... but it doesn't. (Even in what you typed above, it doesn't flip - it's always >.)

Second question (about choice E)
For future, please note the proper formatting to use to indicate exponents: a^2 - b^2
On this one - you're right, that's a rather major typo. The sign shouldn't flip.

Do you understand why (a+b) and (a-b) would have opposite signs IF that statement were typed correctly? Or was your confusion there just because you didn't know why the sign flipped? Let me know.

So explanation for E should say (a+b)(a-b) > 0, therefore the two expressions (a+b) and (a-b) have the same sign. This is possible according to the original inequality, so it is not the one that must be false.