If a b and |a-b| = b-a, which of the following statements

If a ≠ b and |a-b| = b-a, which of the following statements must be true ?

I. a < 0
II. a + b < 0
III. a < b

(A) None
(B) I only
(C) III only
(D) I and II
(E) II and III

Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\);


Thus, according to the above, since \(|a-b| = b-a=-(a-b)\), then \(a-b\leq{0}\) --> \(a\leq{b}\). Since we also know that \(a\neq{b}\), then we have that \(a<b\). So, III is always true.

As for the other options:
I. a < 0 --> not necessarily true, consider a=1 and b=2.
II. a + b < 0 --> not necessarily true, consider a=-2 and b=-1.

Answer: C.

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