Is |4m - 3n| > |3m - n| + |m - 2n| ?

Official Solution:


Is \(|3m - n| + |m - 2n| > |4m - 3n|\)?

One of the properties of absolute values says that \(|x|+|y| \geq |x+y|\). Note that "=" sign holds for \(xy \geq {0}\) (or simply when \(x\) and \(y\) have the same sign). So, the strict inequality (>) holds when \(xy < 0\) (Check here: https://gmatclub.com/forum/tips-and-hint ... l#p1381430). ​For example:

If \(x=2\) and \(y=3\), then \(|x|+|y| = |x+y|\). \(x\) and \(y\) have the same sign, so we have "=" sign.

If \(x=-2\) and \(y=-3\), then \(|x|+|y| = |x+y|\). \(x\) and \(y\) have the same sign, so we have "=" sign.

If \(x=2\) and \(y=-3\), then \(|x|+|y| > |x+y|\). \(x\) and \(y\) have the opposite signs, so we have ">" sign.

If \(x=-2\) and \(y=3\), then \(|x|+|y| > |x+y|\). \(x\) and \(y\) have the opposite signs, so we have ">" sign.

Next, notice that if we denote \(x=3m - n\) and \(y=m - 2n\), then \(x+y=4m-3n\). So, the question becomes: is \(|x|+|y|>|x+y|\)? Thus, the question basically asks whether \(x\) and \(y\), or which is the same \(3m - n\) and \(m - 2n\), have the opposite signs.

(1) \(m > 0\)

Clearly insufficient as no info about \(n\).

(2) \(2n < m\)

This implies that \(m-2n>0\). If \(m=3\) and \(n=1\), then \(3m - n>0\) (so in this case \(3m - n\) and \(m - 2n\) will have the same sign) but if \(m=-4\) and \(n=-3\), then \(3m - n 0\), or which is the same \(5m>0\) and \(m>2n\). Add them: \(6m>2n\). Reduce by 2 and re-arrange: \(3m-n>0\). Thus, both \(m-2n\) and \(3m-n\) are positive, so we have a NO answer to the question. Sufficient.

To elaborate: \(m-2n>0\) and \(3m-n>0\) means that their sum, which is \(4m - 3n\), will also be more than 0. So, "Is \(|3m - n| + |m - 2n| > |4m - 3n|\)?" will become "Is \((3m - n) + (m - 2n) > 4m - 3n\)?". This transforms into "Is \(0 > 0\)?". The answer to this question is NO.


Answer: C


Try NEW Absolute Value PS question.