If |a| > |b|, which of the following must be true?

\(|a| = a\), when \(a \geq 0\) and \(|a| = -a\), when \(a\leq 0\). You cannot simply plug a or -a if you don't know which case you have.


If \(|a| > |b|\), which of the following must be true?


A. \(ab > 0\)

B. \(|a| + b > 0\)

C. \(a + |b| > 0\)

D. \(\frac{|b|}{a} > 0\)

E. \(|a|*b > 0\)


\(|a| > |b|\) means that a is further from 0 than b is. So, we can have one of the following 4 cases:

1. ----a----b----0---------------
2. ----a---------0----b----------
3. ---------b----0---------a-----
4. --------------0----b----a-----

A. \(ab > 0\) --> Not true for cases in which a and b have different signs (cases 2 and 3). Discard.

B. \(|a| + b > 0\) --> true for all cases.

C. \(a + |b| > 0\) --> not true for cases in which a is negative (cases 1 and 2). Discard.

D. \(\frac{|b|}{a} > 0\) --> not true for cases in which a is negative (cases 1 and 2). Discard.

E. \(|a|*b > 0\) --> not true for cases in which b is negative (cases 1 and 3). Discard.

Answer: B.

Hope it helps.