For all non-zero real numbers \(a\) and \(b,\) which of the following must be true?

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For all non-zero real numbers \(a\) and \(b,\) which of the following must be true?

I. \(|a+b|=|a|+|b|\)

II. \(\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}\)

III. \(|ab| = |a|*|b|\)


A. II only
B. III only
C. I and II
D. II and III
E. I, II, and III

The OA is D
Source: — Problem Solving |

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BTGmoderatorLU wrote:
Wed Dec 11, 2024 5:13 pm
For all non-zero real numbers \(a\) and \(b,\) which of the following must be true?

I. \(|a+b|=|a|+|b|\)

II. \(\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}\)

III. \(|ab| = |a|*|b|\)


A. II only
B. III only
C. I and II
D. II and III
E. I, II, and III

Two non zero numbers a and b can be positive or negative in any combination but their absolute values will always be positive. We can choose numbers like a = 3 and b = -2 to make sure that statement I is not always true, hence we may get rid of C and E. Statements II and III are true in all situations, hence D.

The OA is D
Two non zero numbers a and b can be positive or negative in any combination but their absolute values will always be positive. We can choose numbers like a = 3 and b = -2 to make sure that statement I is not always true, hence we may get rid of C and E. Statements II and III are true in all situations, hence D.