For all real numbers a and b, which of the following must be true?

Question

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

I.  ⏐a + b⏐ = ⏐a⏐ + ⏐b⏐

II.  ⏐\frac{a}{b}⏐ = \frac{|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

yashikaaggarwal · Answer

Let A = 3 and B = -2

I. ⏐a + b⏐ = ⏐a⏐ + ⏐b⏐
=> ⏐3 -2⏐ = ⏐3⏐ + ⏐-2⏐
=> 1 is not equal to 5 (Incorrect)

II. ⏐a/b⏐ = |a|/|b|
=> ⏐3/-2⏐ = |3|/|-2|
=> 1.5 = 1.5 (Correct)

III. ⏐ab⏐ = ⏐a⏐ * ⏐b⏐
=> ⏐3*-2⏐ = ⏐3⏐ * ⏐-2⏐
=> 6 = 6 (Correct)

Answer is D

CEdward · Answer

How did you know that the solutions that you tested for 2/3 were enough? What if there were other combinations that don't work?

Cucah · Answer

You can test this by looking only at signs.

I. If a and b have opposite signs (that is one is positive, one is negative), then abs value of |a+b|  
e |a|+|b| . You can plug in any number to prove this

II. If a and b have opposite signs, their value will always be the same and positive in a division.

III. Same as II. Product of two number is negative only if their signs are opposite. But if they are opposite, abs value makes the product positive anyways. Hence, they will always equal their separate abs values.

Posted from my mobile device

BrushMyQuant · Answer

Given that a and b and non-zero real numbers and we need to find which of the following must be true?

Let's take values to prove the options wrong.

I.  ⏐a + b⏐ = ⏐a⏐ + ⏐b⏐  
Now, this becomes FALSE when a and b have opposite signs
=> Lets take a = 2 and b = -1
=> | a + b | = | 2 + -1| = |1| = 1 (Watch  this video  to know about the  Basics of Absolute Value )
|a| + |b| = |2| + |-1| = 2 + 1 = 3 ≠ 1 =>   FALSE

II.  ⏐\frac{a}{b}⏐ = \frac{|a|}{|b|}  
Dividing two numbers and then taking their absolute value or dividing absolute values of these two numbers will result in the same answer
Ex: a = 2, b = -1
=>  ⏐\frac{2}{-1}⏐  = |-2| = 2
 \frac{|2|}{|-1|}  =  \frac{2}{1}  = 2 =>   TRUE

III.  ⏐ab⏐ = ⏐a⏐ * ⏐b⏐  
Multiplying two numbers and then taking their absolute value or multiplying absolute values of these two numbers will result in the same answer
Ex: a = 2, b = -1
=> |2*-1| = |-2| = 2
|2| *|-1| = 2*1 = 2 =>   TRUE

So,  Answer will be D 
Hope it helps!

Watch the following video to learn How to Solve Absolute Value Problems

https://www.youtube.com/watch?v=rBWfl4AmQDI

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For all real numbers a and b, which of the following must be true?

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For all real numbers a and b, which of the following must be true?

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