   Chapter 2.1, Problem 10TFE

Chapter
Section
Textbook Problem

True or FalseLabel each of the following statements as either true or false. | | x | − | y | | ≤ | x − y | for all x and y in Z .

To determine

Whether the statement, “||x||y|||xy| for all x and y in Z” is true or false.

Explanation

Consider four cases to check the validity of the inequality.

Case I: When x>0,y>0.

If x>0,y>0 then |x|=x and |y|=y. Then,

||x||y||=|xy|

Case II: When x>0,y<0.

If x>0,y<0 then |x|=x and |y|=y. Then,

||x||y||=|x+y|

Case III: When x<0,y<0.

If x<0,y<0 then |x|=x and |y|=y

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