   Chapter 2.1, Problem 30E

Chapter
Section
Textbook Problem

For an integer x , the absolute value of x is denoted by | x | and is defined by | x | = { x                 if   0 ≤ x − x             if   x<0 Use this definition for the proofs in Exercises 28 − 30 .Prove that | x + y | ≤ | x | + | y | for all x and y in Z .

To determine

To prove: The inequality |x+y||x|+|y| for any integer x and y.

Explanation

Formula Used:

For any integer x the absolute value is denoted by |x| defined as

|x|={xifx0xifx<0

Proof:

Consider three cases for x:

Case I: If x>0 and xZ+.

|x|=x and |x|=x by the definition of absolute value.

Since x=x, x|x| is trivial.

Now as x>0, x+xZ+x(x)Z+ which means x(x)>0x<x which means

|x|x

Thus, the inequality |x|x|x| holds for this case.

Case II: x=0

Then |x|=0 and |x|=0. This means that |x|x|x| is obvious.

Case III: If x<0, then xZ+

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