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vodka [1.7K]
3 years ago
5

For x, y ∈ R, prove that ||x| − |y|| ≤ |x − y|, (Hint: consider x = y + (x − y)).​

Mathematics
1 answer:
dlinn [17]3 years ago
7 0

Using the triangular inequality (and the given hint!), we have

|x|=|(x-y)+y|\leq|x-y|+|y|\implies|x|-|y|\leq|x-y|

Similarly,

|y|=|(y-x)+x|\leq|x-y|+|x|\implies|y|-|x|\leq|x-y|\implies -|x-y| \leq |x|-|y|

We managed to bound the quantities in this fashion:

-b\leq a\leq b \implies |a|

And thus we have

||x|-|y||\leq |x-y|

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Option C:

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Solution:

Given equation:

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In the chart, x values are given as –2, 0, 2.

To find the value of y for the corresponding x given.

Substitute x  = –2 in the equation of y.

y = |3(–2) + 1|

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Substitute x  = 0 in the equation of y.

y = |3(0) + 1|

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y = 1

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y = |3(2) + 1|

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y = 7

Option C is the correct answer.

The values of y are 5, 1, 7.

The answer image of the table is attached below.

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