# Inequality Proofs

Discussion in 'Algebra' started by nycmathguy, May 29, 2022.

1. ### nycmathguy

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nycmathguy, May 29, 2022

2. ### MathLover1

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MathLover1, May 29, 2022
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3. ### nycmathguy

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nycmathguy, May 29, 2022
4. ### MathLover1

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62.
show that for all real numbers a and b, we have |a| - |b| <= |a-b|

|a|+|b| ≥|a−b|

Now, substitute a=a and b=a−b to get,

|a−b|≥|b|−|a| ......................(1)

Also, we can writ |a|+|b|≥|b−a| as |a−b|=|−(a−b)|=|b−a|

Now, substitute a=b−a and b=b to get, |b−a| ≥ |a|−|b|....................(2)

Now, as |b−a|=|a−b|

We may write (2) as |a−b|≥|a|−|b|.............(3)

because |a−b|=|−(a−b)|=|b−a|

From (1) and (3), we can conclude that

|a−b| ≥ ||a|−|b|| or vice verse |a|−|b|| ≤|a−b|

MathLover1, May 29, 2022
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