Basic Properties of Numbers #2

Brief Explanation: Another basic exercise about absolute value proofs. Quite useful when we get to epsilon-delta proofs. (Exercise taken from Michael Spivak’s Calculus)

The reason why I chose this exercise is to show you a ‘trick’: because sometimes that’s all it takes to solve the exercise. With practice I hope you (and me) get a feel for this kind of situations.

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Question: Prove the following:

Condição Inicial.png

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Comprehending: Like I said before, this post wont be too long because I just want to show you the ‘trick’ required to solve this problem.

First, remember the basic inequality about absolute value:

Desigualdade moduo.png

These two inequalities are quite similar but with different signs. How do we get from one to another?

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Solving:

The solution here lies in being ‘smart’ about the absolute value inequality. How do we get an inequality with the minus sign inside? The answer lies in the following trick:

1.png

This is true, isn’t it? If you add and subtract a number you get the original number! Now that is an expression we can work with. If we apply the absolute value inequality:2.png

(this step was just to make it easier to understand)

3.png

Now, joining the two different conditions we have we get the proof:

4.png

5.png

6.png

And it’s done! Quite easy, right? Once you get the trick, it becomes quite simple.

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One thought on “Basic Properties of Numbers #2

  1. Pingback: The Basic Properties of Frogs #3.145 – Algebraic Can of Worms (A-CoW)

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