Prove Inequality: |x^2-4x+3| < 3

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The discussion centers on proving the inequality |x^2 - 4x + 3| < 3 given the condition |x - 1| < 1. The proof establishes that if |x - 1| < 1, then x is confined to the interval (0, 2). This leads to the conclusion that |x - 3| < 3, and by applying the multiplication of inequalities, it confirms that |x^2 - 4x + 3| < 3 holds true. The proof is validated by the community, affirming its correctness.

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Homework Statement



If |x-1| < 1 then Prove |x^2 -4x + 3| < 3.


Homework Equations





The Attempt at a Solution



proof: Assume |x-1| < 1. Then X has to be between 0 and 2.Because X has to be between 0 and 2 then |x-3|<3,and |x-1||x-3|<3 by multiplication of inequalities.
|x-1||x-3|=|x^2 - 4x + 3| by distribution. Thus, |x^2 -4x + 3| < 3. (QED)

I was wondering if this is sufficient. I was a little unsure if what I did in the second sentence, and the start of the third was ok

Thank you.
 
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