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

[EV33]

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.

 

[HallsofIvy]

Yes, that is exactly right!