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

In summary, the inequality |x^2-4x+3| < 3 means that the absolute value of the expression x^2-4x+3 is less than 3. To prove this inequality, we can use the triangle inequality property and simplify the expression inside the absolute value signs. The steps for solving the inequality include rewriting it, simplifying, and proving its truth. This inequality can have infinitely many solutions and can be applied in real life situations such as in physics, engineering, and mathematics.
  • #1
EV33
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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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  • #2
Yes, that is exactly right!
 

1. What does the inequality |x^2-4x+3| < 3 mean?

The inequality |x^2-4x+3| < 3 means that the absolute value of the expression x^2-4x+3 is less than 3.

2. How do you prove the inequality |x^2-4x+3| < 3?

To prove the inequality |x^2-4x+3| < 3, we can use the triangle inequality property. This states that for any real numbers a and b, |a+b| ≤ |a| + |b|. Therefore, we can rewrite the inequality as |x^2-4x+3 - 0| < |3 - 0|. Simplifying, we get |x^2-4x+3| < 3, which is the original inequality. This proves that the inequality is true.

3. What are the steps for solving the inequality |x^2-4x+3| < 3?

The steps for solving the inequality |x^2-4x+3| < 3 are as follows:

  1. Rewrite the inequality as |x^2-4x+3 - 0| < |3 - 0|.
  2. Simplify the expression inside the absolute value signs.
  3. Apply the triangle inequality property.
  4. Simplify the resulting inequality to the original inequality.
  5. Prove that the inequality is true.

4. Can the inequality |x^2-4x+3| < 3 have more than one solution?

Yes, the inequality |x^2-4x+3| < 3 can have more than one solution. In fact, it has infinitely many solutions. This is because the absolute value of any number can be positive or negative, so there can be many values of x that satisfy the inequality.

5. How can this inequality be applied in real life situations?

This inequality can be applied in many real life situations, such as in physics and engineering. For example, it can be used to determine the maximum and minimum values of a function in a given interval. It can also be used to calculate error bounds in measurements and experiments. Additionally, it can be used to prove the convergence or divergence of infinite series in mathematics.

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