# Thread: Question regarding polynomial inequalities

1. ## Question regarding polynomial inequalities

Hi In my calculus class today, we had a lesson on solving polynomial inequalities. However, after my classmates and I got the answers to the homework wrong and I checked the internet, I suspect my teacher taught us wrong. Before I do anything about it though, I want to check for certain whether or not the method he gave us is wrong.

He told us to factor the polynomial, then treat each one as a separate linear function, and just bring it over across the sign (obviously flipping it if you multiply or divide by a negative).

For example, here's what he showed us to do for this question:

-x^2 + 4x - 4 > 0

-(x-2)^2 > 0

(x-2)^2 < 0

x-2 < 0

Two identical solutions: x < 2

However, I found the correct answer on the internet and in the text to be x = 2. This is just one of many where the answer I got using his method was wrong. Is what he did correct, or should it be done as these sites say:
Solving Polynomial Inequalities
Solving Polynomial Inequalities Analytically

Thanks so much to anyone who takes the time to read this and help

EDIT: It should read greater than or equal to.

2. Originally Posted by starswept
-x^2 + 4x - 4 > 0

-(x-2)^2 > 0

(x-2)^2 < 0

x-2 < 0

Two identical solutions: x < 2
Actually

$\displaystyle (x-2)^2<0\implies|x-2|<0$

3. Originally Posted by starswept
Hi In my calculus class today, we had a lesson on solving polynomial inequalities. However, after my classmates and I got the answers to the homework wrong and I checked the internet, I suspect my teacher taught us wrong. Before I do anything about it though, I want to check for certain whether or not the method he gave us is wrong.

He told us to factor the polynomial, then treat each one as a separate linear function, and just bring it over across the sign (obviously flipping it if you multiply or divide by a negative).

For example, here's what he showed us to do for this question:

-x^2 + 4x - 4 > 0

-(x-2)^2 > 0

(x-2)^2 < 0

x-2 < 0

Two identical solutions: x < 2

However, I found the correct answer on the internet and in the text to be x = 2. This is just one of many where the answer I got using his method was wrong. Is what he did correct, or should it be done as these sites say:
Solving Polynomial Inequalities
Solving Polynomial Inequalities Analytically

Thanks so much to anyone who takes the time to read this and help
this question has no solution. we need the polynomial to be greater than or equal to 0. if it is just greater than, there is no solution

as Krizalid pointed out, you would end up with |x - 2|<0 and there is no solution to that equation, since |x - 2| >= 0 for all x

if the original question was really -x^2 + 4x - 4 >= 0 then the solution is x = 2 as your text says as the inequality |x - 2| =< 0 has x = 2 as the only solution

4. I'm sorry, my mistake. The polynomial should read greater than or equal to, but I guess I didn't paste the symbol properly.

5. Originally Posted by starswept
The polynomial should read greater than or equal to, but I guess I didn't paste the symbol properly.
Same answer expect you have $\displaystyle \geq$