Inequalities/writing set in interval notation

• Oct 4th 2009, 05:46 PM
A Beautiful Mind
Inequalities/writing set in interval notation
$\displaystyle (x-1)^2(x+2)^3\geq 0$

So far I've gathered that the critical numbers are x = 1 and x= -2. I graphed them on the number line. I plugged in test values for each of them.

When I do that, for every test value I've plugged in they've been $\displaystyle \geq 0$.

Now I don't know what that means. Does that mean that since every test proved to be a positive, it'll be the whole thing? $\displaystyle [-2, \infty)$

The test intervals were $\displaystyle (-\infty, -2), (-2, 1), (1, \infty)$.

I'm also confused on this problem:

$\displaystyle \frac{1}{x} - x > 0$

I already know what it turns into.

$\displaystyle \frac{1-x^2}{x}$

That wasn't the hard part. The hard part is figuring out the critical numbers for this.

$\displaystyle 1- x^2$

$\displaystyle (1 + x) (1 - x)$
1 -x+x -x^2

$\displaystyle x+1=0$
$\displaystyle x = - 1$ and $\displaystyle x = 1.$

$\displaystyle x = 0$ is supposed to be a critical number, but I don't know how you get that.
• Oct 4th 2009, 05:55 PM
Jameson
Yup looks good to me.
• Oct 4th 2009, 06:03 PM
A Beautiful Mind
Thanks, I updated it with a second problem. I got 2 of the critical numbers. I dunno how you get zero cause in the back of the book it said 0 was a critical number. Only one I'm missing.
• Oct 4th 2009, 06:11 PM
Jameson
Critical numbers are where the either the numerator or the denominator equal 0. So x=0 for this problem sets the denominator to 0 and could possibly change the direction.

Example: f(x)=1/x

At x=0 there is obviously an asymptote but the direction of the graph is opposite from each side.