# Inequalities/writing set in interval notation

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• Oct 4th 2009, 06:46 PM
A Beautiful Mind
Inequalities/writing set in interval notation
$(x-1)^2(x+2)^3\geq 0$

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 $\geq 0$.

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? $[-2, \infty)$

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

I'm also confused on this problem:

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

I already know what it turns into.

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

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

$1- x^2$

$(1 + x) (1 - x)
$

1 -x+x -x^2

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

$x = 0$ is supposed to be a critical number, but I don't know how you get that.
• Oct 4th 2009, 06:55 PM
Jameson
Yup looks good to me.
• Oct 4th 2009, 07:03 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, 07:11 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.