- steve816

Please help me solve the inequality!

- jamiebookeater

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- steve816

\[x^3-x^2-72x >0\]

- steve816

I solved for x and got x = 0, 9, -8
Not sure what to do from there.

- jim_thompson5910

so you solved x^3 - x^2 - 72x = 0 for x and you got x = 0, x = 9, x = -8 ??

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## More answers

- steve816

Yes

- jim_thompson5910

ok first draw a number line and plot those three values on it
|dw:1440998491371:dw|

- jim_thompson5910

|dw:1440998528778:dw|

- jim_thompson5910

label the different regions A,B,C,D
|dw:1440998552769:dw|

- jim_thompson5910

|dw:1440998579792:dw|

- jim_thompson5910

|dw:1440998592421:dw|

- jim_thompson5910

|dw:1440998604341:dw|

- steve816

Okay

- jim_thompson5910

x^3 - x^2 - 72x factors to x(x-9)(x+8)
then set up a sign chart where the factors are written in a column on the left side
the regions are written along the top row like this
|dw:1440998659579:dw|

- jim_thompson5910

now let's pick a number from region A
this could be any number you want as long as it's less than -8. So let's pick -9
if x = -9, then the sign of x is negative. We put a negative sign/symbol in the first row, first column like this
|dw:1440998809112:dw|

- jim_thompson5910

keep x = -9 the same
what is the value of x-9? that is
x-9 = -9-9 = -18 which is also negative
so place a negative symbol in the "A" column and "x-9" row
|dw:1440998838462:dw|

- jim_thompson5910

if x = -9, then x+8 = -9+8 = -1 which is also negative
|dw:1440998879510:dw|

- jim_thompson5910

I'm going to let y = x^3 - x^2 - 72x
and write "y" in the bottom of the left most column
|dw:1440998908837:dw|

- jim_thompson5910

to find the sign of y for column A (is y positive or negative?) we multiply the signs in column A
negative*negative*negative = positive*negative = negative
so ultimately, y is negative when x < -8, ie in region A
|dw:1440998977212:dw|

- steve816

So I did all the work and got
x <- 8
-8 < x < 0
x > 9
Is this right?

- jim_thompson5910

does the sign chart make sense? if so, then go ahead and fill out the rest of it so I can see what you got

- steve816

I figured out how to use the chart and made sense to me. So I got the answers that I wrote above.

- jim_thompson5910

you should have this sign chart
|dw:1441000190738:dw|

- jim_thompson5910

take a few moments to check what I got to make sure you got the same things

- jim_thompson5910

did you get the same table?

- steve816

Yup!

- jim_thompson5910

ok great, just checking

- jim_thompson5910

notice how we want to solve for x in `x^3 - x^2 - 72x > 0`
which is the same as `y > 0` since I made `y=x^3 - x^2 - 72x`

- jim_thompson5910

look in the chart when y > 0
ie when there's a + sign in the y row
|dw:1441001180490:dw|

- jim_thompson5910

that corresponds to the intervals `-8 < x < 0` and `x > 9`
|dw:1441001210042:dw|

- jim_thompson5910

so the solution to `x^3 - x^2 - 72x > 0` is
`-8 < x < 0` OR `x > 9`
we write \[\Large (-8, 0) \cup (9, \infty)\] if we wanted to state the entire solution set in interval notation
the graph confirms all this
https://www.desmos.com/calculator/6qe6nf0jao

- steve816

Wow just wow *clap clap clap*
Absolutely brilliant!
Thanks for your time and help man, really appreciate it.

- jim_thompson5910

I'm glad it's all clicking now

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