1. steve816

$x^3-x^2-72x >0$

2. steve816

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

3. jim_thompson5910

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

4. steve816

Yes

5. jim_thompson5910

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

6. jim_thompson5910

|dw:1440998528778:dw|

7. jim_thompson5910

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

8. jim_thompson5910

|dw:1440998579792:dw|

9. jim_thompson5910

|dw:1440998592421:dw|

10. jim_thompson5910

|dw:1440998604341:dw|

11. steve816

Okay

12. 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|

13. 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|

14. 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|

15. jim_thompson5910

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

16. 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|

17. 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|

18. steve816

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

19. 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

20. steve816

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

21. jim_thompson5910

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

22. jim_thompson5910

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

23. jim_thompson5910

did you get the same table?

24. steve816

Yup!

25. jim_thompson5910

ok great, just checking

26. 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

27. jim_thompson5910

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

28. jim_thompson5910

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

29. 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

30. steve816

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

31. jim_thompson5910

I'm glad it's all clicking now