Can someone mathematically explain this to me?
Okay, so my little sister is in the 9th grade and said that she had a really confusing problem on a math test. Apparently on the test, they asked her to solve the following inequality and sketch the graph of it:
X^3+4x^2-12x > 0
She told me to do it for her to see if she got the same answer and we did, we ended up both doing this:
Factor out x. -----> x(x^2+4x+12) > 0
Factor completely. ------> x(x+6)(x-2) > 0
When I graphed it, the part of the graph that was over the x-axis was between -6 and 0 and approached infinity after crossing through the x-axis at (2,0).
However, I'm confused. I wrote -6<x<0 or x>2 as an answer based on where the function was higher than the x-axis. But it didn't make sense to me that it mathematically worked when I solved the inequality for (x+6)(x-2) and then the lone x on the outside says x>0 when my answer -6<x<0 doesn't fit that. Yet when you see it visually the function passes through the origin from quadrant 2 into quadrant 4? How does this work? I was finished with all of math before graduating and couldn't even understand how that worked out. Thanks for the help.
14 Comments • Newest first
What you found are the points where the function is equal to zero. At x=-6, x=0, and x=2, the function is equal to zero. To visualize this case, no other point on the graph f(x) gives a y=0 other than these three x values, and so you can say that the interval between them are guaranteed to be either positive or negative.
Make a sign table. You will have the intervals (-inf,-6), (-6,0), (0,2), (2,+inf). Plug in a value in one of these intervals into f(x) to determine if it will be positive or negative. Taking (-inf,-6) for example, I choose x=-10. If I plug it into the factored inequality, I'll get a negative # times negative # times another negative #. This equals a negative number. And so, in the interval (-inf,-6), f(x) is negative. Do the same thing for the other three intervals and you'll get something like this.
http://imgur.com/QX6EwQB
Sorry for sloppy handwriting. Also you made a typo when factoring out the x: x(x^2+4x-12) > 0 should be the right eq.
Edit: Nevermind
How did you graph it? Did you simply plug in integers into the function or did you do first derivative and second derivative tests to find concavities, infliction points and increase/decreases?
@simfel:
Now this works.
http://www.wolframalpha.com/input/?i=x^3%2B4x^2-12x+%3E+0
@yongyong139: This is actually fairly typical for high school students; a lot of them don't realize it, but most pre-university mathematics is highly cumulative in relation to lower-level courses (e.g. pre-calculus in high school is essentially algebra II with a few new tricks for computations).
@burning: try with all lower-case x's. it's interpreting your cubic term as a separate variable.
@yongyong139: This is standard math for 9th grade though. Some kids even take this in middle school.
This is twelfth grade advanced functions where I'm at. Is your sister ap or something?
I threw the formula through Wolfram and it spit out a wacky graph with a semicircle and hyperbolic arcs. It definitely doesn't follow the expected shape of a cubic function.
https://www.wolframalpha.com/input/?i=X^3%2B4x^2-12x+%3E+0
I could barely comprehend graphs went there were shown to me in real life...trying to understand a written description on Basil is freaking impossible.
God, I hate graphs.
math gives me cancer
the factor out part looks misspelled so that could of been counted as a wrong part
https://www.desmos.com/calculator
when x = 1, the equation thing = -7
dunno
I'm almost completely willing to bet that the question was asking for interval notation as opposed to set notation. In most cases, interval notation is used largely because it's far more intuitive and harder to screw up on accident.
As mentioned before, (-6,0)u(2,infinity) or in other words, greater than -6 to less than 0; union; greater than 2 to less than infinity (or just flat out infinity).
The answer should be (-6,0)u(2,infinity) I'm not sure why you're not getting it correct perhaps they want you to answer it in a specific notation.