Algebra II please help!
posted by Erin on .
I just cannot seem to be able to grasp the concept of solving polynomial inequalities. Can someone please explain, step by step, how to solve them? Here's a problem I can't solve. Please use this as an example:
(x2)(x5)<0
I cannot thank you enough for helping me with this. I have a test on this tomorrow and this is the only section I don't get. Thank you for all the help!

First of all, sketch a graph of
y = (x2)(x5)
This will be a parabola because it is a quadratic, a x^2 + b x + c
for large negative x, y is big positive
it is zero at x = 2
then it is negative for a while
it is then zero at x = 5
then it goes big positive
so
negative between x = 2 and x = 5 
Thank you so much, Damon! You really helped me out! : )

I used to teach it with a slightly different approach because it worked well with multiple factors.
for example I will add another factor to your problem
(x2)(x5)(x+2) > 0
the xintercepts of the corresponding function would be x = 2,2, and 5
mark those on the xaxis
there are now 4 sections to my xaxis
1. all values below 2
2. all values between 2 and 2
3. all values between 2 and 5
4. all values greater than 5
now pick an arbitrary value in each section and test it. You don't have to actually evaluate it, just worry about + or  (matching the > or <)
1. let x=5  ()()() =  (does not work)
2. let x = 0 ()()(+) = + works
3. let x = 3 (+)()(+) =  no
4. let x = 6  (+)(+)(+) = + yes
so 2 < x < 2 OR x > 5
for quadratic inequalities, I would also use the method outlined by Damon, but for cubics and higher, the above method works very nicely.
Of course if the function does not factor it would be a mess anyway. 
Thank you, Reiny. My teacher taught me something very similar to your method and now I understand it. Thank you for your help! : )