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:

Question

(x-2)(x-5)<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!

Answer 1

First of all, sketch a graph of
y = (x-2)(x-5)
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

Answer 2

Thank you so much, Damon! You really helped me out! : )

Answer 3

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

(x-2)(x-5)(x+2) > 0

the x-intercepts of the corresponding function would be x = -2,2, and 5

mark those on the x-axis

there are now 4 sections to my x-axis
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.

Answer 4

Thank you, Reiny. My teacher taught me something very similar to your method and now I understand it. Thank you for your help! : )