Posted by **Erin** on Wednesday, December 10, 2008 at 4:13pm.

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:

(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!

- Algebra II- please help! -
**Damon**, Wednesday, December 10, 2008 at 4:26pm
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

- Algebra II- please help! -
**Reiny**, Wednesday, December 10, 2008 at 5:17pm
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.

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