Polynomial Inequality

Date: 06/23/2004 at 16:14:02
From: Lisa
Subject: solve and simplify

I'm trying to solve the inequality (x^2 + x - 12)/(x + 1) >= 0 
(>= means greater than or equal to)

I think I can factor the trinomial to get (x - 3)(x + 4)/(x + 1) >= 0.
Is that correct, and if so, how do I continue solving?

Date: 06/24/2004 at 17:23:29
From: Doctor Ian
Subject: Re: solve and simplify

Hi Lisa,

You've factored the quadratic correctly.  The next thing to do is 
think about what would have to be true in order for 

  (x - 3)(x + 4)
  --------------  >  0
     (x + 1)      -


to be true.  There are several common ways that you can approach this
step, but I'll show you the one I like the best because it really
helps you understand how those three binomial factors all play a part
in the final answer.

If you think about it, all you really need to pay attention to are the
signs of the three factors.  Note that (x + 1) is 0 at x = -1,
positive whenever x is greater than -1, and negative when it's less
than that:

         <--|---|---|---|---|---|---|---|---|---|---|---|-->
           -5  -4  -3  -2  -1   0   1   2   3   4   5   6

  (x + 1)  -----------------0+++++++++++++++++++++++++++++++

What about (x - 3)?  It's 0 at x = 3, positive when x is greater than
3, and negative when x is less than 3:

         <--|---|---|---|---|---|---|---|---|---|---|---|-->
           -5  -4  -3  -2  -1   0   1   2   3   4   5   6

  (x - 3)  ---------------------------------0++++++++++++++
 
So suppose we were just looking at the quotient of those two terms:


         <--|---|---|---|---|---|---|---|---|---|---|---|-->
           -5  -4  -3  -2  -1   0   1   2   3   4   5   6

  (x - 3)  ---------------------------------0++++++++++++++
  (x + 1)  -----------------0++++++++++++++++++++++++++++++


  (x - 3)
  -------  +++++++++++++++++U---------------0++++++++++++++
  (x + 1)

             negative          negative          positive  
             divided by        divided by        divided by
             negative is       positive is       positive is
             positive          negative          positive
          

The 'U' at x = -1 means 'undefined', since dividing by (x + 1) when x
is equal to -1 means dividing by zero.  The 0 at x = 3 means that the
quotient is 0 there since 0 divided by a positive number will be 0. 
Everywhere else on the number line, the quotient is positive or
negative as determined by the signs of the two factors, as shown.

Can you see how to work the remaining factor, (x + 4), into this? 
Once you do, remember that since your original inequality was 'greater
than or equal to zero,' your final answer should include all parts of
the number line where the overall expression is positive or equals zero.

Good luck!  Write back if you are still confused about any part of
this or if I can help further.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/