Polynomial InequalityDate: 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/ |
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