solve the inequality
2/(x^2+6x+9) less than or equal to 1/(x^2-9)

Question

solve the inequality 
2/(x^2+6x+9) less than or equal to 1/(x^2-9)

anonymous · Answer

I got -3,9 too. But when I try 0 in the original equation, it doesn't work?

anonymous · Answer

Sorry. You have to take into account that you multiplied by x^2-9 and x^2+6x+9 in the beginning. x^2-9 is negative on (-3,3) so, the inequality sign changes. x^2+6x+9=(x+3)^2 is always positive, so it never changes sign.

This means we get |x-3| >= 6 on the interval (-3,3). This gives you no solutions, as |x-3| < 6 on (-3,3).
On the remaining domain (-INF, -3) U (3, INF) we get |x-3| <= 6. Note that you have to forget about -3 and 3 because they are zeros of the denominator polynomials, so the inequality is not even defined there.
Getting back to the problem: We know that |x-3| <= 6 has solutions in [-3, 9]. So by intersecting with (-INF, -3) U (3,INF) we get the final answer: (3, 9].