ascii wrote:After you've found the critical points to your inequality, do you need to test every interval created, or can you only test one?

You can test none, if you like!

Instead of testing, you can use what you know about quadratics and their graphed parabolas. For instance, suppose you had an inequality that looked something like "0 < x

^{2} + ..." and you'd found zeroes at -2 and 2. You'd note that this is asking for where the quadratic is positive, which means it's asking for where the parabola is above the x-axis. You'd note that this quadratic has a positive leading coefficient, which means that the parabola opens upward so it's "up" on the "ends". You'd put these together to realize that the graph is above the x-axis on the ends and below in the middle, which would tell you that the solution intervals are the intervals on the ends: (-infinity, -2) and (2, +infinity).

No testing required!