2.  A quadratic inequality is an inequality in the form ax 2 +bx+c >, <, ≤, or ≥ 0.  In most cases, you just need to find the points where ax 2 +bx+c = 0 and use these points as the boundaries of your solution set.

3.  Unfortunately, you won’t always have two solutions to the equation ax 2 +bx+c = 0.  If you have only one solution, there are several possibilities.  If you have no solutions, the solution is either the empty set or all x.

4.  Let’s look at the case where ax 2 +bx+c = 0 at exactly one point.  There are several possibilities for solutions to these inequalities, depending on the type of inequality.  These situations are summarized on a table on the next page.

5.  This table assumes that a > 0. If a < 0, multiply both sides of your equation by -1 and flip the inequality.  Rather than memorizing this, try to understand why each one is the way it is. Type of InequalityType of Solution ax 2 +bx+c > 0Single exception ax 2 +bx+c < 0No solution ax 2 +bx+c ≥ 0All real numbers ax 2 +bx+c ≤ 0Single solution

6.  Say we’re trying to solve the inequality x 2 – 2x + 1 ≤ 0.  We start by finding the points where x 2 – 2x + 1 = 0, as usual. Factoring the equation, we have (x -1) 2 = 0.  This tells us that we have only one point where x 2 – 2x + 1 = 0: x = 1.  Since the x 2 term is positive, x will be greater than 0 at all other points.  This means that we have only one solution to our inequality: x = 1.

7. Note that x 2 – 2x + 1 = 0 at exactly one point and x 2 – 2x + 1 > 0 at all other points. If we had instead been solving x 2 – 2x + 1 < 0, our solution would have been the empty set. In general, graphing the equation makes it easier to tell what your solution set is. Just look for where the equation fulfills the inequality.

8.  If x is never equal to zero, there are two possibilities.  Either the inequality is true for all x, or the inequality is false for all x.  To figure out which type of situation you have, all you have to do is look at the type of inequality you have.

9.  First, make sure a > 0. If a < 0, multiply both sides of your equation by -1 and flip the inequality.  Afterwards, if you have an equality where ax 2 +bx+c 0 or ax 2 +bx+c ≥ 0, the solution to the inequality is all real numbers.

10.  Say we’re trying to solve the inequality -x 2 - 5 < 0.  Multiplying by -1, we can simplify this inequality to x 2 + 5 > 0.  Notice that x 2 +5 ≠ 0 for any real x.  This means that we can determine our solution set based on the type of inequality we’re working with.  We have a “greater than zero” inequality, so our solution set is all real numbers.

11. Here is a graph of x 2 + 5. Note that it’s greater than 0 for all x.