Solving Quadratic Inequalities in One Variable

This week in math we learned how to solve a quadratic inequality in one variable. When doing this in the end we have to make sure the statement stays true.

First to start we have a quadratic inequality as seen below:

Next we have to use the zero product law and make sure there is a 0 on the other side.  In this case this is already done for us.

Now we have to use our factoring rules and factor the following quadratic inequality.  This can be done by simply factoring, using the box, completing the square, or using the quadratic formula.  In this case it is a easy quadratic equation so we can easily factor it:

Now that we know our X intercepts are 2 and 7 we have to test three different numbers. Our first number we will test will be smaller than 2 in this case, the easiest number to choose is 0. Next we will pick a number between 2 and 7, in this example I picked 3 because smaller numbers are easier to deal with. Finally we will pick a number greater than 7 in this case 8, and replace X once more:

Now with the three separate solutions we have we need to create one big solution that stays true to the inequality.  We know that a negative is smaller than a positive so we need to use the signs appropriately while incorporating our X intercepts: