Solve the inequality 𝑥 squared minus 𝑥 minus 12 is less than zero.
This is a quadratic inequality because we want to find when a quadratic expression is less than zero. And the first thing to do when solving a quadratic inequality is to sketch a graph of the quadratic function. There are two features of the graph, which are going to help us solve this inequality: firstly the 𝑥-intercept of the graph, that is, where the graph crosses the 𝑥-axis, and the direction that the graph passes through these 𝑥-intercepts.
First, let’s find the 𝑥-intercepts. The 𝑥-intercepts of the graph of 𝑦 equals 𝑓 of 𝑥 are the values of 𝑥 for which 𝑓 of 𝑥 is equal to zero. In our case, 𝑓 of 𝑥, the function that we’re graphing, is 𝑥 squared minus 𝑥 minus 12. We can factor the left-hand side by inspection. And we find that the two solutions are 𝑥 equals four and 𝑥 equals negative three. And these are the 𝑥-intercepts of the graph. So we mark them on the 𝑥-axis.
We know that the graph passes through these two points. But in which direction does it do so? Our graph is a quadratic graph. And so it’s a parabola in shape. And there are basically two possibilities for how this parabola passes through these two points. It’s either an upward facing parabola or a downward facing one.
How do we decide between these two possibilities? We look at the 𝑥 squared term and see that its coefficient is one, which is positive. And so we have an upward facing parabola. So this is a sketch of our graph. We could add other features to it like the 𝑦-intercept or the vertex of the graph. But we’ll see that we don’t need to.
Let’s turn our attention back to the inequality. So how does this graph help us solve the inequality 𝑥 squared minus 𝑥 minus 12 is less than zero? Well, 𝑓 of 𝑥 is less than zero for the values of 𝑥 where the graph of 𝑦 equals 𝑓 of 𝑥 lies below the 𝑥-axis. And we can see now why we chose those two important features. They are the features that allow us to determine precisely when the graph lies below the 𝑥-axis.
We highlight the part of the graph which lies below the 𝑥-axis. And we ask ourselves for what values of 𝑥 is this graph below the 𝑥-axis. Because we found and marked the 𝑥-intercepts on the graph, we can just read this off. The graph of 𝑓 of 𝑥 is below the 𝑥-axis when 𝑥 is between negative three and four. And we’re using less than signs here and not less than or equal to signs because the end points negative three and four are not solutions of the inequality and so should not be included.
Looking at the graph, we can see that 𝑓 of negative three is equal to zero and not less than zero. And the same is true for 𝑓 of four. This is how you write the solution in inequality notation. But we can also write this in interval notation.
We write that 𝑥 is an element of the interval from negative three to four. And we’re using parentheses and not square brackets here for the same reason we use less than signs and not less than or equal to signs in the inequality notation to show that the end points negative three and four are not included in this interval.