Solve 𝑥 squared minus 𝑥 minus six is less than zero.
So, in order to solve this inequality, what we want to do first is find some critical values. And in order to do that, what I’m gonna do is set our inequality to an equation. So, what I’m gonna do is I’m gonna equate 𝑥 squared minus 𝑥 minus six to zero. So, what we need to do here is solve this quadratic. And what this is gonna give us is the two points where our quadratic crosses the 𝑥-axis.
And the way to solve this quadratic is by using factoring. When we do that, we get 𝑥 minus three multiplied by 𝑥 plus two is equal to zero. And that’s because if we look at the two numbers that need to multiply together to give us negative six, then they’re gonna be negative three and positive two. We could’ve had some others, so we could’ve had negative six multiplied by one, or six multiplied by negative one, or even negative two multiplied by three.
However, they need to sum to negative one. And that’s cause negative one is the coefficient of our 𝑥 term. Well, negative six add one is negative five. Six add negative one is five. Or negative two add three is one. So, none of these give negative one as a result. So therefore, the correct factors are the ones we found, which were negative three and two. And that’s because if you add two to negative three, you get negative one.
Okay, so, we factor it now, which means that we can say that our critical value is gonna be 𝑥 equals three or 𝑥 equals negative two. And we got these by setting each of the parentheses equal to zero. And that’s because we need one of the parentheses to be equal zero because zero multiplied by anything gives the result of zero, which is what we want. An example of this, I’ve shown, is the right-hand parentheses because we got 𝑥 plus two is equal to zero. Then, subtract two from each side. We get 𝑥 is equal to negative two, which is what we had.
Okay, great, we have our critical values. But what next? How do we solve our inequality? Well, if we sketch the quadratic, then what we’ve got is we’ve got points at negative two and three where it intersects the 𝑥-axis because this is our solutions, or our critical values. And we know the graph is a U-shaped parabola because our 𝑥 squared term is positive. And if it’s positive, then it’s a U-shaped parabola. If it’s negative, then it’s an n or inverted U-shaped parabola.
Now, to solve the inequality, we need to look at the inequality sign. And what we’re interested in is when 𝑥 squared minus 𝑥 minus six is less than zero. So therefore, we’re interested in this region here, which is the region which is below the 𝑥-axis, so where 𝑦 is less than zero. So therefore, we can say that the inequality that would represent this is that 𝑥 is greater than negative two but less than three.
And if we want to show this using interval notation, what we’ll have is open parentheses negative two then comma three close parentheses. And we use parentheses because these show that the numbers negative two and three are not included. And that’s cause 𝑥 is greater than negative two and less than three. If it was including the values, so we had 𝑥 was greater than or equal to negative two or less than or equal to three, then instead of the parentheses what we would use is brackets.