Question Video: Solving Quadratic Inequalities in One Variable Algebraically

Find the values of 𝑥 that satisfy 𝑥² − 3𝑥 − 10 ⩽ 0.

04:38

Video Transcript

Find the values of 𝑥 that satisfy 𝑥 squared minus three 𝑥 minus ten is less than or equal to zero. So what we’re gonna do is, we’ll say let’s consider the equation 𝑦 equals 𝑥 squared minus three 𝑥 minus ten. So we’re basically putting all of this lot equal to our 𝑦-coordinate. Now this is a quadratic, and the coefficient of 𝑥 squared is one, so that’s positive. So we know that this would be a positive happy curve. We also know that the constant term on the end is negative ten, so 𝐶 is equal to negative ten. And that’s where it cuts the 𝑦-axis. And it cuts the 𝑥-axis when the 𝑦-coordinate is equal to zero. So since 𝑦 is equal to 𝑥 squared minus three 𝑥 minus ten, what we’re saying is it cuts the 𝑥-axis when 𝑥 squared minus three 𝑥 minus ten equals zero.

And that factors, so 𝑥 squared minus three 𝑥 minus ten factors to 𝑥 plus two times 𝑥 minus five. And now we’ve got it in the format. We’ve got something times something is equal to zero, so one of those things must be equal to zero in order to get a-a result of that product of zero. So either 𝑥 plus two equals zero or 𝑥 minus five equals zero. And that means that 𝑥 must be the equal to negative two to make that equal to zero or 𝑥 must be equal to five to make that equal to zero.

So now we’ve got enough information for us to be able to sketch the curve of 𝑦 equals 𝑥 squared minus three 𝑥 minus ten. Well it cuts the 𝑦-axis at negative ten, and it cuts the 𝑥-axis at negative two and positive five. So that’s maybe about here, negative two; and positive five is over here. Now it’s a quadratic, so that’ll be a symmetric parabola. So the axis of symmetry, because it’s gonna be midway between negative two and positive five, is gonna be sort of here somewhere. And the curve is gonna look something like that. Now as we’ve said at the beginning, 𝑦 is equal to all of this stuff, and what we’re trying to find is the 𝑥-values for which that is less than or equal to zero. So we’re looking on this particular graph for where 𝑦 is less than or equal to zero.

Well 𝑦 is equal to zero here and 𝑦 is equal to zero here, so negative two and negative five are the 𝑥-values that generate a 𝑦-coordinate of zero. And we’re also looking for the region for which 𝑦 is less than zero, so that’ll be everything in between. So that’s all the way round here. So in terms of the 𝑥-values that generate those 𝑦 coordinates, well 𝑥 is negative two, 𝑥 is five, and everything in between. They are the valid 𝑥-coordinates. And for the 𝑥-coordinates we’re not interested in, well look up here, you can see that the 𝑦-coordinate is greater than zero so we’re not interested in that. So in terms of the region we’re not interested in, it’s this region out to infinity here; and it’s not including negative two, but it’s this region out to a negative infinity over here.

So the 𝑥-values we’re looking for is to generate that 𝑦-coordinate of less than or equal to zero are negative two is less than or equal to 𝑥 is less than or equal to five. So that’s in inequality format. In interval format, the ends of the interval we’re looking for are negative two and five, and they are both included. So we need to put the square brackets around those. So that’s in interval format. And using set notation, we can say that we’ve got the set of 𝑥 such that 𝑥 is real where negative two is less than or equal to 𝑥 is less than or equal to five.

So the process that we went through there was we, first of all, we came up with an equation for 𝑦 equals some combination of 𝑥, some function of 𝑥, and then we worked out where that generated a value of zero. And then we were trying to think of, you know, okay we were looking for the function to be less than or equal to zero in this case, or it might be equal to zero or greater than zero in other cases. So you’re then doing those comparisons. Now the bit that’s really important that I was talking about at beginning in terms of your working out is to do this sketch. If you do the sketch, it’s really clear to see whether you’re looking for points above the 𝑥-axis or points below the 𝑥-axis. If you don’t do that, lots of people go through these questions and they-they find out these critical 𝑥-values, but then they just kinda guess at whether we’re going between the 𝑥-values or outside of the 𝑥-values. So this final sketch here is just really helpful in getting it nice and clear in your mind whether you’re looking for points for 𝑦-coordinates above the line or below that line, the 𝑥-axis.

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