Solve the inequality x^2 – 9x – 22 ≥ 0

We are looking for what values of x make this statement true. To start we need to factorise the quadratic.We can do this in two ways, the quadratic formula, or just seeing what numbers work.We'll start with seeing what works:We need two numbers that multiply to make -22. and add to make -9. This will mean they help us solve the quadratic.Since -9 is negative, we know one of our two numbers is negative, whilst the other is positive. 22 is only divisible by 1, 2 and 11, so We can try -2 and 11, and 2 and -11-2 and 11 dont work as -2 + 11 is 9, -9.So we try -11 and 2, which does indeed work: -11 X 2 = -22, -11+2=-9. Why is this important? Because if we put these numbers in brackets like these: (x-11)(x+2) we get that they expand to make the quadratic.So x^2 – 9x – 22 = (x-11)(x+2) This means that when x-11 = 0 or x+2 = 0 the quadratic is also zero.So we can solve these to get x = 11 and x = -2 as the roots of the quadratic. This is very important as it gives us information about when the quadratic is above or below zero.
So to see when, as in inequality, x^2 – 9x – 22 ≥ 0 . We are looking for when it is greater than zero, which means it is positive.We will draw the quadratic on a graph to show ourselves where it is positive.
(a picture would be shown here)
We plot the roots, then we see it is a positive quadratic because the x^2 term is a + not a -.
Then simply, we look at where it is above the x axis. It will be either on: either side of the roots, in between them, for all numbers, or for none.
In this case, it is between the roots, from the graph. And that is our answer.
In this case it is inbetwee