How do you solve inequalities when they involve quadratics? i.e x^2+x-6<0
It is easiest to first solve the quadratic as if it isn't an inequality. This will then help us to sketch the graph of the function which will in turn help us to solve the inequality. So let's change the < into a = and we have x2 + x -6 = 0. The easiest way to solve this is to factorise. To do this we need to find two numbers which add up to 1 and times together to get -6. This can come with practise but for this we have 3 and -2. Therefore we know that (x-2)(x+3)= 0. If this is true, then we know that either x -2 = 0 or x+3 = 0. These are normal linear equations so can be solved to get x = 2 or x = -3. We can now sketch the graph. We can mark our two solutions on the graph since we know when y=0, x=2 or x=-3 (so mark down (2,0) and (-3,0)). We also know our y- intercept is at y=6 (so mark on the graph (0,6)). Then we can join up these points using a parabola or U-shaped quadratic graph. Now back to the inequality. We need to know when our equation is less than zero. The easiest way to do this is to look at the graph. We know that the equation is equal to zero when x=2 or x=-3, so when is the equation less than zero? Well, if we think of the x axis as being the line where it is zero, we just need to see which part of the graph is below the axis. Clearly it is the part where our x value is between -3 and 2, therefore our answer is -3<x<2. If our inequality were the other way around, we could see that the graph is greater than the x axis when x<-3 and x>2.
