Question

How do you solve the inequality `6x^2-5x>6`?

Answer 1

I would start by solving it as a normal equation (2nd degree). So you write:
`6x^2-5x-6>0` (I took the `6` to the left);
Solving your equation you should get two values `x_1=3/2` and `x_2=-2/3`.
Now the tricky bit:
your inequality asks you for values of `x` that make your equation have a value bigger than zero.
They cannot be `x_1 and x_2` because at these point your equation IS equal to zero.
Graphically your function gives the following parabola:

graph{6x^2-5x-6 [-11.96, 13.02, -7.14, 5.34]}

So, basically I have to choose values that are outside the boundaries formed by the two values `x_1 and x_2` to get a value bigger than zero (in the graph the two bits that are ABOVE the `x` axis).!!!
Consider `x_1=3/2=1.5` ok I cannot choose it but what about `2` (which is bigger)?
If I put `x=2` in the equation I get: `6*4-5*2-6=8>0` YES!
Consider now `x_2=-2/3=-0.67` again I cannot choose it but what about `-1`?
If I put `x=-1` in the equation I get: `6*(-1)^2-5*(-1)-6=5>0` YES!
So, OUTSIDE the interval bound by `x_1 and x_2` you can choose `x`.
You express this by writing your solution as:
`-2/3>x>3/2`
Or graphically

Hope it helps