Doubt on Solution to Inequalities

Hi Bunuel

Thanks a lot for those links and your wonderful contribution.

I realized that I am struggling specifically with the below:

For example, let us take the equation (x-1)(x-4)=0

So, when I look at the expression (x-1)(x-4), I know that it is a parabola passing through x=1 and x=4.

But, I don't know whether this is V-shaped parabola or an inverted V-shaped parabola. Currently, the way I am doing this is to substitute a value between the roots (for eg. x=2 in the above case) and if it is negative, I know that the curve is a V-shaped parabola. Hence the curve is +ve outside the points x=1 and x=4 and is -ve between the roots.

So, for the inequality (x-1)(x-4)>0 the solution is 4<x<1 (where is the curve +ve)
So, for the inequality (x-1)(x-4)<0 the solution is 1<x<4 (where is the curve -ve)

If you see the above, the crucial thing is to identify whether the curve is V-shaped or inverted V-shaped?

Currently, I am doing it using substitution method, and I feel it is prone to errors. So, I want to know is there any other way apart from substituting values to find out the shape of the curve?

I am extremely sorry for the repeated questions, but I really need some handholding on this topic. Kindly help me on this.