Can we find a quadratic inequality for each region of the Venn diagram?

The regions are defined as follows.

A: The solution set is a subset of \(x≤1\).

B: The solutions are given by \(a≤x≤b\) where \(a\) and \(b\) are real numbers.

C: The inequality is satisfied by \(x=4\), e.g. \(x=4\) satisfies the inequality \(x≥2\).

If our inequality needs to be satisfied by \(x=4\) and \(a≤x≤b\), what values for \(a\) and \(b\) could we choose?

If \(x=4\) satisfies the inequality, but the inequality cannot be written in the form \(a≤x≤b\), what form will it take?

Some of the regions of the Venn diagram might be empty. Can you find a quadratic inequality to fit this region? If not, can you explain why it is impossible to fill this region. It may be useful to try and sketch a graph of the situation.