Fluency exercise

## Suggestion

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.