Fluency exercise

## Solution

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 $x≥2$.

We will solve the given quadratic inequalities so they can be placed in the Venn diagram. We will start with the more straightforward ones.

$x^2≤9$

$11x≥2x^2$

Does it matter whether we move the terms to the left or the right hand side of the inequality? What would change if we solved $11x - 2x^2 ≥ 0$ instead?

$6x^2-1≥5x$

$3x^2≥21x-30$

Why can we divide both sides by $3$? How does the graph of $y = 3x^2 - 21x + 30$ compare to the graph of $y = x^2 - 7x +10$?

$x^2≤-x$

$-2x^2≤x-6$

$x^2+3≥2$

We didn’t try to solve this algebraically because of what we know about $x^2$. What would happen if we did try to solve it algebraically?

$x^2≤x-2$

What other graph might we have sketched to see there are no solutions?

Here are the given inequalities presented in the Venn Diagram.