Quadratic inequalities

In a nutshell

Sometimes, an inequality will include a quadratic expression. A sketch of the quadratic can be used to help find the required set of solutions.

Solving a quadratic inequality

To solve a quadratic inequality, follow this procedure:

procedure

Example 1

Find the set of values of $$x$$ for which $$x^2-8x+12 \gt 0$$. Give your answer in set notation.

Set the quadratic equal to $$0$$:

$$x^2-8x+12=0$$​​

Solve using the quadratic formula to find the critical values:

$$x = \dfrac{8 \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)}$$​​

$$x=2,\ x=6$$​​

Sketch the function:

​

The question asks for values which make the quadratic greater than $$0$$, therefore take the values above the $$x$$-axis:

$$x\lt2\ or\ x\gt 6$$​​

In set notation:

$$\underline{\{x:x\lt2\}\ \cup\ \{x:x\gt6\}}$$​​

Example 2

Find the set of values of $$x$$ for which $$x^2-6x+5 \gt0$$ and $$2x-1 \gt 3$$. Give your answer in set notation.

Set the quadratic to equal $$0$$:

$$x^2-6x+5=0$$​​

Solve using the quadratic formula:

$$x = \dfrac{6 \pm \sqrt{(-6)^2 - 4(1)(5)}}{2(1)}$$​​

$$x=1,\ x=5$$​​

Sketch the function:

For the quadratic equation to be greater than $$0$$:

$$x \lt1 \ or\ x\gt5$$​​

Solve the linear inequality:

$$\begin{aligned}2x-1&\gt3\\2x&\gt4\\x&\gt2\end{aligned}$$​​

The two inequalities overlap when $$x \gt 5$$:

$$\underline{\{x:x\gt5\}}$$​​