# How do you solve the inequality: (x - 3) (x - 5) > 0?

Aug 28, 2015

$x \in \left(- \infty , 3\right) \cup \left(5 , + \infty\right)$

#### Explanation:

In order for this inequality to be true, you need $\left(x - 3\right)$ and $\left(x - 5\right)$ to either both be positive or both be negative.

For any value of $x > 5$ you will get

$\left\{\begin{matrix}x - 3 > 0 \\ x - 5 > 0\end{matrix}\right. \implies \left(x - 3\right) \left(x - 5\right) > 0$

For any value of $x < 3$ you will get

$\left\{\begin{matrix}x - 3 < 0 \\ x - 5 < 0\end{matrix}\right. \implies \left(x - 3\right) \left(x - 5\right) > 0$

This inequality will thus be satisfied for any value of $x \in \left(- \infty , 3\right) \cup \left(5 , + \infty\right)$.

On the other hand, any value of $x \in \left[3 , 5\right]$ will not be a valid solution.