Question

What are the solutions to the inequality (x-3)(x+5) <0?

Answer 1

You want a product to be negative. A product is negative if and only if the two factors have opposite signs.

So, there are two cases: either $x-3$ is positive and $x+5$ is negative, or vice versa. But since $x-3$ is less than $x+5$, the only possibility is that

$\begin{cases}x-30\end{cases} \iff \begin{cases}x-5\end{cases}$

So, the solution is given by all numbers $x$ such that

$-5$