# How do you solve and write the following in interval notation: (x + 3)(x – 1)(x – 5) < 0?

Jun 21, 2016

Either $x < - 3$ or $1 < x < 5$

#### Explanation:

If $\left(x + 3\right) \left(x - 1\right) \left(x - 5\right) < 0$ (i.e. the product is negative) then options are

(i) All three are negative i.e. $x + 3 < 0$ and $x - 1 < 0$ and $x - 5 < 0$ i.e. $x < - 3$ and $x < 1$ and $x < 5$. Hence $x < - 3$.

(ii) While $x + 3 > 0$ and $x - 1 > 0$, $x - 5 < 0$ i.e. $x \succ 3$ and $x > 1$ and $x < 5$. This is possible only if $1 < x < 5$.

(iii) While $x + 3 > 0$ and $x - 5 > 0$, $x - 1 < 0$ i.e. $x \succ 3$ and $x > 5$ and $x < 1$. This is just not possible.

(iv) While $x - 1 > 0$ and $x - 5 > 0$, $x + 3 < 0$ i.e. $x > 1$ and $x > 5$ and $x < - 3$. This is too not possible.

(v) It is also not possible to have all positive.

Hence the solution is either $x < - 3$ or $1 < x < 5$.

This is also apparent from the following graph.

graph{(x+3)(x-1)(x-5) [-10, 10, -40, 40]}