# How do you solve and write the following in interval notation: x< 5 AND x< 3?

Mar 25, 2017

$x < 3$
$x \in \left(- \infty , 3\right)$

#### Explanation:

As can be seen in the diagram: only the values to the left of $3$ are included in
$\textcolor{w h i t e}{\text{XXX}} x < 3$ AND $x < 5$

In interval notation
$\textcolor{w h i t e}{\text{XXX}} \in$ means "in" or that $x$ is included in the interval that follows;

$\textcolor{w h i t e}{\text{XXX}}$rounded brackets mean that the value beside the bracket is not included in the set (but everything up to that value) is;

$\textcolor{w h i t e}{\text{XXX}}$square brackets mean that the value beside the bracket is included in the set (as well as everything up to that value.

Note that the "infinite" values, $- \infty$ and $+ \infty$ are always not included (always written with round brackets).

Some examples:
$\textcolor{w h i t e}{\text{XXX}} x \in \left[- 2 , + 7\right)$ would mean $- 2 \le x < 7$

$\textcolor{w h i t e}{\text{XXX}} x \in \left(4 , + \infty\right)$ would mean $4 < x$ (with no upper limit)

For the given case
$\textcolor{w h i t e}{\text{XXX}} x < 3$ implies $x < 5$
$\textcolor{w h i t e}{\text{XXX}}$so if both are included in th4e restrictions (AND) then only the $x < 3$ is required
so we would write the interval as:
$\textcolor{w h i t e}{\text{XXX}} x \in \left(- \infty , 3\right)$