Interval Notation - What is the purpose? I understand that you can write an inequality:
$$
x \leq 3
$$
as
$$
x \in (-\infty,  3 ]
$$
in interval notation, but, I don't understand the practicality of doing this. When is this used? How does it help?
 A: In general, the interval notation is cleaner to use. E.g. $x \in [2,3] \cup [5,10]$ is slightly more concise than the statement: "$2 \leq x \leq 3 \text{ or } 5 \leq x \leq 10$". As other answers will most likely point out, the interval notation seems closer to set theoretic notation.
A: As the commenters have mentioned, $(-\infty, 3]$ is a set; it says nothing about any variable until you declare $x \in (-\infty, 3]$ or perhaps $x \notin (-\infty, 3]$.
This notation can be much more easy to read and write sometimes, but more importantly the notation stresses that $(-\infty, 3]$ is, indeed, a set, and that its nature as a set may matter.
For instance, the domain of a function may be $(-\infty, 3]$, and when you construct topological arguments about that function, you often want to be dealing with a subset of the extended real numbers.
Similarly, the $[a,b]$ means the set of all numbers between $a$ and $b$ inclusive; if you have a function acting on this set, it is much better to write $f : [a,b] \to \Bbb R$ rather than $f : \Bbb R \to \Bbb R,\ a \le x \le b$ (the latter notation is actually awful; there is nothing about $x$ in the function statement, and so it is unclear when it need not be so).
This notation also puts useful topological properties on the outside of the notation, rather than somewhere in between. $(a,b)$ is an open set, $[a,b]$ is closed. This helps reinforce the notions of openness or closedness. The other way to write this set would be $\{ x \mid a < x < b \}$ or $\{ x \mid a \le x \le b\}$. It is easier to overlook that the set is opened or closed with the set notation.
This notation also has uses in other areas: defining the support of a random variable; defining the region of convergence of a series or algorithm; defining stability intervals for parameters; etc.
A: You can express the fact that $x\leq 3$ by saying that $x\in(-\infty,3]$. What are the advantages of the latter method?
One reason behind using the $\in$ symbol and intervals is that in modern mathematics we like to express mathematical objects and relations between them in the language of set theory.
Other reason comes in some sense from the first one: expressing inequalities in terms of the set theoretic notions make operating on them easier. For example instead of saying that $x$ satisfies inequality A, inequality B, and so on you can just take intersections of sets associated to those inequalities. So the connective "and" between inequalities is replaced with the intersection of sets. The connective "or" between inequalities is replaced with taking the union of appropriate sets.
When you are visualizing the system of inequalities on an axis you see that basicly you are just marking sets of elements satisfying them on an axis.
Those two methods complement each other when working with inequalities. The latter method make visualizations straightforward.
