First, watch the video below to learn about inequalities. You can use the notes here to follow along with the video and record your thoughts.

We start with a terminology review.

A set is a collection of mathematical objects. We’ll commonly look at sets of numbers like the Natural numbers: $\{ 1, 2, 3, 4, \ldots \}$.

An interval is a collection of Real numbers. For example, $(1, 2)$ is the set of Real numbers between 1 and 2 (but not including 1 or 2). If we want to include the endpoints of an interval, we use brackets, such as $[1, 2]$.

We can describe solutions that exist in an interval by using the notation $x \in (a, b)$. We read this as “$x$ is an element of $(a, b)$” and means that $x$ is some number between $a$ and $b$. For example, $x \in [1, 2]$ means that $x$ is some number between 1 and 2 (and could be one of the two numbers).

We can also describe intervals using inequalities. For example, to describe the set of $x \in (1, 2)$ using inequalities, we would use $1 < x < 2$. This is usually a more natural way for students to read “$x$ is a Real number between 1 and 2.” If we want to include the endpoints of an inequality, we use the symbols $\leq$ or $\geq$.