To describe numbers that lie between endpoints, we may use the following notation.
For values greater than a real number $a$:
use $x>a$ if you do not want to include the endpoint. On the real number line, use an open circle at $a$.
The interval $x>2$.
use $x \geq a$ if you do want to include the endpoint. Use a closed circle at $a$.
The interval $x\geq 2$.
For values less than a real number $a$:
use $x < a$ if you do not want to include the endpoint. Use an open circle at $a$.
The interval $x < 2$.
use $x \leq a$ if you do want to include the endpoint. Use a closed circle at $a$.
The interval $x\leq 2$.
Interval notation, part 1.
You should be comfortable using both interval notation and set notation.
For values greater than a real number $a$:
use $(a,\infty) = \left\{ x \mid x>a \right\}$ if you do not want to include the endpoint. Use an open circle at $a$.
The interval $(2,\infty) = \left\{ x \mid x>2 \right\}$.
use $[a,\infty) = \left\{ x \mid x \geq a \right\}$ if you do want to include the endpoint. Use a closed circle at $a$.
The interval $[2,\infty) = \left\{ x \mid x \geq 2 \right\}$.
For values less than a real number $a$:
use $(-\infty,a) = \left\{ x \mid x < a \right\}$ if you do not want to include the endpoint. Use an open circle at $a$.
The interval $(-\infty,2) = \left\{ x \mid x < 2 \right\}$.
use $(-\infty,a] = \left\{ x \mid x\leq a \right\}$ if you do want to include the endpoint. Use a closed circle at $a$.
The interval $(-\infty,2] = \left\{ x \mid x\leq 2 \right\}$.
