Interval Notation Notes

The solution set of an inequality can be described by using interval notation. The infinity symbols +∞ and ∞ are used to indicate that a set is unbounded in the positive or negative direction, respectively. To indicate that an endpoint is NOT included in the set, a parenthesis, “(” or “)”, is used. A bracket “[” or “]” is used to indicate that the endpoint is included in the set. Parentheses are always used with the symbols +∞ and ∞.

Let’s look at a few examples: Open Interval:   (a, b)  is interpreted as a < x <b  where the endpoints are NOT included. (While this notation resembles an ordered pair, in this context it refers to the interval upon which you are working.) We say that this interval is bounded. (1, 5)

Closed Interval:  [a, b]  is interpreted as a < x < b  where the endpoints are included. We say that this interval is bounded. [1, 5]

Half-Open Interval:  (a, b]  is interpreted as a < x < b  where a is not included, but b is included. We say that this interval is bounded. (1, 5]

Half-Open Interval:  [a, b) is interpreted as a < x < b where a is included, but b is not included. We say that this interval is bounded. [1, 5)

Non-ending Interval:  (a, ∞) is interpreted as x > a where a is not included and infinity is always expressed as being "open" (not included). We say that this interval is unbounded. (1, ∞)

Non-ending Interval: (∞, b] is interpreted as x < b where b is included and again, infinity is always expressed as being "open" (not included). We say that this interval is unbounded. (∞,5] Notice that the interval in unbounded when we have a ∞ or +∞. If the right arrow is shaded I will not count it wrong if you just use ∞ and not +∞.

If both arrows are shaded you would have (∞, ∞) and this would represent all real numbers.

A compound inequality consists of two inequalities joined by the word and or the word or. To solve a compound inequality, containing and is the intersection of the solution sets of the two inequalities. To solve a compound inequality, containing or is the union of the solution set of the two inequalities. Compound inequalities involving the word and are called conjunctions. Compound inequalities involving the word or are called disjunctions.

(∞, 1] or ( 4, ∞ ) I really like to write this as (∞, 1]  ( 4, ∞ )  stands for “or”. Notice that the graph has two intervals.

Do not make interval notation hard, just PRACTICE. Make your own set of rules to help you.