Interval and Set-Builder Notation - Compound Inequalities

Interval Notation and Set-Builder Notation for Intervals

Earlier, we learning about writing inequalities. We saw how to graph linear inequalities. Now, we'll look at intervals, specifically interval notation and set-builder notation.

An interval is one of the simplest infinite sets of reals. It's just the continuous block of real numbers between two endpoints.

There are several types of intervals, among them closed intervals and open intervals. The two types differ in whether they contain their endpoints.

A closed interval is denoted $[a, b]$. It is the set of real numbers between $a$ and $b$, including both $a$ and $b$.

An open interval is denoted $(a, b)$. It is the set of real numbers between $a$ and $b$, but not including $a$ or $b$.

Sometimes, you will see half-open intervals $[a, b)$ or $(a, b]$. These include one of their endpoints. The endpoint with the square bracket is included, the endpoint with a parentheses is not.

Interval Notation and Infinity

Some intervals include all real numbers. This means the endpoints are $-\infty$ and $\infty$. These intervals cannot include infinity, because infinity is not a real number. The correct interval notation for the set of all real numbers is $(-\infty, \infty)$.

(Video) Linear Inequality Interval Notation

by mathman1024

This video by mathman1024 goes over interval notation.

Summary

You find the bounds of the solution and put them in between parentheses $(\ \ \ )$ if the number is not included or brackets $[\phantom{0} ]$ if the number is included. For example if you wanted to use interval notation to express the solutions to the inequality $x>5$ you would write:

$$(5,\infty)$$

This says that any number between $5$ and infinity (not including $5$) is a solution to the inequality. We always use a parentheses with infinity. This is because infinity is not a real number, and therefore can't be included in an interval.

Let's look at some of the examples he goes over on a number line!

Example 1

First, look at the circle. Since it's closed we know that we use a bracket. Next, we find the bounds. This one starts at $7$ and goes on to infinity. So we would write: $$[7,\infty)$$

Let's look at another.

Example 2

In this one we see the first number has an open circle which indicates that we should use an open parenthesis, i.e. $($. The second circle is closed so we use a closed bracket, i.e. $]$. Next, we find the bounds. This one starts at $-12$ (not including it) and ends at $3$. So we would write: $$(-12,3]$$

He does a few more examples but the final one is also very important.

Example 3

This example has two different sets of intervals. The first is from negative infinity to $\frac12$ (so, including all negative numbers) and the second is from $5$ to infinity (i.e., greater than or equal to $5$). When there are multiple intervals you string them together with the union symbol, $\cup,$ which we also use when we are finding the union of sets. Our solution is: $$\bigg(-\infty,\frac12\bigg)\cup[5,\infty)$$

Interval Notation for Compound Inequality

We can use interval notation to represent the solutions of a compound inequality. Before we see how, let's review interval notation.

AND Example

We can represent AND compound inequalities with one pair of numbers in interval notation: $$x>1 \text{ AND } x \le 5 \Rightarrow (1, 5] \text{ in interval notation}$$

OR Example

For OR compound inequalities, we need a union of our interval notation sets.
$$x<1 \text{ OR } x \ge 5 \Rightarrow (-\infty, 1) \cup [5, \infty) \text{ in interval notation}$$