The *real number line* is a horizontal line used to represent the real numbers. Every point on the line represents a different real number, with values increasing from left to right. The arrows on either end indicate that the line extends forever in either direction, which allows it to include all values from to .

Whereas an equation relates two quantities that are of equal value, an *inequality* is a way to mathematically relate two quantities that do not have the same value. An inequality may use one of the following symbols:

- means that is not equal to .
- means that is greater than .
- means that is less than .
- means that is greater than or equal to .
*solution set*of that inequality. Solution sets are often represented by graphs on number lines.

To write with MathQuill, type \neq and then a space. To write , type \le and then a space, and to write , type \ge and then a space.
means that is less than or equal to .
The set of all numbers that satisfy an inequality (i.e. can be plugged in for the variables to make the inequality true) is called the

If we know that *chained inequality* or *chained notation* to combine these two inequalities into one statement: .

An *interval* is a set of real numbers that includes all real numbers between a lower bound and an upper bound. If an interval represents a set of values that a variable, let’s say for now, is allowed to equal, then it might be expressed with *inequality notation*: , , , or
Intervals can also be expressed using what is called *interval notation*. For the intervals written above, we could instead use interval notation to write , , , or . Interval notation thus uses square or round brackets surround the lower and upper bound of the interval, separated by a comma. A round bracket or parenthesis indicates that the bound or endpoint next to it is not included in the interval, while a square bracket means that that number is included.

Let’s take a look at an example. If we have , we could also write that lies in the interval . Here, can equal , since that has a square bracket next to it, but it can’t equal , since that is by a round parenthesis.

The *union* of two sets is the set of all values that belong to either of those two sets, and we denote the union of two sets and by writing . So if we have { } and { }, then { }. In this course, we will most often use unions to join together intervals that make up solution sets.

A *compound inequality* connects two or more inequalities by using the words “and” or “or”. In this course, we’ll limit ourselves to compound inequalities with just two inequalities.

- If the word “and” is used, then a value must satisfy both of the inequalities in order to be in the solution set. Example: If we start out with and , then we can simply to have and , so the solution is , written as a chained inequality. In interval notation, this is . Some values that would satisfy this compound inequality, then, are , , and . Some values that would not satisfy it are , , and .
- If the word “or” is used, then a value may satisfy either or both of the inequalities in order to be in the solution set. Example: Let’s say we begin with or . We could also write this as , where , typed as \or in MathQuill, just means “or”. We can then simplify to have . In interval notation, we would write this as . Some values that would satisfy this compound inequality, then, are , , and . Some values that would not satisfy it are , , and .

The *absolute value* is the magnitude of , that is, the number with any negative sign removed. For example, and . Note that any calculation within the absolute value is done first and then the sign ignored eg .