## Using Interval Notation and Properties of Inequalities

This section is an important foundation for learning how to express intervals and sets fluently using words, set-builder notation, and interval notation. You will use the concepts you learn here to describe the solutions to inequalities. Similarly, we can use sets to describe the behavior of all types of functions. You will be required to use the notation and concepts presented here in many future units in the course, so it is vital to make sure that you achieve mastery of the techniques presented here to be successful in the coming sections on functions.

### Using Interval Notation and Properties of Inequalities

### Learning Objectives

In this section, you will:

- Use interval notation
- Use properties of inequalities.
- Solve inequalities in one variable algebraically.
- Solve absolute value inequalities.

It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.

#### Using Interval Notation

Indicating the solution to an inequality such as can be achieved in several ways.

We can use a number line as shown in Figure 2. The blue ray begins at and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to .

Figure 2

We can use set-builder notation: , which translates to "all real numbers such that is greater than or equal to ". Notice that braces are used to indicate a set.

The third method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to are represented as . This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.

The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be "equaled". A few examples of an interval, or a set of numbers in which a solution falls, are , or all numbers between and , including , but not including , all real numbers between, but not including and and , all real numbers less than and including . Table 1 outlines the possibilities.

Table 1

#### EXAMPLE 1

##### Using Interval Notation to Express All Real Numbers Greater Than or Equal to

Use interval notation to indicate all real numbers greater than or equal to .

##### Solution

Use a bracket on the left of and parentheses after infinity: . The bracket indicates that is included in the set with all real numbers greater than to infinity.

#### TRY IT #1

Use interval notation to indicate all real numbers between and including and .

#### EXAMPLE 2

##### Using Interval Notation to Express All Real Numbers Less Than or Equal to or Greater Than or Equal to

Write the interval expressing all real numbers less than or equal to or greater than or equal to .

##### Solution

We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to . So, this interval begins at and ends at , which is written as .

The second interval must show all real numbers greater than or equal to , which is written as . However, we want to combine these two sets. We accomplish this by inserting the union symbol, , between the two intervals.

#### TRY IT #2

Express all real numbers less than or greater than or equal to in interval notation.

Source: Rice University, https://openstax.org/books/college-algebra/pages/2-7-linear-inequalities-and-absolute-value-inequalities

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