1.14: Real Number Line Graphs
Can you describe the number 13? Can you say what number sets the number 13 belongs to?
Watch This
Khan Academy Inequalities on a Number Line
Guidance
All of the numbers you have learned about so far in math belong to the real number system. Positives, negatives, fractions, and decimals are all part of the real number system. The diagram below shows how all of the numbers in the real number system are grouped.
Any number in the real number system can be plotted on a real number line. You can also graph inequalities on a real number line. In order to graph inequalities, make sure you know the following symbols:
 The symbol > means “is greater than.”
 The symbol < means “is less than.”
 The symbol
≥ means “is greater than or equal to.”  The symbol
≤ means “is less than or equal to.”
The inequality symbol indicates the type of dot that is placed on the beginning point and the number set indicates whether an arrow is drawn on the number line or if a line with an arrow is drawn. The arrow means that the numbers included in the graph continue. The only time that an arrow is not used is when the inequality represents a beginning point and an ending point.
Example A
Represent
The open dot on the four means that 4 is not included in the graph of all integers greater than 4. The closed dots on 5, 6, 7, 8 means that these numbers are included in the set of integers greater than 4. The arrow pointing to the right means that all integers to the right of 8 are also included in the graph of all integers greater than 4.
Example B
Represent this inequality statement on a number line
The inequality symbol says that
Example C
Represent this inequality statement, also known as set notation, on a number line
The inequality statement that was to be represented on the number line had to include the natural numbers greater than 2 and less than or equal to 7. These are the only numbers to be graphed. There is no arrow on the number line.
Concept Problem Revisited
To what number set(s) does the number 13 belong?
The number 13 is a natural number.
The number 13 is a whole number.
The number 13 is an integer.
The number 13 is a rational number.
The number 13 belongs to the real number system.
Vocabulary
 Inequality

An inequality is a mathematical statement relating expressions by using one or more inequality symbols. The inequality symbols are
>,<,≥,≤
 Integer

All natural numbers, their opposites, and zero are integers. A number in the list
…,−3,−2,−1,0,1,2,3…
 Irrational Numbers
 The irrational numbers are those that cannot be expressed as the ratio of two numbers. The irrational numbers include decimal numbers that are nonterminating decimals as well as nonperiodic decimal numbers.
 Natural Numbers

The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are numbers in the list
1,2,3… and are often referred to as positive integers.
 Number Line
 A number line is a line that matches a set of points and a set of numbers one to one. It is often used in mathematics to show mathematical computations.
 Rational Numbers

The rational numbers are numbers that can be written as the ratio of two numbers
ab andb≠0 . The rational numbers include all terminating decimals as well as periodic decimal numbers.
 Real Numbers
 The rational numbers and the irrational numbers make up the real numbers.
 Set Notation

Set notation is a mathematical statement that shows an inequality and the set of numbers to which the variable belongs.
{xx≥−3,x ε I} is an example of set notation.
Guided Practice
1. Check the set(s) to which each number belongs. The number may belong to more than one set.
Number 








5  


1.48  


0  

2. Graph
3. Use set notation to describe the set shown on the number line.
Answers:
1. Before answering this question, review the definitions for each set of numbers. You can find these in the vocabulary.
Number 






5  X  X  X  X  

X  X  X  X  
1.48  X  X  X  X  

X  
0  X  X  X  

X 
2.
The set notation means to graph all real numbers between 3 and +8. The line joining the solid dots represents the fact that the set belongs to the real number system.
3. The closed dot means that 2 is included in the answer. The remaining dots are to the right of 2. The open dot means that 3 is not included in the answer. This means that the numbers are all less than 3. Graphing on a number line is done from smallest to greatest or from left to right. There is no line joining the dots so the variable does not belong to the set of real numbers. However, negative whole numbers, zero and positive whole numbers make up the integers. The set notation that is represented on the number line is
Practice
Describe each set notation in words.

{xx>8,x ε R} 
{xx≤−3,x ε I} 
{x−4≤x≤6,x ε R} 
{x5≤x≤11,x ε W} 
{xx≥6,x ε N}
Represent each graph using set notation
For each of the following situations, use set notations to represent the limits.
 To ride the new tilta whirl at the fairgrounds, a child can be no taller than 4.5 feet.
 A dance is being held at the community hall to raise money for breast cancer. The dance is only for those people 19 years of age or older.
 A sled driver in the Alaska Speed Quest must start the race with no less than 10 dogs and no more than 16 dogs.
 The residents of a small community are planning a skating party at the local lake. In order for the event to take place, the outdoor temperature needs to be above
−6∘C and not above−5∘C .  Juanita and Hans are planning their wedding supper at a local venue. To book the facility, they must guarantee that at least 100 people will have supper but no more than 225 people will eat.
Represent the following set notations on a number line.

{xx>6,x ε N}  \begin{align*}\{xx\le 8, x \ \varepsilon \ R\}\end{align*}
 \begin{align*}\{x3\le x < 6, x \ \varepsilon \ I\}\end{align*}
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Image Attributions
Here you will review the number sets that make up the real number system. You will use the information learned in the review, to identify the number set to which given numbers belong. In addition, you will use the information learned, to graph inequalities on a real number line.