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# 1.10: The Real Number Line

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## Graphing on a Real Number Line

Objectives

The lesson objectives for Graphing on the Real Number Line are:

• Identifying the number set(s) to which numbers belong
• Graphing given sets on a number line

Introduction

In this concept 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.

Guidance

To what number set(s) does the number 13 belong?

The number 13 is a natural number. $N=\{1,2,3,4 \ldots\}$

The number 13 is a whole number. $W=\{0,1,2,3 \ldots\}$

The number 13 is an integer. $I=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\}$

The number 13 is a rational number. $Q=\{\frac{a}{b}, b \ne 0 \}$

The number 13 belongs to the real number system.

Example

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.”

To represent all integers greater than 4, you can write $\{5,6,7,8, \ldots\}$, or write $x > 4$ such that $x$ is an integer. You can also use a number line to represent all integers greater than 4. Remember that the set of integers are negative whole numbers, zero and positive whole numbers. The number 4 is not included in the numbers greater than 4. However, to indicate 4 as a starting point, mark it on the number line with an open dot. If it were included in the answer, the dot would be closed or solid.

Represent $x>4$ where $x$ is an integer, on a number line.

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 $\{x \ge -2 | x \ \varepsilon \ R\}$.

$\{x \ge -2 | x \ \varepsilon \ R\}$ The statement can be read as “$x$ is greater than or equal to -2, such that x belongs to or is a member of the real numbers.” In other words, represent all real numbers greater than or equal to -2.

The inequality symbol says that $x$ is greater than or equal to -2. This means that -2 is included in the graph. A solid dot is placed on -2 and on all numbers to the right of -2. The line is on the number line to indicate that all real numbers greater than -2 are also included in the graph.

Example C

In the above examples, 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.

Represent this inequality statement, also known as set notation, on a number line $\{x|2 < x \le 7, x \ \varepsilon \ N\}$. This inequality statement can be read as $x$ such that $x$ is greater than 2 and less than or equal to 7 and $x$ belongs to the natural numbers. In other words, all natural numbers greater than 2 and less than or equal to 7.

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.

Vocabulary

Inequality
An inequality is a mathematical statement relating expressions by using one or more inequality symbols. The inequality symbols are $>,<,\ge,\le$
Integer
All natural numbers, their opposites, and zero are integers. A number in the list $\ldots, -3, -2, -1, 0, 1, 2, 3 \ldots$
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 non-terminating decimals as well as non-periodic 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 $1, 2, 3\ldots$ 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 $\frac{a}{b}$ and $b \ne 0$. 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. $\{x|x \ge -3, x \ \varepsilon \ I\}$ 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 $N$ $W$ $I$ $Q$ $\overline{Q}$ $R$
5
$-\frac{47}{3}$
1.48
$\sqrt{7}$
0
$\pi$

2. Graph $\{x|-3 \le x \le 8, x \ \varepsilon \ R\}$ on a number line.

3. Use set notation to describe the set shown on the number line.

1. Before answering this question, review the definitions for each set of numbers. You can find these in the vocabulary.

Number $N$ $W$ $I$ $Q$ $\overline{Q}$
5 X X X X
$-\frac{47}{3}$ X X X X
1.48 X X X X
$\sqrt{7}$ X
0 X X X
$\pi$ X

2. $\{x|-3 \le x \le 8, x \ \varepsilon \ R\}$

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

$\boxed{\{x|-2 \le x < 3, x \ \varepsilon \ R\}}$

Summary

In this lesson you revisited the real number system. You reviewed the sets of numbers that made up the real number system and concentrated on the types of numbers that were included in each set. These sets included the natural numbers, whole numbers, integers, rational numbers and irrational numbers. Given a group of numbers, you learned how to assign each number to the number set(s) to which it belonged.

You also learned how to represent a set notation on a number line. You now know the meaning of a closed dot and an open dot when it is graphed on a number line. You also learned how to represent the various number sets on a number line. From a number line graph, you learned how to write the set notation that describes the set shown on the number line.

Problem Set

Describe each set notation in words.

1. $\{x|x > 8, x \ \varepsilon \ R\}$
2. $\{x|x \le -3, x \ \varepsilon \ I\}$
3. $\{x|-4 \le x \le 6, x \ \varepsilon \ R\}$
4. $\{x|5 \le x \le 11, x \ \varepsilon \ W\}$
5. $\{x|x \ge 6, x \ \varepsilon \ N\}$

Represent each graph using set notation

For each of the following situations, use set notations to represent the limits.

1. To ride the new tilt-a whirl at the fairgrounds, a child can be no taller than 4.5 feet.
2. 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.
3. A sled driver in the Alaska Speed Quest must start the race with no less than 10 dogs and no more than 16 dogs.
4. 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 $-6^\circ C$ and not above $-5^\circ C$.
5. 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.

1. $\{x|x>6, x \ \varepsilon \ N\}$
2. $\{x|x\le 8, x \ \varepsilon \ R\}$
3. $\{x|-3\le x < 6, x \ \varepsilon \ I\}$

Describe each set...

1. $\{x|x > 8, x \ \varepsilon \ R\}$ All real numbers greater than 8.
1. $\{x|-4 \le x \le 6, x \ \varepsilon \ R\}$ All real numbers between -4 and 6.
1. $\{x|x \ge 6, x \ \varepsilon \ N\}$ All natural numbers greater than or equal to 6.

Represent each graph...

1. $\{x|< -6
1. $\{x|x \le -8, x \ \varepsilon \ R\}$
1. $\{x|-7 \le x < 1, x \ \varepsilon \ R\}$

For each of the following situations...

1. $\{x|x \le 4.5, x \ \varepsilon \ R\}$
1. $\{x|10 \le x \le 16, x \ \varepsilon \ N\}$
1. $\{x|100 \le x \le 225, x \ \varepsilon \ N\}$

Represent the following...

1. $\{x|x > 6, x \ \varepsilon \ N\}$
1. $\{x|x \le 8, x \ \varepsilon \ R\}$
1. $\{x|-3 \le x < 2, x \ \varepsilon \ I\}$

Summary

In this lesson you revisited the real number system $(R)$. You reviewed the sets of numbers that make up the real number system and concentrated on the types of numbers that were included in each set. These sets included the natural numbers $(N)$, whole numbers $(W)$, integers $(I)$, rational numbers $(Q)$ and irrational numbers $(\overline{Q})$. Given a group of numbers, you learned how to assign each number to the number set(s) to which it belonged. When you assigned a number to its number set(s), you had to be careful not to put it in the wrong set. All whole numbers are integers but not all integers are whole numbers.

You also learned that a mathematical statement that shows an inequality and the set of numbers to which the variable belongs is known as set notation. Set notation can be described in words or can be represented on a number line. A closed dot on the number line means that the number is included in the set notation and an open dot means that the number is not included. A closed dot is the result of the inequality symbol $\le$ or $\ge$. An open dot is the result of the inequality sign < or >. You also learned how to represent the various number sets on a number line.

From a number line graph, you learned how to write the set notation that describes the set shown on the number line. You also learned how to write set notation to represent various real life situations. The set notations that represent real life situations can all be drawn on a number line.

## Comparing Real Numbers

Objectives

The lesson objectives for Comparing Real Numbers are:

• Ordering real numbers from least to greatest
• Representing real numbers on a number line
• Using technology to simplify the process

Introduction

In this concept you will revisit the number sets that make up the real number system. You will also apply the skills you have learned for changing fractions to decimal numbers. In addition, you will learn to order real numbers from least to greatest and to place these numbers on a number line. When placing numbers on a number line, you will learn helpful hints to make the process easier. Finally, you will learn to order the numbers using your TI83 calculator.

Guidance

Order the following real numbers from least to greatest.

$\frac{22}{7},1.234 234 \ldots, - \sqrt{7}, -5, -\frac{21}{4}, 7,-1.666,0,8.32,\frac{\pi}{2},-\pi,-5.38$

As you examine the above numbers, you can see that there are natural numbers, whole numbers, integers, rational numbers and irrational numbers. These numbers, as they are presented here, would be very difficult to order from least to greatest. The simplest way to order the numbers is to express them all in the same form – all fractions or all decimal numbers. Since you all have a calculator, use the calculator to express every number as a decimal number. Watch your signs – don't drop any of the negative signs.

Now that all the numbers are in decimal form, make two lists of decimal numbers – negatives and positives. The most places after the decimal point in the given numbers is 6. The decimal numbers that you determined with your calculator need only have 6 numbers after the decimal point.

Given Pos. Value Positives Given Neg. Value Negatives
$\frac{22}{7}$ 3.142857 $-\sqrt{7}$ -2.645751
1.234 234... 1.234234 -5 -5
7 7 $-\frac{21}{4}$ -5.25
0 0 -1.666 -1.666
8.32 8.32 $-\pi$ -3.141592
$\frac{\pi}{2}$ 1.570796 -5.38 -5.38

The negative numbers with the greatest magnitude go left on the number line since they are the smallest of the numbers. Now arrange the numbers from least to greatest using the numbers you were given in the problem.

$-5.38, -\frac{21}{4}, -5, -\pi,\sqrt{7}, -1.666, 0, 1.234234, \frac{\pi}{2},\frac{22}{7}, 7, 8.32$

Example A

Draw a number line and place these numbers on the line.

${\color{red}\sqrt{\frac{2}{5}}}, {\color{blue}0.6467},{\color{red}-\frac{3}{5}},{\color{red}\frac{1}{8}},{\color{green}0},{\color{red}\sqrt{0.5}},{\color{blue}-2.34},{\color{red}\pi},{\color{red}\frac{2 \pi}{3},{\color{green}-1}},{\color{green}2}$

$\sqrt{\frac{2}{5}}=0.6324 \quad -\frac{3}{5}=-0.6 \quad \frac{1}{8}=0.125 \quad \sqrt{0.5}=0.7071 \quad -\pi=-3.1416 \quad \frac{2 \pi}{3}=2.0944$

Start by placing the ${\color{green}\mathbf{integers}}$ on the line first. Next place the ${\color{blue}\mathbf{decimal \ numbers}}$ on the line.

Use your calculator to convert ${\color{red}\mathbf{the \ remaining \ numbers}}$ to decimal numbers. Place these on the line last.

It is impossible to place decimal numbers in the exact location on the number line. However, place them as close as you can to the appropriate spot on the line. Use your estimating skills when doing an exercise like this one.

Example B

For each given pair of real numbers, find another real number that is between each of the pairs.

i) $-2,1$

ii) $3.5,3.6$

iii) $\frac{1}{2},\frac{1}{3}$

iv) $-\frac{1}{3}, -\frac{1}{4}$

The answers to these will vary.

i) The number must be greater than -2 and less than 1. $\boxed{-2, {\color{blue}0},1}$

ii) The number must be greater than 3.5 and less than 3.6. $\boxed{3.5, {\color{blue}3.54},3.6}$

iii) The number must be greater $\frac{1}{3}$ than and less than $\frac{1}{2}$. Write each fraction with a common denominator. $\frac{1}{2}=\frac{3}{6},\frac{1}{3}=\frac{2}{6}$. If you look at $\frac{2}{6}$ and $\frac{3}{6}$, there is no fraction, with a denominator of 6, between these values. Write the fractions with a larger common denominator. $\frac{1}{2}=\frac{6}{12}, \frac{1}{3}=\frac{4}{12}$. If you look at $\frac{4}{12}$ and $\frac{6}{12}$, the fraction $\frac{5}{12}$ is between them. $\boxed{\frac{1}{3},{\color{blue}\frac{5}{12}},\frac{1}{2}}$

iv) The number must be greater than $-\frac{1}{3}$ and less than $-\frac{1}{4}$. Write each fraction with a common denominator. $-\frac{1}{3}=-\frac{4}{12},-\frac{1}{4}=-\frac{3}{12}$. If you look at $-\frac{3}{12}$ and $-\frac{4}{12}$, there is no fraction, with a denominator of 12, between these values. Write the fractions with a larger common denominator. $-\frac{1}{3}=-\frac{8}{24},-\frac{1}{4}=-\frac{6}{24}$. If you look at $-\frac{6}{24}$ and $-\frac{8}{24}$, the fraction $-\frac{7}{24}$ is between them. $\boxed{-\frac{8}{24}, {\color{blue}-\frac{7}{24}}, -\frac{6}{24}}$

Example C

Order the following fractions from least to greatest.

$\frac{2}{11},\frac{7}{9},\frac{8}{7},\frac{1}{11},\frac{5}{6}$

The fractions do not have a common denominator. This makes it almost impossible to arrange the fractions from least to greatest. To determine the common denominator, may take some time. Let’s use the TI83 to order these fractions.

The fractions were entered into the calculator as division problems. The decimal numbers were entered into List 1.

The calculator has sorted the data from least to greatest.

The data is sorted. The decimal numbers and the corresponding fractions can now be matched from the screen where they were first entered.

$\frac{1}{11},\frac{2}{11},\frac{7}{9},\frac{5}{6},\frac{8}{7}$

Guided Practice

1. Arrange the following numbers in order from least to greatest and place them on a number line.

$-3.78, -\frac{11}{4},-4, \frac{\pi}{2}, -\sqrt{6},-1.888,0,0.2424,\pi,\frac{21}{15},2,1.75$

2. For each given pair of real numbers, find another real number that is between each of the pairs.

i) $-3,-5$

ii) $-3.4,-3.5$

iii) $\frac{1}{5},\frac{3}{10}$

iv) $-\frac{3}{4},-\frac{11}{6}$

3. Use technology to sort the following numbers:

$\sqrt{\frac{3}{5}},\frac{15}{38},-\frac{7}{12},\frac{1}{4},0,\sqrt{8},-\frac{13}{21},-\pi,\frac{3 \pi}{5},-6,3$

1. $-3.78, -\frac{11}{4},-4, \frac{\pi}{2}, -\sqrt{6},-1.888,0,0.2424,\pi,\frac{21}{15},2,1.75$

2. i) $-3,-5$

ii) $-3.4,-3.5$

iii) $\frac{1}{5},\frac{3}{10}$

iv) $-\frac{3}{4},-\frac{11}{6}$

i) The number must be greater than -5 and less than -3. $\boxed{-5,{\color{blue}-4},-3}$

ii) The number must be greater than -3.5 and less than -3.4. $\boxed{-3.5,{\color{blue}-3.45},-3.4}$

iii) The number must be greater than $\frac{1}{5}$ and less than $\frac{3}{10}$. Write each fraction with a common denominator. $\frac{1}{5}=\frac{2}{10}$. If you look at $\frac{2}{10}$ and $\frac{3}{10}$, there is no fraction, with a denominator of 10, between these values. Write the fractions with a larger common denominator. $\frac{1}{5}=\frac{4}{20},\frac{3}{10}=\frac{6}{20}$. If you look at $\frac{4}{20}$ and $\frac{6}{20}$, the fraction $\frac{5}{20}=\frac{1}{4}$ is between them. $\boxed{\frac{1}{5}, \frac{{\color{blue}1}}{{\color{blue}4}}, \frac{3}{10}}$

iv) The number must be greater than $-\frac{3}{4}$and less than $-\frac{11}{16}$. Write each fraction with a common denominator. $-\frac{3}{4}=-\frac{12}{16}$. If you look at $-\frac{12}{16}$ and $-\frac{11}{16}$, there is no fraction, with a denominator of 16 between these values. Write the fractions with a larger common denominator. $-\frac{3}{4}=-\frac{24}{32},-\frac{11}{16}=-\frac{22}{32}$. If you look at $-\frac{24}{32}$ and $-\frac{22}{32}$, the fraction $-\frac{23}{32}$ is between them. $\boxed{-\frac{3}{4},-{\color{blue}\frac{23}{32}},-\frac{11}{16}}$

3. $\sqrt{\frac{3}{5}},\frac{15}{38},-\frac{7}{12},\frac{1}{4},0,\sqrt{8},-\frac{13}{21},-\pi,\frac{3 \pi}{5},-6,3$

The numbers have been sorted. The numbers from least to greatest are:

$-6,-\pi,-\frac{13}{21},-\frac{7}{12},0,\frac{1}{4},\frac{15}{38},\sqrt{\frac{3}{5}},\frac{3 \pi}{5},\sqrt{8},3$

## Summary

In this lesson you learned to order real numbers from least to greatest. To facilitate the process, you learned that changing the numbers to decimal numbers made the ordering less difficult.

You also learned to represent the numbers on an appropriate number line. The given integers could be placed on the line first and then any given decimal numbers were located on the number line. The remaining numbers were changed to decimals and placed on the number line last. Placing the numbers on the number line gave you a visual image of the order of the real numbers.

You were also shown the key strokes to use to order the numbers by using technology. The numbers were entered into the calculator and were recorded as decimal numbers. You then sorted the numbers on the calculator to order them. The one thing that you had to remember was to track the numbers as you entered them into the calculator. This step was necessary to match the given numbers with the corresponding decimal numbers.

To apply the concept of ordering numbers, you then solved problems that asked you to find a real number between a pair of given numbers.

Problem Set

Arrange the following numbers in order from least to greatest and place them on a number line.

1. $\{0.5,0.45,0.65,0.33,0,2,0.75,0.28\}$
2. $\{\frac{1}{2},-2,0,-\frac{1}{3},3,\frac{2}{3},-\frac{1}{2}\}$
3. $\{0.3,-\sqrt{2},1,-0.25,0,1.8,-\frac{\pi}{3}\}$

For each given pair of real numbers, find another real number that is between each of the pairs.

i) $8,10$

ii) $-7.6,-7.5$

iii) $\frac{1}{7},\frac{4}{21}$

iv) $-\frac{3}{5},-\frac{1}{2}$

Use technology to sort the following numbers:

1. $\{-2,\frac{2}{3},0,\frac{3}{8},-\frac{7}{5},\frac{1}{2},4,-3.6\}$
2. $\{\sqrt{10},-1,\frac{7}{12},3,-\frac{5}{4},-\sqrt{7},0,-\frac{2 \pi}{3},-\frac{3}{5}\}$

Arrange the following...

1. $\{\frac{1}{2},-2,0,-\frac{1}{3},1,\frac{2}{3},-\frac{1}{2}\}$

For each given pair...

i) The number must be greater than 8 and less than 10. $\boxed{8,{\color{blue}9},10}$

iii) The number must be greater than $\frac{1}{7}$ and less than $\frac{4}{21}$. Write each fraction with a common denominator. $\frac{1}{7}=\frac{3}{21}$. If you look at $\frac{3}{21}$ and $\frac{4}{21}$, there is no fraction, with a denominator of 21, between these values. Write the fractions with a larger common denominator. $\frac{1}{7}=\frac{6}{42},\frac{4}{21}=\frac{8}{42}$. If you look at $\frac{6}{42}$ and $\frac{8}{42}$, the fraction $\frac{7}{42}=\frac{1}{6}$ is between them. $\boxed{\frac{1}{7},{\color{blue}\frac{1}{6}},\frac{4}{21}}$

Use technology...

1. $\Big \{\sqrt{10},-1,\frac{7}{12},3,-\frac{5}{4},-\sqrt{7},0,-\frac{2 \pi}{3},-\frac{3}{5} \Big \}$ The numbers in order from least to greatest are: $\Big \{-\sqrt{7},-\frac{2 \pi}{3},-\frac{5}{4},-1,-\frac{3}{5},0,\frac{7}{12},3,\frac{7}{12} \Big \}$

Summary

In this lesson you learned to order real numbers from least to greatest. To avoid using the guessing method, numbers that were not integers or were not given as decimal numbers, were changed to decimal numbers. When you did this, you were less likely to make an error when ordering the numbers.

You also learned to represent the numbers on an appropriate number line. The given integers could be placed on the line first and then any given decimal numbers were located on the number line. The remaining numbers were changed to decimals and placed on the number line last. If a number line marled in intervals of 10 is used, the decimal numbers can be placed approximately where they belong. It is impossible to place the decimal numbers exactly where they belong. An approximate location is acceptable. This visual representation of the ordered numbers made the process more meaningful.

You were also shown the key strokes to use to order the numbers by using technology. As you entered the numbers into the calculator, they were converted to decimal numbers and stored in a list on the calculator. The numbers were sorted within seconds when the SORT command was used. The one thing that you had to remember was to track the numbers as you entered them into the calculator. This step was necessary to match the given numbers with the corresponding decimal numbers.

To apply the concept of ordering numbers, you then solved problems that asked you to find a real number between a pair of given numbers. When fractions were used, it was often necessary to write equivalent fractions using a common denominator and not a least common denominator.

## Date Created:

Jan 16, 2013

Jun 04, 2014
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