Can you describe the number 13? Can you say what number sets the number 13 belongs to?
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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 \begin{align*}\geq\end{align*} means “is greater than or equal to.”
- The symbol \begin{align*}\leq\end{align*} 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 points are used.
Example A
Represent \begin{align*}x>4\end{align*} where \begin{align*}x\end{align*} is an integer, on a number line.
Solution:
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 \begin{align*}\{x \ge -2 | x \ \in \ R\}\end{align*}.
Solution: The statement can be read as “\begin{align*}x\end{align*} 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 \begin{align*}x\end{align*} 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
Represent this inequality statement, also known as set notation, on a number line \begin{align*}\{x|2 < x \le 7, x \ \in \ N\}\end{align*}.
Solution: This inequality statement can be read as \begin{align*}x\end{align*} such that \begin{align*}x\end{align*} is greater than 2 and less than or equal to 7 and \begin{align*}x\end{align*} 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.
Concept Problem Revisited
To what number set(s) does the number 13 belong?
The number 13 is a natural number because it is in the set \begin{align*}N=\{1,2,3,4 \ldots\}\end{align*}.
The number 13 is a whole number because it is in the set \begin{align*}W=\{0,1,2,3 \ldots\}\end{align*}.
The number 13 is an integer because it is in the set \begin{align*}Z=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\}\end{align*}.
The number 13 is a rational number because it is in the set \begin{align*}Q=\{\frac{a}{b}, b \ne 0 \}\end{align*}.
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 \begin{align*}>,<,\ge,\le\end{align*}.
- Integer
- All natural numbers, their opposites, and zero are integers. A number in the list \begin{align*}\ldots, -3, -2, -1, 0, 1, 2, 3 \ldots\end{align*}
- 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 both 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 \begin{align*}1, 2, 3\ldots\end{align*} 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 \begin{align*}\frac{a}{b}\end{align*} and \begin{align*}b \ne 0\end{align*}. 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. \begin{align*}\{x|x \ge -3, x \ \in \ Z\}\end{align*} 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 | \begin{align*}N\end{align*} | \begin{align*}W\end{align*} | \begin{align*}Z\end{align*} | \begin{align*}Q\end{align*} | \begin{align*}\overline{Q}\end{align*} | \begin{align*}R\end{align*} | |
---|---|---|---|---|---|---|---|
5 | |||||||
\begin{align*}-\frac{47}{3}\end{align*} | |||||||
1.48 | |||||||
\begin{align*}\sqrt{7}\end{align*} | |||||||
0 | |||||||
\begin{align*}\pi\end{align*} |
2. Graph \begin{align*}\{x|-3 \le x \le 8, x \ \in \ R\}\end{align*} on a number line.
3. Use set notation to describe the set shown on the number line.
Answers:
1. Review the definitions for each set of numbers.
Number | \begin{align*}N\end{align*} | \begin{align*}W\end{align*} | \begin{align*}Z\end{align*} | \begin{align*}Q\end{align*} | \begin{align*}\overline{Q}\end{align*} |
---|---|---|---|---|---|
5 | X | X | X | X | |
\begin{align*}-\frac{47}{3}\end{align*} | X | X | X | X | |
1.48 | X | X | X | X | |
\begin{align*}\sqrt{7}\end{align*} | X | ||||
0 | X | X | X | ||
\begin{align*}\pi\end{align*} | X |
2. \begin{align*}\{x|-3 \le x \le 8, x \ \in \ R\}\end{align*}
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 –3 is included in the answer. The remaining dots are to the right of –3. The open dot means that 2 is not included in the answer. This means that the numbers are all less than 2. 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 \begin{align*}\{x|-3 \le x < 2, x \ \in \ Z\}\end{align*}.
Practice
Describe each set notation in words.
- \begin{align*}\{x|x > 8, x \ \in \ R\}\end{align*}
- \begin{align*}\{x|x \le -3, x \ \in \ Z\}\end{align*}
- \begin{align*}\{x|-4 \le x \le 6, x \ \in \ R\}\end{align*}
- \begin{align*}\{x|5 \le x \le 11, x \ \in \ W\}\end{align*}
- \begin{align*}\{x|x \ge 6, x \ \in \ N\}\end{align*}
Represent each graph using set notation
- .
- .
- .
- .
- .
For each of the following situations, use set notations to represent the limits.
- To ride the new tilt-a 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 \begin{align*}-6^\circ C\end{align*} and not above \begin{align*}-1^\circ C\end{align*}.
- 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.
- \begin{align*}\{x|x>6, x \ \in \ N\}\end{align*}
- \begin{align*}\{x|x\le 8, x \ \in \ R\}\end{align*}
- \begin{align*}\{x|-3\le x < 6, x \ \in \ Z\}\end{align*}