7.13: Inequalities on a Number Line
Have you ever wanted to make a certain amount of money babysitting?
Kara has figured out that if she works a certain number of hours that she will make enough money for her vacation. Kara earns six dollars an hour and she charges an additional two dollars for bus fare when she babysits. Kara wants to make more than thirty dollars for any babysitting job that she does.
She wants to figure out how many hours she needs to work to earn $30.00 or more for each babysitting job. Here is an equation to help you figure this out.
\begin{align*}6h + 2 \ge 32\end{align*}
This equation shows six dollars times the number of hours plus the two dollars bus fare is greater than or equal to thirty - two dollars. Thirty - two includes the two dollars for bus fare.
Do you know how to figure this out?
Can you graph the inequality on a number line?
You will learn how to do this in this Concept.
Guidance
In the last few Concepts you have been learning about equations and about balancing an equation. Let’s think about equations. An equation is a number statement with an equal sign. The equal sign tells us that the quantity on one side of the equation is equal the quantity on the other side of the equation. We can solve an equation by figuring out the quantity that will make the equation a true statement.
\begin{align*}x+5=12\end{align*}
If we think about this equation, we can use mental math and we know that the unknown quantity is equal to 7. If we substitute 7 in for \begin{align*}x\end{align*}, we will have a true statement.
\begin{align*}7 + 5 &= 12\\ 12 &= 12\end{align*}
Our equation is balanced because one side is equal to the other side. Notice that there is one answer for \begin{align*}x\end{align*} that makes this a true statement.
What is an inequality?
An inequality is a mathematical statement that can be equal, but can also be unequal.
We use the following symbols to show that we are working with an inequality.
> means greater than
< means less than
\begin{align*}\ge\end{align*} means greater than or equal to
\begin{align*}\le\end{align*} means less than or equal to
How do we apply these symbols?
Well, if you think about it, we use the symbols to make a true statement. Here is an inequality.
\begin{align*}x + 3 > 5\end{align*}
There are many possible answers that will make this a true statement. We need the quantity on the left side of the inequality to create a sum that is greater than five. Notice that the sign does not have a line under it. We want a quantity that is greater than five not greater than or equal to five on the left side of the inequality.
To make this true, we can choose a set of numbers that has a 3 or greater in it.
\begin{align*}x = \{3, 4, 5 \ldots .\}\end{align*}
We don’t need to worry about solving inequalities yet, the key thing to notice is that there are many possible answers that will make an inequality a true statement.
We can use graphs to help us understand inequalities in a visual way.
Graphing inequalities on a number line can help us understand which numbers are \begin{align*}s\end{align*} solutions for the inequality and which numbers are not solutions.
Here are some tips for graphing inequalities on a number line.
- Use an open circle to show that a value is not a solution for the inequality. You will use open circles to graph inequalities that include the symbols > or < .
- Use a closed circle to show that a value is a solution for the inequality. You will use closed circles to graph inequalities that include the symbols \begin{align*}\ge\end{align*} or \begin{align*}\le\end{align*}.
Write these hints in your notebook and then continue with the lesson.
Graph this inequality \begin{align*}x > 3\end{align*}
To complete this task, first draw a number line from –5 to 5.
The inequality \begin{align*}x > 3\end{align*} is read as “\begin{align*}x\end{align*} is greater than 3.” So, the solutions of this inequality include all numbers greater than 3. It does not include 3, so draw an open circle at 3 to show that 3 is not a solution for this inequality. Then draw an arrow showing all numbers greater than 3. The arrow should face right because the greater numbers are to the right on a number line.
The graph above shows the solutions for the inequality \begin{align*}x > 3\end{align*}.
The graph above can help you see which numbers are solutions for \begin{align*}x > 3 \end{align*} and which are not. For example, the arrow includes the numbers 3.5, 4, and 5. If you continued the number line, the arrow would also include the numbers 10 and 100. So, all those numbers––3.5, 4, 5, 10, and 100––are possible values for \begin{align*}x\end{align*}.
Graph this inequality \begin{align*}x<-1\end{align*}.
First, draw a number line from –5 to 5.
The inequality \begin{align*}x<-1\end{align*} is read as “\begin{align*}x\end{align*} is less than -1.” So, the solutions of this inequality include all numbers less than -1. It does not include -1, so draw an open circle at -1 to show that -1 is not a solution for this inequality. Then draw an arrow showing all numbers less than -1. The arrow should face left because the lesser numbers are to the left on a number line.
The graph above shows the solutions for the inequality \begin{align*}x<-1\end{align*}.
Think about what you have learned about inequalities. Then, answer the following true/false questions.
Example A
True or false. An open circle on a graph means that the number is not included in the solution set.
Solution: True
Example B
True or false. An inequality can never be equal.
Solution: False
Example C
True or false. A closed circle on a graph means that the number is included in the solution set.
Solution: True
Here is the original problem once again.
Kara has figured out that if she works a certain number of hours that she will make enough money for her vacation. Kara earns six dollars an hour and she charges an additional two dollars for bus fare when she babysits. Kara wants to make more than thirty dollars for any babysitting job that she does.
She wants to figure out how many hours she needs to work to earn $30.00 or more for each babysitting job. Here is an equation to help you figure this out.
\begin{align*}6h + 2 \ge 32\end{align*}
This equation shows six dollars times the number of hours plus the two dollars bus fare is greater than or equal to thirty dollars.
Do you know how to figure this out?
Can you graph the inequality on a number line?
Now we can solve the inequality first.
\begin{align*}6h + 2 \ge 32\end{align*}
\begin{align*}6h \ge 30\end{align*}
\begin{align*}h \ge 5\end{align*}
If Kara works at least five hours babysitting, she will earn at least thirty dollars.
Here is a graph of the data.
Notice that we included 5 in the graph.
This is the solution to this dilemma.
Vocabulary
Here are the vocabulary words in this Concept.
- Equation
- a number sentence with an equal sign where the quantity on one side of the equals is the same as the quantity on the other side of the equals.
- Inequality
- a number sentence where one side is not necessarily equal to the other side. There are several possible answers that will make an inequality a true statement.
Guided Practice
Here is one for you to try on your own.
Graph this inequality \begin{align*}x \ge 0\end{align*}.
Answer
First, draw a number line from –5 to 5.
The inequality \begin{align*}x \ge 0\end{align*} is read as “\begin{align*}x\end{align*} is greater than or equal to 0.” So, the solutions of this inequality include zero and all numbers that are greater than 0. Draw a closed circle at 0 to show that 0 is a solution for this inequality. Then draw an arrow showing all numbers greater than 0.
The graph above shows the solutions for the inequality \begin{align*}x \ge 0\end{align*}.
Video Review
Here is a video for review.
- This is a Khan Academy video on graphing inequalities on a number line.
Practice
Directions: For problems 1-4, graph each inequality on the given number line.
1. \begin{align*}x < -3\end{align*}
2. \begin{align*}x > -5\end{align*}
3. \begin{align*}n \le 2\end{align*}
4. \begin{align*}1 \le n\end{align*}
Directions: For problems 5-15, solve each inequality and then graph its solution on the given number line.
5. \begin{align*}x + 3 > 9\end{align*}
6. \begin{align*}n \div 2\end{align*}
7. \begin{align*}n + 4 < 6\end{align*}
8. \begin{align*}x - 1 > 4\end{align*}
9. \begin{align*}x - 3 \le 5\end{align*}
10. \begin{align*}x + 6 > 7\end{align*}
11. \begin{align*}2x \le 4\end{align*}
12. \begin{align*}3x > 9\end{align*}
13. \begin{align*}4x + 1 < 9\end{align*}
14. \begin{align*}2x - 1 > 5\end{align*}
15. \begin{align*}3x + 2 \le 8\end{align*}
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Image Attributions
Here you'll learn to graph inequalities on a number line.