7.6: Solving Inequalities
Introduction
The Movie Tickets
Marc and Kara have made several friends while swimming at the town pool. One rainy day, they decide to invite the group to go to a movie. Everyone is very excited. Grandma is so pleased that Kara and Marc have been making friends that she offers to treat.
“How many of you are going?” she asks Kara.
“Well, there are eight of us altogether,” Kara said.
“Alright, let me see,” Grandma says while reaching into her wallet. “I have forty-eight dollars to contribute. Here it is.”
“Thank you Grandma,” the twins say, smiling.
Later at the movie theater, Kara takes out the money. They have $48.00 to spend on tickets. Last minute, one of the friends has brought her brother along. Kara isn’t sure that they have enough money for the brother too. If each ticket is $6.00, how many tickets can she buy without going over the $48.00?
This problem requires an inequality. The total must be less than or equal to $48.00 without going over. The movie is going to start soon. Pay attention to the work in this lesson and you will know how to help Kara purchase the tickets when you complete it.
What You Will Learn
By the end of this lesson, you will have learned how to complete following:
- Graph inequalities on a number line.
- Recognize equivalent inequalities.
- Solve inequalities and graph solutions.
- Model and solve real-world problems involving inequalities.
Teaching Time
I. Graph Inequalities on a Number Line
In the last few lessons you have been learning about equations and about balancing an equation. Let’s review a short summary of solving 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.
Example
\begin{align*}x+5=12\end{align*}
If we think about this equation, we can use mental math and calculate 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 only 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 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 the same way we used the equals sign before: to make a true statement. Let’s look at an example.
Example
\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. In fact, it is kind of cool to realize that there are an infinite number of possible answers that work.
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 solutions for the inequality and which numbers are not solutions.
Here are a couple of rules 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.
Example
Graph the 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 to the 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*}.
Example
Graph the 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*}.
Example
Graph the inequality \begin{align*}x \ge 0\end{align*}.
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*}.
7M. Lesson Exercises
Answer the following true/false questions.
- An open circle on a graph means that the number is not included in the solution set.
- An inequality can never be equal.
- A closed circle on a graph means that the number is included in the solution set.
Take a few minutes to check your answers with a friend.
II. Recognize Equivalent Inequalities
Sometimes, you may need to rewrite an inequality as an equivalent inequality in order to better understand it. This means that an inequality can be written in two different ways so that we can understand it. Look at the example below.
Example
Graph the inequality \begin{align*}4 \ge x\end{align*}.
Draw a number line including several numbers larger and smaller than the 'starting value', 4. A good choice might be from –5 to 5.
The inequality \begin{align*}4 \ge x\end{align*} is read as “4 is greater than or equal to \begin{align*}x\end{align*}.” This inequality will be easier to understand if we rewrite it so that the variable is listed first. If we list the \begin{align*}x\end{align*} first, we will be describing the relationship in reverse, so we must reverse the inequality symbol. That means changing the “greater than or equal to” symbol \begin{align*}(\ge)\end{align*} to a “less than or equal to symbol” \begin{align*}\le\end{align*}.
\begin{align*}4 \ge x\end{align*} is equivalent to \begin{align*}x \le 4\end{align*}.
Logically, if 4 is greater than or equal to \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} must be less than or equal to 4.
The inequality \begin{align*}x \le 4\end{align*} is read as “\begin{align*}x\end{align*} is less than or equal to 4.” So the solutions of this inequality include 4 and all numbers that are less than 4. Draw a closed circle at 4 to show that 4 is a solution for this inequality. Then draw an arrow showing all numbers less than 4.
The graph above shows the solutions for the inequality \begin{align*}x \le 4\end{align*}.
Example
Write an equivalent inequality for \begin{align*}x \ge 5\end{align*}.
To write an equivalent inequality, we reverse the terms. If \begin{align*}x\end{align*} is greater than or equal to five, then five is less than or equal to \begin{align*}x\end{align*}. Let’s rewrite this.
The equivalent inequalities are \begin{align*}x \ge 5\end{align*} and \begin{align*}5 \le x\end{align*}.
7N. Lesson Exercises
Write an equivalent inequality for each example.
- \begin{align*}12 \le y\end{align*}
- \begin{align*}x>7\end{align*}
- \begin{align*}a<4\end{align*}
Check your answers with a partner.
III. Solve Inequalities and Graph Solutions
Now that you have developed an understanding of inequalities, we can solve them and graph the solution sets.
We can solve an inequality using all of the skills we would use to solve an equation with only one exception: if we multiply or divide by a negative number, we need to switch the direction of the > or < symbol. Once we solve the inequality, we can graph the solution. Let’s look at an example.
Example
Solve this inequality and graph the solution: \begin{align*}n-4 \le 3\end{align*}.
Solve the inequality just as you would solve an equation, by using inverse operations. Since the 4 is subtracted from \begin{align*}n\end{align*}, add 4 to both sides of the inequality to solve it. You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol. The symbol should stay the same.
\begin{align*}n-4 & \le 3\\ n-4 +4 & \le 3+4\\ n+(-4+4) & \le 7\\ n+0 & \le 7\\ n & \le 7\end{align*}
Now, graph the solution. The inequality \begin{align*}n \le 7\end{align*} is read as “\begin{align*}n\end{align*} is less than or equal to 7.” So, the solutions of this inequality include 7 and all numbers that are less than 7.
Draw a number line from 0 to 10. Add a closed circle at 7 to show that 7 is a solution for this inequality. Then draw an arrow showing all numbers less than 7.
The solution for this inequality is \begin{align*}n \le 7\end{align*}, and its graph is shown above.
Example
Solve this inequality and graph its solution: \begin{align*}-2n<14\end{align*}.
Solve this inequality as you would solve an equation, by using inverse operations. Since the -2 is multiplied by the \begin{align*}n\end{align*}, divide both sides of the inequality by -2 to solve it. Since this involves multiplying both sides of the inequality by a negative number, the sense of the inequality will change and you will need to reverse the inequality symbol. This means changing the inequality symbol from a “less than” symbol (<) to a “greater than” symbol (>).
\begin{align*}-2n & < 14\\ \frac{-2n}{-2} & > \frac{14}{-2}\\ 1n & > -7\\ n & > -7\end{align*}
Now, graph the solution. The inequality \begin{align*}n>-7\end{align*} is read as “\begin{align*}n\end{align*} is greater than -7.” So, the solutions of this inequality include all numbers that are greater than -7.
Draw a number line from –10 to 0. Add an open circle at -7 to show that -7 is not a solution for this inequality. Then draw an arrow showing all numbers greater than -7.
The solution for this inequality is \begin{align*}n>-7\end{align*}, and its graph is shown above.
Sometimes, you will need to take more than one step to solve an inequality. You can think of these problems in the same way that you thought about two-step equations.
Example
\begin{align*}\frac{n}{3}+9 \ge -9\end{align*}.
Solve this inequality as you would solve an equation, by using inverse operations. First, try to get the term with the variable, \begin{align*}\frac{n}{3}\end{align*}, by itself on one side of the inequality. Since the 9 is being added to \begin{align*}\frac{n}{3}\end{align*}, subtract 9 from both sides of the inequality. You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol during this step.
\begin{align*}\frac{n}{3}+9 & \ge -9\\ \frac{n}{3}+9-9 & \ge -9-9\\ \frac{n}{3}+0 & \ge (-9+-9)\\ \frac{n}{3} & \ge -18\end{align*}
There is a second step you must take to find the solution. Since \begin{align*}n\end{align*} is divided by 3, you must multiply both sides of the inequality by 3 to find its solution. This involves multiplying by a positive number, 3, so you do not need to reverse the inequality symbol. Be careful! It is true that you will need to multiply 3 by –18 to find the solution. However, since you are not multiplying both sides of the inequality by a negative number, you do not reverse the inequality symbol.
\begin{align*}\frac{n}{3} & \ge -18\\ \frac{n}{3} \times 3 & \ge -18 \times 3\\ \frac{n}{3} \times \frac{3}{1} & \ge -54\\ \frac{n}{\cancel{3}} \times \frac{\cancel{3}}{1} & \ge -54\\ \frac{n}{1} & \ge -54\\ n & \ge -54\end{align*}
The solution for this inequality is \begin{align*}n \ge -54\end{align*}.
7O. Lesson Exercises
Solve each inequality.
- \begin{align*}x-4<10\end{align*}
- \begin{align*}2y+4 \ge 12\end{align*}
- \begin{align*}-4x \le 16\end{align*}
Check your answers with a friend. Then continue with the next section.
IV. Model and Solve Real-World Problems Involving Inequalities
Now that you know how to graph and solve inequalities, let's take a look at some ways we can use inequalities to solve problems. We can use inequalities to represent some real-world problem situations, too.
Let's take a look at some key words that can help us write inequalities to represent real-world problems.
> | < | \begin{align*}\ge\end{align*} | \begin{align*}\le\end{align*} |
---|---|---|---|
greater than more than | less than fewer than | greater than or equal to at least | less than or equal to at most |
The key words above provide clues about which inequality symbol you should use to represent a problem situation. While key words can be a helpful guide, it is important not to rely on them totally. It is always most important to think about what translation of the problem makes the most sense. This is especially important because the same key words may mean different things. For example, the key words “more than” may mean you should use a > symbol or they may mean you should write an addition expression.
Example
Antonio is buying milk for a breakfast event. Each container of milk costs $3. At most, he can spend $12 on milk for the event.
a. Write an inequality to represent, \begin{align*}c\end{align*}, the number of containers of milk he can buy.
b. Could Antonio buy 4 containers of milk for the event? Explain.
Consider part a first.
Use a number, an operation sign, a variable, or an inequality symbol to represent each part of the problem. Since each container of milk costs $3, you can find the total cost, in dollars, of the milk he buys by multiplying 3 by the number of containers. The key words “at most” indicate that you should use a \begin{align*}\le\end{align*} symbol.
\begin{align*}& \underline{\text{Each container}\ldots \text{costs} \ \$3}. \ \underline{\text{At most}}, \ \text{he can spend} \ \underline{\$12}\ldots\\ & \qquad \qquad \qquad \quad \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \quad \quad \ \ \downarrow\\ & \qquad \qquad \quad \quad \ c \times 3 \qquad \qquad \quad \le \qquad \qquad \qquad \qquad 12\end{align*}
You may also want to consider that the value of \begin{align*}c\end{align*} must be an integer greater than or equal to 0. Think about why that is for a moment.
The reason that the value of \begin{align*}c\end{align*} must be an integer greater than or equal to zero is because Antonio cannot buy a negative number of containers nor can he buy a fraction of a container. Neither of those situations makes sense in real life. When using inequalities to represent real-life situations, you should always think about which values would make sense for the variable and which values would not make sense.
Next, consider part \begin{align*}b\end{align*}.
Solve the inequality to help you.
\begin{align*}c \times 3 & \le 12\\ \frac{c \times 3}{3} & \le \frac{12}{3}\\ c \times \frac{3}{3} & \le 4\\ c \times 1 & \le 4\\ c & \le 4\end{align*}
According to the inequality above, the number of containers, \begin{align*}c\end{align*}, that he can buy must be less than or equal to 4.
Since 4 is a solution for this inequality, and since 4 is an integer that is greater than 0, he could buy 4 containers of milk.
Example
At the Stereo Store, Erika bought a $9 set of headphones and a DVD that was on sale for half its regular price. Erika spent more than $15 on these two purchases.
a. Write an inequality to represent \begin{align*}d\end{align*}, the regular price of a DVD at the store.
b. List three possible values of \begin{align*}d\end{align*}.
Consider part a first.
Use a number, an operation sign, a variable, or an inequality symbol to represent each part of the problem. The key words “more than”, in this case, indicate that you should use a > symbol.
\begin{align*}& \underline{\$9} \ \text{set of headphones} \ \underline{\text{and}} \ a \ DVD \ldots \text{for} \ \underline{\text{half the regular price}}\ldots \text{spent} \ \underline{\text{more than}} \ \underline{\$15} \ldots\\ & \ 9 \qquad \qquad \qquad \qquad \ \ + \qquad \qquad \qquad \qquad \qquad \quad \frac{d}{2} \qquad \qquad \qquad \qquad \quad \quad \ > \qquad 15\end{align*}
Next, consider part \begin{align*}b\end{align*}.
Solve the inequality to help you.
\begin{align*}9+\frac{d}{2} & > 15\\ 9-9+\frac{d}{2} & > 15-9\\ 0+\frac{d}{2} & > 6\\ \frac{d}{2} & > 6\\ \frac{d}{2} \times 2 & > 6 \times 2\\ \frac{d}{\cancel{2}} \times \frac{\cancel{2}}{1} & > 12\\ \frac{d}{1} & > 12\\ d & > 12\end{align*}
According to the inequality above, the regular price of a DVD, \begin{align*}d\end{align*}, is more than 12 dollars.
So, three possible values of \begin{align*}d\end{align*} are $12.50, $13, and $20. These are only 3 possible answers. You could choose any amount that is greater than 12 dollars.
Now let’s go back to our introduction problem and help Tony with his dilemma on the train.
Real Life Example Completed
The Movie Tickets
Here is the original problem once again. Use what you have learned about inequalities and about solving inequalities to help you with the movie tickets. Reread the problem before you begin and underline any important information.
Marc and Kara have made several friends while swimming at the town pool. One rainy day, they decide to invite the group to go to a movie. Everyone is very excited. Grandma is so pleased that Kara and Marc have been making friends that she offers to treat.
“How many of you are going?” she asks Kara.
“Well, there are eight of us altogether,” Kara replies.
“Alright, let me see,” Grandma says, reaching into her wallet. “I have forty-eight dollars to contribute. Here it is.”
“Thank you Grandma,” the twins say, smiling.
Later at the movie theater, Kara takes out the money. They have $48.00 to spend on tickets. Last minute, one of the friends has brought her brother along. Kara isn’t sure that they have enough money for the brother too. If each ticket is $6.00, how many tickets can she buy without going over the $48.00?
To figure this out, we need to write an inequality. The original group consisted of 8 kids without the extra brother. Each ticket is $6.00. We can use this in our inequality. If we have money left over, then we will know whether or not we can pay for the brother too.
We need the total to be less than or equal to $48.00.
Here is the inequality.
\begin{align*}6x \le 48\end{align*}
Now we can solve the inequality. The \begin{align*}x\end{align*} represents the number of tickets that we can purchase if they are $6.00 each.
\begin{align*}6x & \le 48\\ \frac{6x}{6} & \le 48\\ x & \le 8\end{align*}
This shows that the kids can purchase less than or equal to 8 tickets with the $48.00. They don’t have enough to pay for the brother too.
Kara tells the group this information and they all chip in with their own money. All of the kids are able to attend the movie thanks to the generosity of the group.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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.
- Equivalent Inequalities
- Two inequalities that are written differently, but still express the same number relationships.
Technology Integration
Khan Academy Inequalities on a Number Line
Khan Academy Solving Inequalities
James Sousa, Solving One-Step Linear Inequalities in One Variable
James Sousa, Graph Basic Inequalities and Express Using Interval Notation
James Sousa, Example of Solving One Step Inequalities by Adding and Subtracting
James Sousa, Example of Solving One Step Inequalities by Division
James Sousa, Example of Solving One Step Inequalities by Multiplication
Other Videos:
- http://www.mathplayground.com/howto_InequalitiesA.html – This is a video that explains how you can solve an inequality.
Time to 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 5-6, 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 (-4) \ge 2\end{align*}
Directions: Solve each inequality.
7. \begin{align*}x+4<10\end{align*}
8. \begin{align*}x-3 \ge 7\end{align*}
9. \begin{align*}b+5 \le 15\end{align*}
10. \begin{align*}a-7 \ge 14\end{align*}
11. \begin{align*}4y>20\end{align*}
12. \begin{align*}6x \le 18\end{align*}
13. \begin{align*}-4y< -12\end{align*}
14. \begin{align*}-5x< -20\end{align*}
15. \begin{align*}\frac{x}{2}=10\end{align*}
16. \begin{align*}\frac{x}{5} \le 6\end{align*}
17. \begin{align*}2x+5 \ge 7\end{align*}
18. \begin{align*}3y-2 \le 4\end{align*}
19. \begin{align*}3a-7>11\end{align*}
20. \begin{align*}2b+9<39\end{align*}
Directions: Solve each problem.
21. Emma bought a fruit smoothie at a juice shop for \begin{align*}t\end{align*} dollars, including tax. Emma paid with a $10 bill. She received less than $5 in change.
a. Write an inequality to represent \begin{align*}t\end{align*}, the number of dollars, including tax, that Emma paid for the fruit smoothie.
b. List three possible values of \begin{align*}t\end{align*}.
22. Kiet has 16 juice boxes for a family picnic and needs to buy more. Juice boxes are sold in packages of 8.
a. Write an inequality to represent \begin{align*}p\end{align*}, the number of packages of juice boxes Kiet needs to buy in order to have at least 40 juice boxes total for the picnic.
b. If Kiet buys 4 packages of juice boxes, will that be enough?
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